1.1 The Status of Logic

Sciences such as biology, chemistry, and physics (i.e. what goes ordinarily under the label natural sciences),Footnote ⁴ primarily aim to provide us with new insights and explanations about previously unknown phenomena or to offer more comprehensive explanations of already known phenomena. It is, indeed, standard to maintain that through discoveries in these scientific fields, we uncover what are often considered new substantive truths about the world. In other words, the sciences provide us with new and substantive information about certain natural phenomena. What about logic? Is logic akin to the sciences?

Facing this question, one may think that there’s some pressure to lean toward a simple affirmative answer to the question: “Is logic a science?” that comes naturally from putting together two prima facie intuitive thoughts: (i) that logic and mathematics are closely linked –in particular, that logic is part of mathematics, and (ii) that mathematics is naturally associated with the sciences. A straightforward way of making the link between logic and mathematics explicit is to consider deductivism in mathematics. As discussed by Maddy (Reference Maddy2022: 9), deductivism holds that a mathematical sentence “p” should be understood as asserting the proposition that p can be inferred from a suitable set of axioms through a deductive process. For example, the sentence “2 + 2 = 4” is interpreted by proponents of deductivism as stating the proposition that 2 + 2 = 4 logically follows from the axioms of arithmetic. In this view, logic is an integral part of mathematics, suggesting that if mathematics is a science, then logic should be considered a science as well.

As always in philosophy, things may, of course, not be that simple. Although deductivism offers one specific example, it nevertheless illustrates that there are substantive assumptions that have been made in order to establish the link between logic and the sciences. However, one may have reasons not to accept a particular view of the connection between logic and mathematics – that is, deductivism in our example. Alternatively, one may indeed argue that the association between mathematics and the (natural) sciences is not that straightforward. In fact, whether, and to what extent, mathematics is continuous with the natural sciences is certainly an open and interesting question. In this respect, when it comes to the relationship between logic and the sciences a critical question remains: How significantly does logic resemble sciences like biology, physics, and chemistry?

In contrast with the intuitive link between logic and the sciences via mathematics, in several conceptions logic is seen as markedly different from the sciences. First, logic is often taken to provide formal tools that serve as a neutral and impartial arbiter in evaluating scientific and philosophical disputes. Second, logic is often viewed as insensitive to empirical evidence, in terms of both justification and revision. Third, logic is primarily considered a normative discipline, and not a descriptive one which is supposed to be in the business of offering explanations about the world. For instance, logical expressivists claim that since logic has an expressive role – for example, in Brandom’s version “to make explicit the inferential relations that articulate the semantic contents of the concepts expressed by the use of ordinary, nonlogical vocabulary” (Brandom Reference Brandom2018: 70) – logical sentences are not representational: they are not meant to make statements about the world and cannot fulfill the function of explaining facts beyond its own realm. In short, conceptions of logic such as logical expressivism see a clear divide between logic and the sciences.

In the philosophy of logic, such a standard divide has been widely discussed. The discussion follows three partially overlapping trends: (i) an epistemological one, where philosophers have focused on whether logic can be justified and/or revised based on empirical evidence; (ii) a methodological one, which particularly addresses whether logical theories should be selected using broadly abductive methods, similar to those employed in selecting scientific theories; (iii) a metaphysical one concerning whether logic’s subject matter is about some, perhaps very general and structural, aspects of reality. The complexity of determining whether and to what extent logic is a science requires establishing criteria to demarcate what counts as logic and what counts as science. This important issue will be addressed in Sections 3 and 4.

Before we delve into the discussion about logic’s status as a science, it’s important to clarify some terminology. In this debate, “logic” is generally used in a restrictive sense, referring specifically to deductive logic rather than to various non-deductive logics like inductive or abductive logic. Throughout this Element, we will adhere to this narrower definition of “logic,” focusing primarily on the relationship between deductive logic and the sciences. Furthermore, we will adopt what has been the traditional twentieth-century view of logic as a discipline concerned with reasoning – what Priest refers to as the canonical application of logic (see Priest Reference Priest2006).Footnote ⁵ Roughly, by logic in the sense of a discipline we mean the interpreted logical theory (or the set of interpreted logical theories) – that is, in which each logical constant has an intended interpretation – which is (are) accepted and employed by logicians.

1.2 Quine’s Philosophy of Logic and the Rise of Logical Anti-Exceptionalism

The question of whether logic should be considered a science has been a recurring theme throughout many historical discussions on the nature of logic, as evidenced in the works of philosophers like Leibniz, Hobbes, Descartes, Kant, Boole, Frege, Husserl, and Russell, among others. However, this question gained sharper focus and more refined treatment following the influential contributions of Bertrand Russell and Willard Van Orman Quine to epistemology and to the philosophy of logic. In some of their writings (e.g., Russell Reference Russell1918; Quine Reference Quine1951, Reference Quine1986a), they contend that logic bears substantial resemblance to the sciences, particularly the natural sciences, based on epistemological, metaphysical, and methodological considerations. As Bertrand Russell famously claimed, “logic is concerned with the real world just as truly as zoology, though with its more abstract and general features” (Russell Reference Russell1919: 169).

Quine famously took logic to be continuous with the natural sciences. And many prominent philosophers of logic and philosophical logicians currently sympathize with the Quinean thesis that logic, as a discipline, should be considered methodologically and epistemologically akin to the natural sciences.Footnote ⁶ In the fairly recent debate within the philosophy of logic, this thesis is known as anti-exceptionalism about logic. While the label in connection to logic is due to Ole Hjortland (Hjortland Reference Hjortland2017), the inspiration for AEL is from the kind of philosophical methodology developed by Timothy Williamson in his book The Philosophy of Philosophy (Williamson Reference Williamson2007). Our discussion of AEL will necessarily encompass multiple facets. Indeed AEL challenges the notion that logic’s methodology, epistemology, and subject matter possess an exceptional status and advocates for a more intimate amalgamation of logic with the natural sciences. Additionally, AEL posits a parallel connection between the laws of logic and those governing the world, akin to the relationship seen in scientific laws. Lastly, AEL, drawing inspiration from Quine’s concept of evidential holism, implies that the justification of logic mirrors the approach used for substantiating scientific theories.

Two broad ways of characterizing AEL have been proposed and discussed in recent works. The first suggests understanding AEL in terms of continuity with the natural sciences (AEL-as-continuity), while the second casts AEL out in terms of rejection of some or all of the characteristics that traditionally have been attributed to logic (AEL-as-tradition-rejection). Let’s briefly describe these two ways of understanding AEL.

As the very label suggests, AEL-as-continuity is the thesis that there is a significant continuity between logic and the sciences on a variety of aspects (crucially, as said, on methodological, epistemological, and metaphysical aspects). In his 2017 paper, Ole Hjortland introduced the position precisely in terms of such a continuity. In an often-cited passage which is typically used as a sort of AEL-manifesto, Hjortland claims that “Logic isn’t special. Its theories are continuous with science; its method continuous with scientific method. Logic isn’t a priori, nor are its truths analytic truths. Logical theories are revisable, and if they are revised, they are revised on the same grounds as scientific theories” (Hjortland Reference Hjortland2017: 631, our italics).

Susane Haack, anticipating by several years the recent anti-exceptionalist trend, claims that “[L]ogic is a theory, a theory on a par, except for its extreme generality, with other, ‘scientific’ theories; and according to which choice of logic, as of other theories, is to be made on the basis of an assessment of the economy, coherence and simplicity of the overall belief set” (Haack Reference Haack1974: 26).

The second way of characterizing AEL is in terms of a rejection of what may be taken to be the traditional conception of logic which sees logic as having a set of very special features which make logic an exceptional discipline.Footnote ⁷ What are these features? According to the way in which Hjortland and Martin introduce the position in some of their joint works, logic possesses some or most of the following features: it is an absolutely general, purely formal, and normative discipline which deals with truths that are both analytic and necessary, the justification of which is non-inferential, a priori, and empirically non-revisable. We may call a conception of logic which endorses all these special characteristics a full-blooded exceptionalist conception. In short, full-blooded exceptionalism.Footnote ⁸

We will provide a detailed characterization of full-blooded exceptionalism in Section 1.3. Before proceeding, though, we would like to clarify what we take to be the conceptual relationship between these two ways of characterizing AEL. Arguably, a conception of logic which sees logic in strict continuity with the natural sciences is ipso facto a conception of logic which rejects most (if not all) of the features that full-blooded exceptionalism – the purest exemplar of a traditional conception of logic – attributes to logic. In this respect, characterizing AEL in terms of continuity with the sciences entails a characterization of logic that rejects most (or all) of the features that the tradition attaches to logic (in other words, AEL-as-continuity entails AEL-as-tradition-rejection). In this respect, accepting a traditional conception – that is, accepting something in the vicinity of what we call “full-blooded exceptionalism” – means ipso facto to reject the idea that there is a significant continuity between logic and the sciences. Things are not so straightforward in relation to the converse entailment relation. Arguably there are ways of being anti-traditionalist about logic which do not necessarily see logic in strict continuity with the sciences. If that’s correct, the characterization of AEL in terms of tradition-rejection does not by itself commit us to establish a continuity between logic and the sciences. In this respect, having both characterizations of AEL on board allows us to see that a commitment to an anti-exceptionalist thesis about logic comes somehow in degrees. For instance, an endorsement of a strict continuity between logic and the sciences matches what may be called a radical form of anti-exceptionalism (borrowing the label from Da Costa & Arenhart (Reference Da Costa and Arenhart2018) we may call it “full-blooded anti-exceptionalism”). However, there may be progressively milder forms of AEL based on a rejection of some of the traditional features of logic. Be that as it may, regardless of whether we take AEL as continuity or as tradition rejection, insofar as the former entails the latter and thus an acceptance of the traditional view of logic as exemplified by full-blooded exceptionalism entails a full rejection of the continuity thesis, it is helpful to dig deeper on what exactly are these characteristics traditionally associated with logic which AEL rejects. We do that by providing a characterization of full-blooded exceptionalism, to which we now turn.

1.3 On Full-Blooded Exceptionalism

Full-blooded exceptionalism is a view that takes logic to be fully general, formal, normative, analytic, necessary, a priori, and non-revisable.Footnote ⁹ We do not claim that full-blooded exceptionalism as such is a view that has been actually endorsed in the history of logic. We do, however, take it as a placeholder for a position that collects all the features that have been historically attributed to logic by a variety of philosophers and logicians. In this respect, its role in this Element is that of providing a neat paradigm of a view that exemplifies the various features that recent anti-exceptioanlist views have rejected (if not all, at least most of them). Let us analyze the features of full-blooded exceptionalism in some detail.

We begin with Generality, a feature that has been almost universally associated with logic.Footnote ¹⁰ An easy way to understand the generality of logic is to argue that, unlike the laws of biology, chemistry and physics, logical laws are typically conceived as wholly general, applying to all entities with no restrictions. In this regard, logical laws are considered to have the broadest scope of all laws, applying universally to everything without exception. On a minimalist interpretation, the generality of logical laws could be seen as exceptional only in a quantitative sense – having a higher, perhaps the highest, degree of generality – rather than in a qualitative sense – that is, as marking a substantive difference in nature between logic and the natural sciences. This conception of generality is best understood when combined with the, perhaps controversial, idea that quantification in logical truths (such as “for every x, either x is F or x is not F”) is taken to be absolutely unrestricted. – that is, as requiring the existence of an all-inclusive domain of quantification.Footnote ¹¹

Historically, however, the notion of generality has been tied to various ideas traditionally associated with logic, giving it a less minimalist interpretation and aligning it more with a substantive sense of exceptionalism concerning the nature of logic. These ideas include, among others, the Kantian perspective that logical laws are universal and necessary and constitutive of rationality; Frege’s thought that his logical system (the Begriffsschrift) is like the language Leibniz sketched, a lingua characteristica; Wittgenstein’s notion of logical truths in the Tractatus, which posits that logical truths lack proper meaning as they do not limit the realm of possibilities; and finally, Carnap’s belief in logical truths as analytic truths.

In the Kantian perspective, logic is a general art of reason (canonical Epicuri) dealing exclusively with general, namely universal and necessary, laws of thought. Logic is based on a priori principles, from which it is possible to derive all its rules, understood as rules to which all knowledge should conform. Such principles are independent of any content and are therefore determinable a priori.

Frege too viewed logic as a fully general discipline. As van Heijenoort points out (van Heijenoort Reference van Heijenoort1967), Frege’s belief in the superiority of his Begriffsschrift over Boole’s algebraic logic was rooted precisely in the higher degree of generality of his system which was taken to provide a universal language for science.

Within the perspective of the early Wittgenstein’s conception of logic, the generality feature is conceived as an essential trait of logic. In the Tractatus the idea that logical propositions are tautologies which say nothing and thus do not deal with any subject matter is coupled with the idea that they represent the scaffolding of the world (Wittgenstein Reference Wittgenstein1921: 6.124). From such a point of view, it is clear that logic is quite distinct from the natural sciences (as well as from psychology or any social science) and not because it had a subject matter that could not be investigated by these sciences, but rather because logic did not have a subject matter at all. In line with the broader neo-positivist conception of logic which conceived of logical laws as lacking inherent meaning, Wittgenstein took the propositions of logic to be tools to demonstrate how language functions. According to this conception, the primary purpose of logical propositions is to illustrate how language operates as well as to reveal the relationships occurring among linguistic expressions of various kinds. This is achieved by means of an appropriate formalization of a sentence or an argument of natural language in a logical language: and the idea is that this process uncovers aspects of the nature and structure of natural language, aspects that might otherwise remain concealed by the grammatical surface of natural language expressions. Carnap, building upon this foundation, emphasizes that the validation and legitimacy of logical propositions are rooted in the rules that govern language. These rules include the formation of expressions and the procedures for inference and transformation.

Generality is thus the first element which is characteristic of full-blooded exceptionalism about logic – an element, as we will see, that is generally kept also in various anti-exceptionalist conceptions.

The second feature of exceptionality that we would like to discuss is the Formality feature. The notion of generality of logic has been linked, for example, by Kant and, more recently by Gila Sher, to a second crucial characteristic of full-blooded exceptionalism, namely that of the formality of logic, which is often understood in terms of topic-neutrality.Footnote ¹² In his PhD thesis John MacFarlane distinguishes three senses of formality.Footnote ¹³ He says that logic is formal:

Although the first sense of formality is not very prominent in contemporary debates,Footnote ¹⁴ it played a central role in Kant’s philosophy. In fact, Kant is typically considered the founder of the tradition of seeing logic as formal. As MacFarlane points out,Footnote ¹⁵ in Kant’s framework of transcendental idealism the first sense of formality illustrates more properly the generality of logic, rather than its formality, and is taken to entail the third sense of formality – which is Kant’s favorite way of interpreting in what sense logic is said to be formal.

Besides Kant’s own view on the issue, and moving to a more contemporary perspective, the second and the third senses of formality are those that have received more attention in recent literature. These two senses of formality have been clearly and directly linked to the notion of topic-neutrality: while the second sense offers an understanding of topic-neutrality in ontological terms where logic’s characteristic notions and laws are taken to be indifferent to the particular identities of objects (they are all the same, logicwise), the third sense takes topic-neutrality as a semantic notion, where logic is said to abstract completely from the semantic contents of statements and inferences. As MacFarlane argues, while these three senses of formality may line up neatly in certain philosophical frameworks in others they can come apart.Footnote ¹⁶ Consequently, the first sense of formality does not imply either the second or third sense. In this regard, these three senses of formality are at least partially independent.Footnote ¹⁷

As mentioned at the beginning of this section, it is customary to connect generality and formality via the concept of topic-neutrality – to the point that these three notions become highly connected and difficult to disentangle. Gila Sher has recently emphasized the intimate link between generality, formality, and topic-neutrality:

Besides its link to the notion of generality, the formality of logic understood as topic-neutrality has a long tradition which counts many prominent logicians such as De Morgan, Frege, Russell, and Tarski. They write:

These are just a few passages that illustrate the centrality of the notion of formality in logic. Its importance is strengthened by the fact that formality, especially in the second sense specified by MacFarlane, has been indicated as the chief criterion for demarcating logic by many logicians with quite different philosophical views on the nature of logic. In particular, formality in the second sense has been formalized as a criterion of logicality in terms of the notion of permutation invariance (now understood in terms of isomorphism invariance)Footnote ¹⁸: logical terms are not altered by arbitrary permutations of the domain of discourse.Footnote ¹⁹

Let us now introduce a third feature which is at the core of exceptionalist conceptions of logic, namely the Aprioricity feature of logic. In delving into full-blooded exceptionalism, a fundamental philosophical quandary centers on the nature of our epistemic access to logical laws – in particular on whether our justification and/or knowledge of logical laws can be fundamentally obtained by a priori means,Footnote ²⁰ as exceptionalists would claim. We take that the real epistemological hallmark of full-blooded exceptionalism is the thesis that knowledge and justification of logical principles are immune to any kind of empirical defeater (be it an underminer or an overrider). This does not mean that a full-blooded exceptionalist excludes the possibility of any kind of revision. Full-blooded exceptionalists should allow for the possibility of revising logical principles – unless they adopt a strong constitutivist thesis (as in Leech Reference Leech2015) according to which certain basic logical principles cannot be rationally doubted because they are normatively constitutive of thought. However, such revisions would have to occur through a priori means, for example, by considerations pertaining to semantic paradoxes interpreted as providing a priori defeaters of a certain principles of classical logic. As we will see in the following sections, the question whether logic is open, at least in principle, to revision on the basis of empirical evidence is a recurrent and crucial theme in the exceptionalism/anti-exceptionalism debate – at least since the influential works of Hilary Putnam on quantum logic.Footnote ²¹

A fourth crucial respect in which logic is taken to be exceptional has to do with the analytic character of its principles. Many historical figures such as Kant, Frege, Carnap, and other logical empiricists,Footnote ²² took logical statements to be analytic – that is, true in virtue of meaning (or the conventions of language) alone. As Ayer puts it, “the criterion of an analytic proposition is that its validity should follow simply from the definition of the terms contained in it” (Ayer Reference Ayer1936: 82).

For Carnap, and more generally for logical empiricists, the analyticity of logic served a double purpose. On the one hand, it helped to keep a broadly naturalistic outlook of our scientific knowledge where logic (and mathematics) can find their place into a natural scientific whole. On the other hand, it was instrumental to account for the idea that logical principles are necessary without having to rely on a spooky conception of metaphysical necessity. Related to these points, the methods of logic offer a precise and comprehensive tool to clarify the negative claims about metaphysics. Logic demonstrates that the supposed claims of metaphysics involve concepts that cannot be defined through empirical means. Therefore, it precisely explains the grounds for dismissing such claims: not because they are untrue or subjective, but because they are, strictly speaking, devoid of meaning.Footnote ²³ Interestingly, however, assessing metaphysics as meaningless based on linguistic grounds raises questions about the status of logical (and mathematical) claims, which also do not concern empirical reality and are meant to get their truth in a uniquely non-empirical way. As Quine claims:

Quine notoriously rejected what he called “the linguistic doctrine of logical truth” and the underlying distinction between analytic and synthetic statements. As a consequence of Quine’s famous attacks on the notion of analyticity (see Quine Reference Quine1951) and on the notion of truth by convention (Quine Reference Quine and Whitehead1936, Reference Quine and Schilpp1963), the philosophical community has mostly accepted the fact that no sound notion of metaphysical analyticity could be concocted.Footnote ²⁴ For instance, Boghossian agrees with Quine that the traditional conception of analyticity is fundamentally flawed. However, in a series of influential papers (Boghossian Reference Boghossian1996; Boghossian & Williamson Reference Williamson2020), he develops an epistemic notion of analyticity. He argues that this new conception of analyticity stands independently from the traditional (metaphysical) notion of analyticity and is not vulnerable to Quine’s wide-ranging criticisms. According to Boghossian, a sentence is “epistemically analytic if grasp of its meaning can suffice for justified belief in the truth of the proposition it expresses” (Boghossian Reference Boghossian2003: 15). When applied to logic, the idea is that being willing to infer according to modus ponens is meaning-constitutive of the ordinary concept “if, then,” and this fact explains our being entitled to reason in accordance with modus ponens, even if the inference is, as Boghossian puts it “blind – unsupported by any positive warrant” (Boghossian Reference Boghossian2003: 23).Footnote ²⁵ Boghossian’s concept of epistemic analyticity has faced various criticisms, with some of the most influential coming from Williamson in a series of papers, collected in Boghossian and Williamson (Reference Williamson2020).

Finally, the last feature of full-blooded exceptionalism that we would like to discuss is Normativity. An exceptionalist conception of the normativity of logic, which would fit well with full-blooded exceptionalism, takes logic to be a normative discipline. As Frege claimed: “Like ethics, logic can also be called a normative science” (Frege Reference Frege, Long and White1979: 128). Setting aside interpretive questions about what Frege truly believed regarding the normativity of logic (a complex issue, as Frege appears to endorse seemingly contradictory claims),Footnote ²⁶ one could argue for the strong thesis that logic, like ethics, is a normative discipline, and its laws, much like ethical laws, should be understood in normative-deontic terms. One way to cast this view out is to take full-blooded exceptionalists to be committed to the view that logical principles (unlike principles or laws of any other science, e.g., physics) are intrinsically normative for reasoning. More precisely, in this view logical principles are taken to be the source of a kind of normativity which provides reasoners with normative guidance as to what to believe or disbelieve in the context of reasoning. Roughly, the idea is that logic, by its very nature, engenders some constraints (the precise character and scope of which are a matter of debate) on what doxastic attitude to have in relation to premises and conclusion of a logically valid argument. To give an example, one may think that logic, or, better, its core notion of validity, exerts something like the following constraint: that a subject ought not to disbelieve a conclusion C which, as a matter of logic, follows from a set of premises P1, … Pn, under the condition that she believes all the premises.Footnote ²⁷ Moreover – the exceptionalist about logical normativity typically maintains – such constraints are entirely sourced in the nature of logic (and logical principles) and not in some extrinsic factors, like truth, rationality, or knowledge.

Historically, Kant, Frege, and Carnap in different ways argued that logic’s normative constraints extend beyond limiting just reasoning and encompass all forms of thinking. This is usually specified by claiming that logical laws are normatively constitutive of thinking. Normative constitutivism is the thesis that no kind of cognitive practice whose correctness conditions allow for breaches of logical laws can count as thinking. While this does not exclude forms of illogical thinking as instances of thinking, it excludes that any kind of illogical thinking is correct thinking: in other words, no correct thinking can be illogical. Normative constitutivism is one way in which an exceptionalist conception of the normativity of logic can be developed.Footnote ²⁸

Alternatively, one may subscribe to a more moderate understanding of logic’s normative role as extrinsic and instrumental on the achievement of truth. In this second, more moderate understanding of logic’s normativity, logic would retain a normative role with respect to thinking and reasoning but without being a normative discipline (and in this sense it would be different from ethics, and more akin to the sciences). As MacFarlane observes, even if logical laws are not prescriptive in their content, they imply prescriptions about asserting, thinking, judging, inferring.Footnote ²⁹ What would be distinctive of the logical laws, even in this picture, is their scope: they “prescribe universally the way in which one ought to think if one is to think at all” (Frege Reference Frege1893: xv). In this sense they should be considered more general than laws of the sciences like for example physics. Indeed, unlike the laws governing sciences such as geometry or physics, the laws of logic transcend specific objects or properties associated with any particular discipline. They operate on a more fundamental level, independent of specific entities or attributes that might be explored within a particular field of study (see Ricketts Reference Ricketts1985: 4–5). One way to cast this out is to follow the Fregean tradition and conceive of rules of inference as principles endowed with normative force, where their normative force derives from the meaning of the logical constants. In this picture an expression’s meaning is equated with its role in determining the circumstances under which a statement containing that expression would be deemed true. As for logical constants, the meaning is represented, in the simplest and most typical case, by the way the truth value of the entire sentence depends on the truth values of the components, and the formulation of the logical rules assumes that their meaning is already fixed. In this context, rules of inference are deemed normative not because they dictate a specific course of epistemic action, as it were, but rather because they elucidate something imbued with normative features: the disposition to make inferences based on meaning.Footnote ³⁰

1.4 Full-Blooded Exceptionalism and Natural Sciences. A First Comparison

Now, even conceding that, as Rossberg and Shapiro observe, “[E]very form of inquiry, scientific or otherwise, is different, in crucial ways, from every other” (Rossberg & Shapiro Reference Rossberg and Shapiro2021: 6430) it is certainly hard to deny that such a full-blooded exceptionalist conception of logic is in sharp contrast with the most reasonable conception of physics and other natural sciences – indeed, with any reasonable conception of the recognized sciences (perhaps, with the exception of mathematics, which may be taken, not so unreasonably, to enjoy a similar degree of exceptionality as logic does).

To illustrate in more detail, let’s focus on physics. First, considering generality, physics certainly has a high degree of generality, but its subject matter is nevertheless restricted to physical matter (perhaps even to anything which has physical existence). Frege, for example, observes that physics concerns itself with laws governing phenomena like weight and heat (Frege Reference Frege, P. Geach, Stoothoff and McGuinness1918). Thus, the generality achieved by physics and fundamental physical laws falls short of the kind of absolute generality that, according to full-blooded exceptionalism, logical laws enjoy.

Second, physics, as the other recognized sciences, is not purely formal. Whether a certain law is a law of physics clearly depends on the nature of the objects and properties it is about. In this respect, the degree of invariance under permutation that physical laws enjoy is lower than that enjoyed by the laws of logic. Consider the laws of special relativity. These laws are invariant under permutations of inertial reference-frames. They remain invariant in all such frames. In other words, they are not affected by replacements of one inertial frame by another.Footnote ³¹ However, they are not invariant under a larger class of permutations.Footnote ³² Laws of physics (e.g., laws of thermodynamics) apply to all physical entities, but not to all kinds of entities – for example, they do not apply to numbers.

Third, physics deals with synthetic truths and our epistemic access to such truths is distinctively a posteriori since it heavily and essentially relies on empirical evidence and observation, latu sensu. In particular, physical theories are taken to be revisable on the basis of empirical evidence: a principle that is taken to be a law, but it is discovered by empirical observation not to be a law of physics should be abandoned.

Fourth, physics is taken to be descriptive: its primary, if not sole, task is to provide an accurate description of the physical reality. In this respect our best physical theory has normative consequences in the sense that believing things that are contrary to it is incorrect. However, this is taken to have nothing to do with the fact that it is physics we are talking about (or that the laws in question are physical law) but rather with the general normative principle that having false beliefs (about anything) is incorrect (under the plausible assumption that truth and falsity exert normative constraints on belief).Footnote ³³

Let us wrap up this section by highlighting, once more, that full-blooded exceptionalism is a rather peculiar position in the philosophy of logic. Furthermore, the accuracy of this description as a historically held position remains controversial. It may well be that historical figures such as Aristotle, Kant, Frege, the early Wittgenstein, perhaps more controversially Carnap, and more contemporary figures such as Michael Dummett, Hartry Field, Bob Hale, Saul Kripke, Jessica Leech, Gil Sagi, and Crispin Wright, among others, have adopted, to a variable degree, a somewhat exceptionalist conception of logic without subscribing to a full-blooded version of it. Be that as it may, as mentioned earlier, our primary objective in briefly introducing this radical, full-blooded, exceptionalist perspective on logic is dialectical. It serves to enhance our understanding of various anti-exceptionalist views in logic. In fact, given the number of aspects involved in the characterization of an exceptionalist stance toward logic, it seems clear that one may adopt an exceptionalist stance to a higher or lower degree, as it were. As a consequence of this flexibility in characterizing logical exceptionalism, it is hard to individuate a sharp boundary between what counts as an exceptionalist conception of logic and what counts, instead, as an anti-exceptionalist conception.

Section 2 will delve into the most radical objections to the core features of an exceptionalist conception of logic which we have outlined in this section. We will directly engage with the primary source of these objections, focusing on Quine and his philosophy of logic.

2.1 Quine as the Precursor of Logical Anti-Exceptionalism

During the twentieth century, most of the core features of the full-blooded exceptionalist conception of logic illustrated in Section 1.3 have been radically criticized by a variety of leading philosophers such as Bertrand Russell, Willard Van Orman Quine, Hilary Putnam, Penelope Maddy, Gila Sher, Timothy Williamson, Gillian Russell, and still others. However, among these figures, nobody more than Quine contributed to the establishment of an anti-exceptionalist conception of logic as one of the most prevalent perspectives in contemporary debates within the philosophy of logic. As Ben Martin and Ole Hjortland write: “It is only with the advance of philosophical naturalism and Quine’s writings that anti-exceptionalism received serious consideration in the philosophy of logic” (Martin & Hjortland Reference Martin, Hjortland, Lasonen-Aarnio and Littlejohn2024).

An interesting feature of Quine’s conception of logic in the light of current debates within the philosophy of logic, in particular in relation to logical anti-exceptionalism, is that it merges the two senses of anti-exceptionalism introduced in Section 1.2. On the one hand, Quine emphasizes the similarity between logic and the sciences, arguing that their differences in epistemology and methodology are more about degree than substance. On the other hand, as a likely result of this view, Quine separates logic from certain traditional characteristics associated with it, particularly its being analytic and a priori. In this sense, and perhaps unsurprisingly, Quine’s original stance toward logic brings together elements of LAE-as-continuity and elements of LAE-as-tradition-rejection. But let’s proceed with a closer discussion of Quine’s conception of logic in the broader context of Quine’s philosophy. A word of caution: although Quine consistently adhered to a broadly anti-exceptionalist view of logic, his philosophical positions evolved over the course of his career. Some of the more radical theses from his early period were softened in his later, more mature, work. For the sake of brevity and clarity, we will set aside a historically and exegetically precise account of Quine’s conception of logic. Instead, we will focus on the aspects of his views on logic that are most relevant to our discussion.

2.2 Quine’s Reaction against the Logical Empiricist Conception of Logic

As John Burgess points out, “[V]irtually all Quine’s philosophical writings, early and late, pertain directly or indirectly to logic, mathematics, or both” (Burgess Reference Burgess, Harman and Lepore2014: 279). This is to say that Quine’s views on the nature of logic (and mathematics) were pivotal in developing his philosophy. These views stemmed from a reaction against logical positivism, which was the prominent form of empiricism at the time when Quine began his career in philosophy. At the core of logical positivism there was a neat distinction between synthetic statements – namely, those statements that are open to empirical (dis)confirmation – and analytic statements – typically the statements of logic and mathematics that are true in virtue of meaning alone and thus fully immune to empirical (dis)confirmation. Logical positivists thus conceived of logic (and mathematics) as sharply distinct from the sciences from a broadly epistemological point of view. A clear statement of the logical positivist view on logic is given by Rudolph Carnap. Commenting on the approach of the Vienna Circle to the philosophy of logic and mathematics, Carnap writes:

Thus, while scientific statements and theories deal with empirical (broadly observational) matters, logic (and logical theories) do not deal with empirical matters at all, their role being, roughly, that of tools needed to clarify the structure of our scientific knowledge of the world and systematize it.

Quine’s opposition to the form of empiricism adopted by logical positivists led him to explicitly reject a number of core features of what we have called full-blooded exceptionalism about logic – features such as aprioricity, analyticity, necessity, and normativity. Even more broadly, Quine shattered some of the most resilient traditions and distinctions in philosophy. He rejected both rationalism and foundationalism by subscribing to a kind of philosophical naturalism coupled with a holistic model of knowledge. While his commitment to naturalism requires that all kinds of enquiries take empirical evidence to be the ultimate arbiter of a theory, his commitment to holism rejects the idea that single hypotheses are directly confirmed or disconfirmed by experience. Instead, it is the entire theory that faces the “tribunal of experience” collectively.

These commitments to naturalism and holism led Quine to argue that some distinctions which were deeply entrenched with the philosophical tradition – distinctions such as those between a priori/a posteriori knowledge, between analytic and synthetic truths, and between necessity and contingency – all collapse. In particular, the rejection of the distinction between analytic and synthetic statements is especially impactful in relation to the philosophy of logic.Footnote ³⁴ As Quine writes:

If logical statements are not devoid of empirical content they are exposed to the possibility of empirical confirmation as well as to the possibility of empirical disconfirmation.Footnote ³⁵ Regarding empirical confirmation, as Crispin Wright puts it, the Quinean idea is “that the epistemic good standing of logical principles is properly earned in the same way as the confirmation of all empirical-scientific laws. We are justified in accepting such principles by, and only by, their participation in ongoing successful scientific theory” (Wright Reference Wright2021: 334). In this sense, the justification of logic, as well as the justification of any other scientific theory, is inferential and inseparable from the justification of the rest of our web of knowledge. And, perhaps even more controversially, Quine explicitly adhered to the thesis that no logical principle is immune to the possibility of revision. In an often cited passage from “Two Dogmas,” Quine writes: “No statement is immune to revision. Revision even of the logical law of the excluded middle has been proposed as a means of simplifying quantum mechanics; and what difference is there in principle between such a shift and the shift whereby Kepler superseded Ptolemy, or Einstein Newton, or Darwin Aristotle?” (Quine Reference Quine1951: 40).Footnote ³⁶

This is one of Quine’s most direct statements of the idea that in line with any kind of scientific hypotheses, and all the rest of our knowledge, logical laws are in principle open to revision (on the basis of experience). A couple of pages later, Quine returns to the issue of the revisability of logic, contextualizing it within his holistic picture of meaning and confirmation:

This amounts to a straight rejection of one of the most central epistemological tenets of full-blooded exceptionalism. If our best logical theory, let’s say a system of classical logic explicitly schematized in propositional form, runs into some recalcitrant experience when used in our scientific endeavors, say, because of the discovery of certain quantum phenomena, then adjusting one of its principles, say, one of the distributivity principle, is perfectly fine if such an adjustment is in fact the most effective way of ironing out the wrinkles. This is in line with, in fact an instance of, Quine’s holistic approach according to which whenever our scientific practice encounters a situation of recalcitrance, it is in principle rational to hold responsible not just the empirical scientific premises, but any aspects of our knowledge of the world, including the underlying logic involved in deriving the consequence of the specific scientific practice under scrutiny. Clearly, Quine’s form of logical anti-exceptionalism, particularly as presented in his 1951 work, retains a significant element of empiricism, albeit “without the dogmas.” In essence, Quine still views sensory experience as the boundary condition for science. But, as Williamson aptly notes, “that experience has any such unique privilege is not a truism, unless ‘experience’ is just another word for learning” (Williamson Reference Williamson, Birman and Weiss2024: 416).

However, despite the fact that in Quine’s holistic picture nothing is in principle immune to revision, it is still the case that in the practice of reassessing our knowledge on the basis of recalcitrant evidence, logic and mathematics enjoy some privileged status. As a matter of fact, Quine takes the vulnerability to revision to be a matter of degree: “It is at a minimum in logic and mathematics, because disruptions here would reverberate so widely through science. […] Basic laws of physics, such as those of physical geometry or of conservation, are a little more vulnerable. There is a grading off. Toward the observational periphery of the fabric of science, vulnerability increases” (Quine Reference Quine, Hahn and Schilpp1986b: 620).

This is known in the literature as the maxim of minimal mutilation, and it consists in the thought that when revising belief systems or theoretical systems to accommodate new information, recalcitrant data, or solve inconsistencies, one should make the least number of changes necessary to resolve the issues, thus mutilating the original belief system as little as possible. In Quine’s own words: “If revisions are seldom proposed that cut so deep as to touch logic, there is a clear enough reason for that: the maxim of minimum mutilation. The maxim suffices to explain the air of necessity that attaches to logical and mathematical truth” (Quine Reference Quine1986a: 100).

Thus, according to Quine, our reluctance to revise logical principles in the face of recalcitrant evidence, even in those situations where revising logic may not sound too far-fetched as – someone may claim – in the case of quantum phenomena, is due to the privileged position that logic occupies in the web of belief – namely, right at the center of it. Given this centrality of logic in the web of belief, a request for revision of our logical theory should be the very last resort, when no other adjustments in the web seem to tame the recalcitrant observations. Because of its core position and role within our total theory of the world, revising logic would be extremely costly (a change in logic would require a change in almost anything else). That’s why Quine didn’t think that the various phenomena (linguistic, such as vagueness or other logico-semantic paradoxes, and non-linguistic, such as quantum phenomena) that some have taken to put pressure on some principles of classical logic were regarded by Quine as not sufficiently strong to require a revision of classical logic.

Additionally, as the quote earlier makes clear, the maxim of minimal mutilation also explains the air of necessity that surrounds the truths of logic. Clearly, the very expression that Quine uses – namely, “air of necessity” – vividly suggests that for Quine there’s no distinct kind of logical necessity that attaches to logical truths. Rather, as Haack points out (Haack Reference Haack1975: 238) the practical immunity to revision of classical logic is the closest Quine gets to the thesis that the laws of classical logic are necessary. This is a notion of necessity which falls considerably short of the strong sense of necessity attached to logical laws by advocates of full-blooded exceptionalism. After all, it could happen that in response to recalcitrant experience, we opt for revising our logic rather than revising something else, like physics. And there’s no reason to exclude that in principle someday experience will lead us to revise our logic.

Thus, we have seen that Quine rejected some of the core ideas of logical exceptionalism namely: that logical statements are analytic; that they cannot be justified or revised on the basis of empirical evidence; and that they enjoy a distinctive kind of necessity. We should now add that, as a consequence of his general naturalistic and holistic approach to philosophy and, more specifically, the epistemology of logic, Quine rejects the idea that logic is normative in any interesting sense. There’s no set of prescriptions or proscriptions (how we ought or ought not to reason) that are distinctively logical – namely, proscriptions or prescriptions that are not derived from or grounded in, for instance, the norms provided by truth and falsity. This does not mean that Quine’s epistemology is completely free of any normative considerations, but these considerations have only the rather minimal function of “warning us against telepaths and soothsayers” (Quine Reference Quine1990: 19) – and thus against the employment of methods of investigation (and reasoning) that are unreliable (broadly non-truth-conducive). Insofar as logic deals with truths, logic has a normative role to play in preventing us from accumulating falsehoods in reasoning from true premises in accordance with logical principles. This is fully in accordance with anti-normativist theses, like the one of Gillian Russell (Russell Reference Russell2020), according to which logic isn’t normative in any interesting sense: its normative role is purely ancillary to the normative role of truth in inquiry.

2.3 Some Elements of Continuity with Logical Exceptionalism

Although Quine’s conception of logic is in sharp contrast, both methodologically and epistemologically, with some of the core features of full-blooded exceptionalism, there are nevertheless some points on which Quine’s view is in continuity with a traditional conception.

One of these features has to do with the generality of logic – a feature that, with Quine, the vast majority of contemporary anti-exceptionalists keep on board.Footnote ³⁷ Being an anti-exceptionalist about the scope of logic would be to adhere to a kind of logical particularism, for which Quine did not express any sympathy. As Quine writes, “Trivially […] the logical truths are true by virtue of any circumstances you care to name – language, the world, anything” (Quine Reference Quine1986a: 96). Relatedly, logic (as well as mathematics) can be contrasted with the rest of science in terms of its “versatility: [its] vocabulary pervades all branches of science” (Quine Reference Quine, Hahn and Schilpp1986c: 402).

Let’s dig deeper on this point of continuity between Quine’s conception of logic and logical exceptionalism. First of all, observe that Quine claims that the best way to regiment a theory, for example a scientific theory, is using the syntax of classical first-order logic with identity. The reason is that he considers the syntax of classical first-order logic with identity to be clear, powerful, simple, and, especially, transparent – namely, what you see is what you get in terms of what is expressed by a sentence regimented in first-order logic. Secondly, Quine accepts that classical first-order logic is topic-neutral. As Quine expresses the point, logic “has no objects it can call its own; its variables admit all values indiscriminately” (Quine Reference Quine1995: 52). Indeed, even if for Quine a priori arguments may not resolve in a definite manner ontological questions, these matters find their resolution through the selection of a canonical notation, which, in turn, is guided by the pursuit of overall systematic simplicity. Let us take a look at the two aforementioned elements of Quine’s conception of logic.

As for the first reason, for Quine, the syntax of classical first-order logic with identity is a powerful tool for regimenting a theory. Using it, we clarify its ontological commitments. Indeed, we owe to Quine the formulation of the most well-known criterion for identifying a theory’s T ontological commitments. The process involves the following steps:

1. 1. We first translate the theory T into the canonical notation of a first-order logical language with identity.

2. 2. We then consider the set of all statements that are formal consequences of the theory T – namely, the theorems of T.

3. 3. Further, we select those members of this set that begin with (at least) one existential quantification (i.e., with an existential quantifier “∃” followed by a variable, let’s call it “x”; the resulting expression “∃x” reads as “there is at least one object x”) whose scope is the rest of the statement. These are the statements that reveal the ontological commitment of a theory T.

4. 4. We then address the following question: what things must be in the domain as values of variables for the theorems of T that begin with an existential quantifier to be true?

According to Quine, the answer to this last question provides the ontological commitment of a theory T, exemplified with the Quinean well-known slogan: “To be is, purely and simply, to be the value of a bound variable” (Quine Reference Quine1948: 32). For example, consider a theory T that has only the following statement in English as its axiom:

We can translate the statement (S) into the notation of first-order logic:

One of the consequences of our axiom will be the statement “∃x (x is a dog)” and this statement is a theorem of T that begins with the existential quantifier “∃.” If we ask the question: “What things must fall in the domain as variable values in order for theorems of T that begin with a quantifier to be true?” one answer will be “dogs.” Thus, we conclude that theory T will be ontologically committed to dogs. It should be noted that according to Quine regardless of what is exactly the domain of quantification, one always speaks of existence in the same sense, and there is nothing to prevent one from using the same notion of existence in relation to a mixed domain of dromedaries and numbers, or even an all-encompassing domain of all things.

Classical first-order logic with identity is the syntactic tool to qualify the ontological commitments of a theory. From this point of view, it does not depart from many classical conceptions of logic. His reasons to privilege classical first-order logic in evaluating the ontological commitment of a theory are that classical logic has a complete proof procedure and it is ontologically innocent (Hylton Reference Hylton2007: 265ff). Indeed, Quine seems committed to the view that only classical first-order logic is logic proper due to its completeness, while extensions of logic are classified as mathematics (Quine Reference Quine1986a: 91). In particular, classical logic has a complete proof procedure for validity and for inconsistency. And either procedure suffices, since a formula is valid if and only if its negation is inconsistent. The completeness of first-order logic shows how to give a syntactic account of first-order logical implication, without presuppositions. Specifically, as Hylton argues, it is possible to define logical truth and validity without presupposing truth and interpretation (Hylton Reference Hylton2007: 266). As Quine himself points out, “[completeness] shows that we can define logical truth by mere description of a proof procedure, without loss of any traits that made logical truth interesting to us in the first place” (Quine Reference Quine1986a: 57).Footnote ³⁸

Consider that in his perspective classical first-order logic permits multiple distinct yet equivalent approaches to logical truths. Within classical logic, one can define logical truth through model theory, substitution, or proof theory – and the latter method is specifically viable because classical logic admits a complete proof procedure. Quine’s preference leans toward the substitutional definition rather than the model-theoretic one, primarily because the latter relies on set theory (Quine Reference Quine1986a: 55).

Classical first-order logic is also considered by Quine ontologically innocent; and this brings us back to the second element of continuity between Quine’s view on logic and the full-blooded exceptionalist perspective. The reason why Quine thinks that logic is ontologically innocent is not because it avoids commitments to any kind of entity; rather, it is because it avoids commitments to entities that are problematic – such as intensional entities which Quine in some of his earlier works famously labeled “creatures of darkness.”Footnote ³⁹

What first-order logic does require is the acceptance of at least one object in existence, without imposing any presuppositions regarding the nature of this object. In this respect it is essential to establish a clear definition of entities that can be deemed as non-problematic. Classical first-order logic is viewed by Quine as ontologically unproblematic. Quine has two reasons for that. First, predicates (and other non naming expressions) do not require a corresponding entity to be meaningful. Thus, the admission of properties to make sense of truths is, in general, dispensable. Quine invites us to reflect on the fact that properties or propositions lack explanatory power and clear identity criteria. It is precisely for this reason that they are not admissible entities. Take, for example, the case of propositions. Quine’s argument for their exclusion runs in the following way: No entity is admissible if we are not able to express the truth-conditions of statements of the form “a is identical with b” in a determinate way. Propositions are identical if the statements expressing them are synonymous. This is the only way to have an identity criterion for propositions. But, on the one hand, there is insufficient behavioral evidence for synonymy. On the other hand, behavioral evidence is the only evidence available for synonymy. If so, there is no identity criterion for propositions. As a consequence there are no propositions. Objects, on the contrary, exhibit the opposite virtues: they have explanatory power and clearer identity criteria. Secondly, Quine emphasizes that classical first-order logic “has no objects it can call its own; its variables admit all values indiscriminately” (Quine Reference Quine1995: 52). This perspective suggests that classical logic’s appeal lies in its neutrality toward the subject matter.

In sum, although Quine’s views on logic do not amount to the most radical kind of anti-exceptionalism to date, it is indisputable that Quine played a crucial role in challenging some of the core characteristics of the traditional conception of logic. Moreover, Quine explicitly endorsed the view that there is a significant continuity, methodological and epistemological, between logic and the sciences – their differences being more a matter of degree than of substance. The recent and ongoing debates in the epistemology of logic demonstrate the profound and extensive influence of Quine’s conception of logic.

3.1 Demarcation Problems: Logic and Science

Our primary objective in this section and the subsequent two is to explore the demarcation issue in both logic and science, delving into the nuances and intricacies that characterize and differentiate these fields. Although we cannot reasonably hope to say anything innovative on the general issue of demarcation in the context of this Element, we believe that a systematic discussion of how to set the boundary between science and non-science, on the one hand, and between logic and non-logic, on the other hand, may help us put into sharper focus the main question at the core of this Element – namely the question of whether and to what extent logic is a science. Our hope is that this brief discussion will advance our understanding, particularly in the context of the anti-exceptionalism debate.

It seems fair to say that the debate on logical anti-exceptionalism has, until now, been characterized by limited and sometimes vague discussions about the supposed scientific status or the anti-exceptional nature of logic. In fact, in examining the current state of the debate, it is clear that many logicians and philosophers of logic readily suggest that logic closely resembles a science. However, they often hesitate to delve into discussions about what it precisely means for logic to be considered a science. Regardless of the current state of the debate in the philosophy of logic, we maintain that the question of whether, and to what extent, logic qualifies as a science is not only philosophically significant but also important and inherently complex. Consequently, it warrants a more focused and detailed discussion. That being said, it is important to clarify that the primary aim of discussing demarcation criteria in relation to logical anti-exceptionalism is not to argue that a sensible anti-exceptionalist stance on logic requires a strict set of criteria for distinguishing science from non-science. Rather, the goal is to show that a more nuanced understanding of these criteria and their interconnections offers a clearer picture of what it means to be an anti-exceptionalist about logic, and to what extent.

To begin with, we take it that when we inquire into whether and to what extent logic is a science, we end up engaging with three, undoubtedly intricate, issues at once, namely: (i) the issue of what counts as logic (i.e.: how do we demarcate logic from non-logic?); (ii) the issue of what counts as science (i.e.: how do we demarcate science from non-science?); (iii) the issue of whether logic meets (some or all of) the demarcation criteria for being a science (i.e.: does logic qualify as a science?). The plan for this section and the subsequent two is thus to discuss these three questions and to provide some useful indication concerning how to address them in relation to the aim of advancing the debate on anti-exceptionalism about logic. Let’s start with the issue of how to demarcate science from non-science.

3.2 Demarcation Problems in Science

“Science”Footnote ⁴⁰ is one of those terms that we, as competent users of a natural language living in a society heavily relying on technological and scientific knowledge, are all acquainted with. We inherently grasp an intuitive, though somewhat vague, distinction between what qualifies as science and what does not. Indeed, it is rare to find someone who would argue against physics being a science, while categorizing divination as one. However, the task of establishing a clear-cut criterion to distinguish what qualifies as science from what does not has proven to be extremely challenging. This challenge has led many philosophers to abandon the demarcation issue for an extended period, viewing it as unsolvable and therefore not worth pursuing. More recently, the research on demarcation made a timid comeback especially in relation to the spreading of science denialism and pseudoscientific disinformation. In this regard, we find ourselves in agreement with Imre Lakatos’ assertion that “the demarcation between science and pseudoscience is not merely a problem of armchair philosophy: it is of vital social and political relevance” (Lakatos Reference Lakatos1978: 1). In this section, our aim is to highlight the key elements of the discussion surrounding the demarcation problem throughout the twentieth century.

Before delving into this task, it is important to note from the outset that the type of demarcation quest prevalent in debates within the philosophy of science operates at a broader level of granularity compared to the kind of demarcation we will be addressing in Section 4 in relation to logic. While our task there will be to delineate what (deductive) logic is (and is about) in order to demarcate at least part (albeit an important part) of the discipline of logic from other disciplines, the task that is before us in this section is concerned with the more general issue of delineating the broad field of scientific disciplines – as opposed to both pseudo-scientific and non-scientific disciplines. The goal, in this respect, is not to establish criteria for distinguishing between disciplines like physics, biology, and chemistry (just to mention three paradigmatic examples of scientific disciplines), but rather to identify the criteria by which all three are recognized as sciences.

With this in hand, let’s proceed with discussing the issue of demarcation in science. There has been an important debate during the twentieth century about the so-called “demarcation problem.”Footnote ⁴¹ The project behind this discussion was the rather ambitious one of providing a set of individually necessary and jointly sufficient conditions for deciding whether a certain theory or practice is scientific or unscientific (which includes both non-scientific as well as pseudo-scientific theories and practices). Success in finding such a set of necessary and sufficient conditions would provide us with a sharp demarcation criterion capable of determining, for any given candidate discipline whether it counts as scientific or not.

This ambitious project was particularly prominent during the heyday of logical positivism and logical empiricism and found its clearest expression throughout the influential work of Karl Popper. Given their staunch opposition to (traditional, mostly German) metaphysics, logical positivists of the Vienna Circle were interested primarily in providing a criterion for distinguishing scientific (and thus meaningful) claims from metaphysical (and thus meaningless) claims. Within this project, logic as a tool for the logical analysis of language had a prominent role. As Carnap sharply puts it:

Following the Wittgensteinian thesis that the meaning of a sentence is the method of its verification,Footnote ⁴² logical positivists proposed verificationism as the main criterion of demarcation between science and metaphysics.Footnote ⁴³ The thought, in simple words, was that where empirical verification, including in-principle verification, is not possible, we are not dealing with science but with metaphysics. Notoriously verificationism, both as a demarcation criterion as well as a constraint on meaning, encountered insurmountable objections chief among which was the failure to classify as scientific all kinds of universally quantified generalizations. This is of course a major shortcoming since many statements that we would certainly classify as scientific take the form of a universal quantification (e.g., the statement that all electrons are negatively charged). However, statements like these cannot be verified, since a verification for such claims would require a method for checking every instance of the generally quantified statement, which is clearly not feasible (and in certain cases not even possible). Another problematic set of cases is given by negative existential generalizations (statements such as “there exists no sphere of uranium with a diameter of 1 mile” or “there is no phlogiston in nature”). All these statements would be classified as non-scientific, which is evidently an unacceptable consequence of the verificationist criterion of demarcation since these are clearly true scientific statements.

After this first, important but ultimately unsuccessful, attempt, the discussion of demarcation was then taken up by Karl Popper who, like the logical positivists, was highly invested in the quest of devising a strict criterion for distinguishing science from non-science. As a matter of fact he thought that demarcating science from non-science constituted the “key to most of the fundamental problems in the philosophy of science” (Popper Reference Popper1962: 42). Popper rejected the logical positivists’ idea that verifiability provides the demarcation criterion and suggested falsifiability instead. The thought was that in order to count as scientific a theory must issue predictions which should be capable of conflicting with possible or conceivable observation – that is, for any scientific theory or hypothesis it should be possible to come up with observations which would prove the theory or hypothesis wrong (see Popper Reference Popper1962: 39). Moreover, Popper made it clear that the kind of falsifiability referred to in his demarcation criterion “only has to do with the logical structure of sentences and classes of sentences” (Popper Reference Popper, Seiffert and Radnitzky[1989] 1994: 82). In this respect, a theoretical statement (a prediction) is falsifiable if and only if it logically contradicts some (empirical) statement about a possible observable event, where the modality involved is logical modality (Popper Reference Popper, Seiffert and Radnitzky[1989] 1994: 83). It’s worth noticing that Popper’s falsification procedure takes the form of a modus tollens: roughly, a recalcitrant observation entails the negation of the working hypothesis. Thus, Popper’s demarcation criterion heavily relies on the validity of certain logical principles (in particular modus tollens, and, more fundamentally, modus ponens and absurdum).

Popper’s account was criticized on several counts. A principled criticism came from considerations related to the so-called Duhem-Quine thesis according to which theoretical hypotheses cannot be directly falsified by recalcitrant observations, since they only entail observation-statements when taken together with auxiliary hypotheses. Thus, when observations do not accord with the predictions of the targeted theoretical hypothesis we face a choice: either you reject (or amend) the hypothesis (and thus the theory) or you reject (or amend) at least some of the auxiliary hypotheses. A second principled set of criticism points to a misclassification issue in that it seems to rule out what are widely, but perhaps controversially, considered legitimate sciences (such as psychology and sociology) while classifying what we intuitively would consider pseudosciences (e.g., astrology) as scientific.Footnote ⁴⁴ For example, many psychological theories deal with complex human behavior, which is influenced by numerous, sometimes non-quantifiable factors, making strict falsification difficult. Similarly, sociological theories often aim to understand broad social patterns that are challenging to test in controlled, falsifiable experiments. In contrast, astrologists may formulate rather bold predictions about individual behaviors or events, which could then be tested and potentially falsified in a rather straightforward way. A third, more specific, set of issues, shared with verificationism, concerns negative existential generalizations which were classified as unscientific in Popper’s account. Despite these difficulties, Popper’s influence within the scientific community was so prominent for many years that even contemporary discussions of the scientific status of certain theories (e.g., string theory) make implicit reference to some form of falsificationist ideas.Footnote ⁴⁵

Imre Lakatos carried forward the quest for a precise demarcation criterion, refining the approach initially proposed by Popper by means of what he labeled “sophisticated (methodological) falsificationism.” While Lakatos believed there was merit in Popper’s approach to demarcation, he was critical of several key aspects of it. In particular, he considered Popper’s view too restrictive since it predicts that a big part of everyday scientific practice is unscientific – which is arguably an unwelcome result. He thus proposed a somewhat radical revision of Popper’s account. Let’s briefly explore the key differences between Lakatos’ and Popper’s approaches to the demarcation of science.

For one thing, Lakatos, differing from Popper, adopted a more holistic approach to the demarcation issue. He suggested that the demarcation criterion should not be applied to an isolated hypothesis, statement, or theory, but rather considered in a broader context. More specifically, in applying the demarcation criterion we should take into account entire research programs which are characterized as a series of theories with a common hard core of fundamental ideas and a shared set of methodological rules. The hard core is enclosed by a protective belt of auxiliary hypotheses, observation statements as well as statements describing initial and boundary conditions. The methodological rules guide the research within a research program by means of a positive and a negative heuristic. The positive heuristic directs researchers toward fruitful avenues of inquiry, suggesting paths to explore. Conversely, the negative heuristic advises which paths should be avoided in conducting research. While the hard core is irrefutable by fiat, what goes into the protective belt can always be changed with the aim of safeguarding the core of the program. This led Lakatos to give up falsifiability as the sole demarcation criterion which can effectively discriminate between science and non-science. In principle a research program can be falsifiable, in some sense of the term, but qualify, intuitively, as unscientific, and, conversely, a research program can be scientific but unfalsifiable. In opposition to Popper, Lakatos believed that it is permissible to protect the core of a research program from empirical refutation. Additionally, Lakatos had a more lenient approach to demarcation compared to Popper in another vital aspect: he did not view the discovery of an inconsistency within a research program as a definitive reason for condemning a research program as unscientific. As Lakatos puts the point: “The discovery of an inconsistency – or of an anomaly – [need not] immediately stop the development of a programme: it may be rational to put the inconsistency into some temporary, ad hoc quarantine, and carry on with the positive heuristic of the programme” (Lakatos Reference Lakatos1978: 58).

Equipped with these insights, Lakatos offers a more flexible framework for demarcation, effectively aligning the criteria for what constitutes science with the characteristics of a good (i.e., progressive) scientific program. A research program is assessed as progressive if the old theories are subsequently replaced with new theories which, while preserving the same core theses, predict novel and hitherto unexpected facts, some of which are then confirmed by observation. Given this, contrary to Popper’s dogmatic approach, Lakatos deemed it as rational to ignore anomalies so long as a research program is progressing. On the contrary, a research program is considered degenerating if its successive theories do not yield novel predictions or if the new predictions they do offer are proven false. This distinction underscores the dual importance of theoretical innovation and empirical validation in evaluating the scientific merit of a research program. In this way the science–pseudoscience spectrum sees an internal articulation that ranges from highly progressive research programs at one end of the spectrum to highly degenerative ones at the other.

Undoubtedly, Lakatos’ proposal effectively improved on Popper’s account of demarcation. In particular, it gives us a more realistic picture of demarcation than what is delivered by Popper’s account. Moreover, it helped significantly in dealing with the problematic aspect of falsificationism as the sole demarcation criterion, especially with the issue of negative existential generalizations that were treated as unscientific in Popper’s model.

However, despite its numerous merits, Lakatos’ proposal is not immune to criticisms. In their critical piece on Lakatos’ “The Methodology of Scientific Research Programme” (in Lakatos Reference Lakatos1978), Catherine Elgin and Jonathan Adler (Elgin & Adler Reference Elgin and Adler1980) put forward a series of criticisms to Lakatos’ account of demarcation. For one thing, they object that Lakatos made real progress over Popper in relation to the search for a demarcation criterion. They write: “The distinction between progressive and degenerating problemshifts is a distinction between good and bad science (or perhaps, promising and unpromising science). It cannot solve the original problem of differentiating science from non-science” (Elgin & Adler Reference Elgin and Adler1980: 414).

Moreover, they put forward a series of criticisms to Lakatos’ idea that the hard core of a scientific program is practically irrefutable by empirical evidence. The core of the criticism lies in the realization that while “fallibilism constrains us to recognize that when we take evidence to count against the hard core of a programme we may be wrong […] it does not compel us to conclude that evidence can never tell against the hard core” (Elgin & Adler Reference Elgin and Adler1980: 416). As a matter of fact, the criticism goes, we cannot exclude that after careful investigation we are forced to conclude that none of the auxiliary hypotheses in the protective belt is wrong. But, if that turns out to be the case, we would be forced to conclude that something in the hard core is wrong.

A second influential line of criticism comes from William Newton-Smith who has argued that Lakatos’ proposal is theoretically wanting because it leaves out important conceptual aspects of science in the evaluation of what counts as a progressive (and thus good) research program. The concern, according to Newton-Smith, is that “[A]ny model of science must leave room for the differential assessment of theories in terms of their power to avoid conceptual difficulties and not just in terms of their power to predict novel facts and explain known facts” (Newton-Smith Reference Newton-Smith1981: 89).

This is an important shortcoming for a demarcation criterion that has the ambition to be fully comprehensive and adequate not only in demarcating science from non-science but also in distinguishing good science from bad science. In this respect not even Lakatos’ highly refined proposal was deemed sufficiently accurate to provide a sharp and univocal demarcation criterion.

Due to the numerous unsuccessful attempts to establish a clear and practical criterion for demarcation, skepticism about the feasibility of the project began to spread within the community of philosophers of science. In particular, the search for a demarcation criterion lost much of its traction after Larry Laudan’s Reference Laudan, Cohen and Laudan1983 paper “The Demise of the Demarcation Problem.” In this critical piece, Laudan severely doubted the philosophical significance of searching for a demarcation criterion claiming that the question – what makes a theory scientific? – “is both uninteresting and, judging by its checkered past, intractable” (Laudan Reference Laudan, Cohen and Laudan1983: 125). At the core of his argument is the thought that given the “epistemic heterogeneity of the activities and beliefs customarily regarded as scientific,” searching for a demarcation criterion is a futile quest. He proposed instead to focus on the question: What makes a belief well founded (or heuristically fertile)? – which he considered both philosophically interesting and tractable. This approach effectively shifts our investigation from the specific field of philosophy of science to a broader inquiry encompassing insights from both epistemology and cognitive psychology.

For a few decades Laudan’s skepticism toward the search for a demarcation criterion dominated the scene in the philosophy of science and thus what Popper considered one of the core issues in the discipline was, for better or worse, mostly ignored. More recently, philosophical discussions over demarcating science from non-science (including pseudoscience) made a comeback.Footnote ⁴⁶ The new impulse to the debate on demarcation sprang from the basic thought that having some indication on how to distinguish scientific from non-scientific theories and discipline not only remains a reasonable and perhaps philosophically interesting endeavor but it also has quite significant implications for our society. Consider, for instance, the enormous impact that scientific results have, directly or indirectly, on policy makers, education, as well as decisions concerning the distribution of research fundings.Footnote ⁴⁷ When deciding between funding medical or naturopathic research, governments need criteria to justify preferring the former over the latter. This preference could be based on the understanding that medical research, unlike naturopathic research, is scientifically grounded. Another prominent reason for the importance of a philosophical investigation on the issue of demarcation concerns the broad phenomenon of science denialism which is due, for a significant part, to a profound mistrust in scientific institutions which is motivated primarily by some deep misunderstanding of what science is and how it works.

Be that as it may, in the 1970s and 1980s, a shift occurred from single to multicriteria approaches.Footnote ⁴⁸ Historically, multicriteria approaches to the demarcation issue can be traced back to the work of Karl Gustav Hempel and Thomas Kuhn,Footnote ⁴⁹ and have been more recently (and explicitly) advocated by Sven Hansson and Martin Mahner,Footnote ⁵⁰ among others. In general, the post-Laudan science-demarcation trendFootnote ⁵¹ has relaxed the demands for what should count as philosophically interesting and practically useful demarcation criteria either by dropping the strict project of isolating a set of individually necessary and jointly sufficient criteria for scientificity or by supplementing such a strict definition with a set of additional criteria which are discipline-specific in order to become fully operative. Multicriteria approaches to demarcation can be articulated in a variety of ways. In one articulation, generally speaking, the primary task of a multicriteria approach is to come up with a list of features (arguably a stratified list with some primary and some secondary elements) that a discipline has to satisfy in order to count as a scientific discipline. This set of features should be strict and precise enough to leave out any discipline that we would confidently judge not to be a science and, on the other hand, general and malleable enough to be able to capture the important variety of special sciences. Quite predictably, there will be controversial cases and the decision about those may require appealing to some extra factors. In a somewhat more flexible interpretation of the multicriteria approach, determining whether a theory T qualifies as scientific involves checking if it meets any of the agreed-upon sufficient conditions, or if it fails to meet any of the necessary conditions for being considered scientific. Sebastian Lutz seems to suggest something in this ballpark when he criticizes Laudan’s claim that “without conditions which are both necessary and sufficient, we are never in a position to say this is scientific: but that is unscientific” (Laudan Reference Laudan, Cohen and Laudan1983: 119). Lutz in fact argues: “But Laudan’s claim is false: To be able to say that a is scientific (Sa) while b is not (¬Sb), all that is needed is one sufficient condition ϕ that is fulfilled by a […] and one necessary condition ψ that is not fulfilled by b […]. Laudan’s demand that ϕ and ψ be one and the same is supererogatory” (Lutz Reference Lutz2011: 126)

In a recent paper, Ilmari Hivonen and Janne Karisto go even further and claim:

We agree that traditional approaches to the demarcation problem based on finding a set of necessary and sufficient conditions have little chances of success. For this reason, we will adhere to the recent trend of considering a multicriteria approach to demarcation as the most promising framework to effectively demarcate science from non-science. Our proposal is to adopt, within a multicriteria approach, an abductivist model concerning theory choice according to which rival theories are chosen on the basis of their score in relation to a variety of factors (or, more aptly, theoretical virtues). Many are the theoretical virtues proposed and discussed in the literature. However, given the somewhat restricted scope of our project, we will focus on discussing those core criteria that seem relevant for assessing the scientificity of logic (which, intuitively, may well be among the controversial cases). Core demarcation criteria that have been discussed in the literature include generality, empirical adequacy, revisability on the basis of (possibly empirical) evidence, explanatory power, predictive success, modeling capacity, and amenability to the use of abductive methods for theory choice – in particular, scientific strength and simplicity. A further important criterion which is typically omitted in discussion about demarcation and theory choice criteria – a glaring omission, as Gila Sher would put itFootnote ⁵² – is truth. While this list isn’t exhaustive, it is nonetheless representative of the most important demarcation criteria within a multicriteria approach. Furthermore, while there are different methods for weighing and aggregating these criteria, we will not take a definitive position on this issue in this volume. Instead, in Section 5, we will offer a concise overview of these criteria, applying them to address the question: Is logic a science? Section 4 will delve into the subject of demarcation in logic.

5.1 Core Elements for the Comparison between Logic and Science

Thus far, we have examined approaches for distinguishing, on the one hand, the discipline of logic from other fields and, on the other hand, differentiating scientific disciplines from non-scientific ones. While we haven’t arrived at definitive conclusions about demarcation (as previously mentioned, that is not among the aims of this Element), we have nevertheless gathered sufficient elements to advance our project. Specifically, we can now discuss what it takes for logic, as a discipline, to share enough characteristics with paradigmatic scientific disciplines, such as biology, chemistry, and physics, to be reasonably considered a science in its own right. Recognizing the difficulty of establishing a strict demarcation criterion based on necessary and sufficient conditions, we will adopt a multicriterial approach. Specifically, we will utilize the criteria outlined at the end of Section 3. Let us begin with a brief recap of these criteria and their relevance to logic.

Let’s start with the truth criterion. A fundamental assumption of any scientific discipline is that it engages in truth-apt discourse. In other words, the statements that make up a scientific theory are intended to express truth-apt contents (propositions). Furthermore, truth is taken to play two distinct, albeit interconnected, normative functions in relation to scientific enquiry. On the one hand, truth is the aim of scientific inquiry in the sense that any specific scientific discipline is in the business of discovering truths about the world. On the other hand, truth is the chief criterion for assessing the correctness of our scientific inquiries in the sense that a scientific statement is correct just in case it expresses a true proposition.Footnote ⁷⁸ How substantive these normative functions that truth exerts on scientific inquiries are will depend on a set of issues pertaining, on the one hand, on how to interpret the nature of truth and, on the other hand, how to understand the kind of normativity at issue.Footnote ⁷⁹ Deciding on these issues will impact on the question of whether and to what extent logic satisfies the truth criterion. For instance, even if we grant that logical theories deal in truth-apt discourse (a view that may be challenged by logical non-cognitivists and logical expressivists), if we endorse some form of substantive correspondentist account of the nature of truth which is the one that may be reasonable to assume in relation to scientific inquiries, things may look controversial in the case of logic. To claim that the truth of logical statements lies in their correspondence with facts commits to endorsing a contentious metaphysics of substantive logical facts (e.g., facts which are mind-independent and determine, in some causal or metaphysical sense, the truth of our logical statements). By contrast, if we adopt a minimalist conception of the nature of truth, things would look much less controversial but also perhaps less interesting in relation to the comparison between logic and science since satisfaction of the truth criterion would be rather cheap. In a minimalist conception, the nature of truth is relatively insubstantial, serving primarily logical and expressive functions, such as indirect endorsement and generalization.Footnote ⁸⁰ There is no need, then, to postulate the existence of substantial logical facts or a metaphysical or causal relationship between logical facts and logical statements. This means that once the truth-aptness of logical statements is granted, logic would be uncontroversially considered a science according to the truth criterion under a minimalist view of truth and its normativity. By contrast, by adopting a correspondence account of truth it would be significantly more controversial to consider logic a science under the truth criterion.Footnote ⁸¹

The second criterion to discuss is generality. A scientific discipline worth its name aims at issuing true, law-like generalizations about the world. The degree of generalization will of course vary depending on which specific scientific discipline we are talking about: for instance, the kind of generalizations reached by research in fundamental physics presumably will be broader in scope than those reached via research in ornithology. In this respect, the generality criterion should not be seen as in tension with the thesis that different sciences may vary in their degree of generality – a thesis that could in principle be reconciled with certain versions of scientific (and logical) particularism (see Payette & Wyatt Reference Payette, Wyatt, Wyatt, Pedersen and Kellen2018). This is because there’s a principled distinction between two theses: the thesis that a certain science (e.g., logic) has an universal scope of application versus the thesis that a science, in relation with its proper subject matter, seeks for generality. While the former is incompatible with particularism (and perhaps, in general, also problematic when it comes to the natural sciences), the latter is perfectly compatible with particularism. The kinds of generalizations issued by logic are law-like, and, arguably, they are the most general ones. These could take the form of metalinguistic generalizations about validity facts, as it were, like in: all the arguments based on modus ponens are valid. Alternatively, they could take the form of objectual generalizations about very abstract general patterns in the mostly non-linguistic world like in: for any given fact F, either F obtains or F doesn’t obtain. Thus, when it comes to generality logic, as individuated by either the model-theoretic or the proof-theoretic criteria, is on the side of the sciences (at least on many accounts of the metaphysics and epistemology of logic, both on the exceptionalists and the anti-exceptionalist camps).

Third we may consider adequacy (or, the ability of being confirmed by evidence) as a second feature of comparison. We may distinguish between two notions of adequacy: factual and empirical adequacy.Footnote ⁸² While empirical adequacy entails factual adequacy, the converse does not hold. Factual adequacy is the claim that the adequacy conditions of a logical theory depend on worldly facts which are not necessarily empirical. Empirical adequacy is the stronger claim that the adequacy conditions of a logical theory depend on strictly empirical facts. What is the relationship between facts and empirical facts is a matter of controversy which depends on complex issues in the metaphysics of facts.Footnote ⁸³ As it may be a matter of controversy whether the requirement of a logical theory to meet a factual adequacy criterion is itself already a challenge to an exceptionalist conception of logic. The adequacy conditions for paradigmatic sciences like physics, chemistry, and biology clearly require empirical adequacy. As a result, when considering the continuity between logic and the sciences in terms of adequacy, the notion of empirical adequacy becomes especially important. Therefore, we will focus on the criterion of empirical adequacy in the following discussion.

In general, empirical adequacy (which could be understood according to a variety of models like the hypothetico-deductive model, the inference to the best explanation model, Bayesian models, etc.) is taken to be the primary ground for testing the success or unsuccess of a scientific theory (via confirmation or disconfirmation of its predictions). Claiming continuity between logic and science on the count of empirical adequacy is certainly more controversial than claiming continuity with respect to generality. What this would require is, at minimum, the thesis that logic is about aspects of the world that are empirically testable. Some proposals in this direction take the empirical data in logic to be considered judgments about which inferences formulated in the vernacular, as Priest has it, are acceptable (or good, or reasonable, etc.) and which aren’t.Footnote ⁸⁴ Another possibility would be to take data in logic to concern very general or structural features of the world which are somehow empirically detectable.Footnote ⁸⁵ A third recognized source of empirical evidence within the purview of logic is data about language use. If one of the main tasks of logic as a discipline is that of formalizing natural language expressions in order to reveal (or disambiguate between) their logical form(s) then the linguistic judgments of competent users of the language may well count as empirical data. One case in point here is the debate about which one between material implication and relevant implication is more suitable to capture the way in which the expression “if … then … ” is typically used in English.Footnote ⁸⁶ In fact, relevant implication has been devised as an attempt to avoid the so-called paradoxes of material and strict implication.Footnote ⁸⁷ An additional example is given by Penelope Maddy who takes the existence in the world of KF-structureFootnote ⁸⁸ – namely, objects that enjoy and fail to enjoy properties, that stand and fail to stand in relations, where some situations involving these objects stand as ground to other situations as consequent – to exert a kind of evolutionary pressure on us which led to the development of a rudimentary logic of classical inferences involving conjunction, disjunction, negation, and quantification, capable of reliably tracking the structural features of the world.Footnote ⁸⁹

A fourth feature that, as we have seen, is closely related to the second one, is the revisability on the basis of (empirical) evidence. It is quite uncontroversial to claim that scientific theories whose predictions are in conflict with observation should be revised, possibly avoiding ad hoc amendments. Thus, revisability on the basis of empirical evidence is another hallmark of science. The applicability of this criterion to logical theories is an issue as contentious as the applicability of empirical confirmation to logic. In fact, it demands an analogous commitment to the idea that logic concerns empirically testable matters, as emphasized in the discussion of the previous criterion. The debate over the potential for empirical revisability probably reached its peak in the 1970s, particularly in relation to certain quantum phenomena that seem to require the revision of some principles of classical logic. More specifically, some philosophers, prominently Hilary Putnam,Footnote ⁹⁰ have suggested in several papers that, given a certain interpretation of Heisenberg’s uncertainty principle, the possibility of superposition in a quantum system requires the revision of classical distributivity principles (in particular, it requires the revision of the distributive principle of conjunction over disjunction).Footnote ⁹¹ Under the assumption that it makes as much sense to speak of “physical logic” as of “physical geometry,” Putnam drew an analogy between the case of classical logic and quantum physics, and what happened to Euclidean geometry which was revised in light of its incompatibility with general relativity. Even though Putnam’s argument did not find much agreement among logicians and philosophers of logic,Footnote ⁹² it offers an interesting case study to appreciate the possibility of empirical revisability of logic.

A fifth significant aspect relates to the explanatory power of a theory concerning the targeted phenomena. It is a hallmark of scientific disciplines to provide explanations of natural phenomena. The kind of phenomena may vary to a significant extent (perhaps, in tandem with the appropriate model of explanation),Footnote ⁹³ depending on the specific science we are dealing with. For instance, while quantum physics provides explanations of natural phenomena at the atomic and subatomic scale, biology provides explanations of phenomena associated with living organisms and their vital processes. Does logic provide explanations? And if, so, what does logic explain? It seems reasonable to claim that logic is not in the business of providing causal explanations of natural phenomena. This already creates a stark contrast with the sciences, where causal explanations are ubiquitous and of crucial importance. But there are other kinds of explanations that logical theories may offer. One natural option within the common understanding of logic as the science of validity is to say that logic does indeed explain facts about the validity or invalidity of arguments formulated in natural language.Footnote ⁹⁴ This view is held, for instance, by Graham Priest:

And it is echoed by Payette and Wyatt:

A different kind of explanatory role that logic may play has to do with providing informative generalizations which capture certain kind of observations of worldly state of affairs which we may call logical in character – for example, that either an object has a property or it lacks it; that if something is actually the case then it’s also possible, and so on.

A sixth relevant criterion is amenability to be used as a modeling device. Sciences make wide use of models which are thus taken to be of central importance in many scientific contexts. There are multiple kinds of models, but many scientific models are representational. Among well-known representational models in science we can mention the Watson and Crick three-dimensional double helix model of DNA, the billiard ball model of a gas, the Bohr model of the atom. Among others, Roy Cook, Michael Glanzberg, and Stewart ShapiroFootnote ⁹⁵ take logic to align with the sciences in this respect: formal systems are taken to provide (mostly representational) models of targeted phenomena (phenomena arising, for instance, from natural language uses and reasoning). As Shapiro nicely puts it in his Vagueness in Context:

A well-known example frequently discussed in the literature is the use of logic as a representational model for vagueness, a phenomenon widely regarded as pervasive in natural languages. In this context, we can employ precise mathematical tools to construct fruitful representations (i.e., models) of linguistic phenomena such as vagueness or any kinds of indeterminacies. By doing so we offer a way to represent and predict the behavior of vague terms without implying precision where it does not naturally exist.

Finally, in exploring the scientific standing of logic, another crucial factor is the methodological underpinnings of theory selection. It’s commonly asserted that the choice of theories in science is predominantly guided by an abductive methodology, interpreted in a broad sense.Footnote ⁹⁶ For instance, Williamson claims:

To choose between rival scientific theories that exhibit comparable merits, particularly in terms of their explanatory and predictive power, scientists often rely on abductive criteria. These criteria include simplicity, non-ad-hocness, and strength. Some logicians, especially Ole Hjortland, Gillian Russell, and Timothy Williamson, following a broadly naturalist methodology coming from Quine, have argued that theory choice in logic should follow a similar abductive strategy. There are some core abductive criteria particularly useful to gain a better grasp of the extent to which an abductive methodology that works in the sciences can be carried over to the case of logic. Strength and simplicity are, in this perspective, paradigmatic ones. How should we understand these theoretical virtues?

Let us start with the notion of strength. A distinction can be drawn between logical and scientific strength.Footnote ⁹⁷ In broad terms, the concept of a theory’s logical strength concerns primarily the deductive power of the theory, while its scientific strength is largely related to the amount of informational content the theory delivers. More specifically, Williamson defines logical strength standardly as follows: “In one standard logical sense, a theory T is stronger than a theory T* if and only if T entails T* but T* does not entail T: every theorem of T* is a theorem of T, but not every theorem of T is a theorem of T*” (Williamson Reference Williamson and Armour-Garb2017: 336).Footnote ⁹⁸

Scientific strength, on the other hand, is standardly based on a notion of informativeness and specificity: scientifically stronger theories are more informative and specific. Indeed, they give better and more precise explanations. But what to make of this idea in the case of logic?Footnote ⁹⁹ The thought is that a scientifically stronger logical theory is one that gives us more informative and precise answers to our questions – for example, questions like: does excluded middle hold for any sentence of our language? If not, for which sentences does it fail? Addressing questions like these can allow us to compare various logics – paradigmatically, classical and non-classical logics. Typically, non-classical logics restrict the range in which a contested logical principle is valid (to all those cases that are not deemed problematic); in all other cases they are fine in claiming that the principle is valid. Take, for example, the Logic of Paradox (LP) – roughly a kind of paraconsistent logic in which we have true contradictions but the consequence relation is not explosive (Priest Reference Priest1979). In LP Priest observes that, in a semantically closed theory, using modus ponendo ponens (MPP) and absorption (P→(P→Q))⊢(P→Q) a version of Curry’s paradox is derivable.Footnote ¹⁰⁰ In LP, (A → B) is defined as ( $\neg$ A $\vee$ B) (the material conditional), which suffices to establish that MPP can’t in general be valid. For, if A is a dialetheia (a proposition both true and false), ( $\neg$ A $\vee$ B) is true even if B is not. MPP is labeled in LP as a quasi-valid rule, a rule that is valid provided that all truth-values involved are classical (i.e., solely true or solely false). Now, however, it appears that classical logic is more informative than LP. When asked, “Which instances of the contested principle hold?” classical logic responds: all, whereas non-classical logics (such as LP) respond: not all. Clearly, an “all” answer provides more information than a “not all” answer.

Such a lack of information may be mitigated by providing a principled reason for sharply individuating the class of statements for which the contested principle is taken to fail (e.g., a principled way of detecting vagueness in natural language expressions that would allow us to predict whether for a given statement S excluded middle would hold). However, even assuming that such a principled reason were available in all relevant cases, the resulting proposal would, firstly, complicate the theory, potentially adding ad hoc elements; and secondly, it would invariably depend on an extra-logical principle, necessitating further philosophical justification in the background, increasing thus the complexity of the theory.

After defining the concepts of logical and scientific strength as described, an interesting question emerges: how are the notions of logical and scientific strength related? Regarding this issue, there is considerable disagreement. Williamson claims that logical strength entails scientific strength for the reason that more deductive power yields more information. Moreover he believes that both logical and scientific strength are virtues. Russell (in Russell Reference Russell2018) agrees with Williamson that scientific strength is a virtue but she rejects Williamson’s claim that logical strength implies scientific strength and, moreover, she believes that logical strength should be considered neither a virtue nor a vice. In the model developed by Incurvati and Nicolai which employ translations between theories in accordance with suitable information-preserving constraints, we have that both scientific and logical strength are considered virtues.Footnote ¹⁰¹ Moreover, in their view, scientific strength entails logical strength but not vice versa, since not all translations involved in the relation of logical strength are adequate for scientific strength. Hjortland (in Hjortland Reference Hjortland2017), adopts a more radical stance. He goes along with Williamson’s characterization of logical strength but argues that logical weakness, and not logical strength, should be considered a virtue in a theory since it allows the theory to draw more, and more fine-grained, distinctions.

As evidenced by this short description of the current state of the art, there’s ample disagreement on how to properly characterize the notion of strength as well as how to conceive of the relationship between logical and scientific strength. Although this disagreement is internal, as it were, to the abductivist camp, it nevertheless, and quite predictably, has significant consequences on the outcome of abductive comparison between rival logical theories.

As mentioned earlier, a second important element of theory comparison has to do with the comparative simplicity of rival theories. How should we understand simplicity?Footnote ¹⁰² Here the state of the debate within the philosophy of logic is less developed than in the case of the notion of strength. One intuitive way of characterizing simplicity is by means of avoidance of gerrymandered concepts or ad hoc hypotheses to account for phenomena that could be equally accounted for with more joint-carving concepts and avoiding the postulation of ad hoc hypotheses. Ceteris paribus, if two theories can equally explain a given set of phenomena but one theory, T1, uses more (or more gerrymandered) theoretical resources than the other theory T2, then T2 is favored over T1.Footnote ¹⁰³ So for example, in science simple theories often have greater predictive power than more complex ones, making it easier to foresee outcomes and design experiments. Or, again, consider simplicity in its relation with the notion of elegance:Footnote ¹⁰⁴ Scientists appreciate elegant theories that can explain a wide range of phenomena with minimal assumptions or complexity, rather than theories that are more complex and considered less elegant. Thus, a simpler theory can be chosen for pragmatic reasons (it is easier to handle) or for aesthetic reasons (it’s more elegant). However, perhaps the most significant – and to some extent, controversial – reason for favouring a simpler theory concerns considerations regarding the relationship between simplicity and truth: ceteris paribus, a simpler theory is taken to be more plausible (more probable, or with a higher expected degree of predictive accuracy) than its less simple rivals.

The concept of simplicity poses notably intricate challenges in the philosophy of science, challenges that extend into the philosophy of logic as well. As Quine has pointed out, there may be a tension between simplifying the ontology and simplifying the ideology (as he calls it): roughly, postulating the existence of more entities may make the formulation of the theory simpler,Footnote ¹⁰⁵ while simplifying the ideology making the formulation of the theory simpler, may require adding complexity to the ontology. And Ole Hjortland, writing about the issue of simplicity as a theoretical virtue in logic writes:

Another interesting trade-off issue discussed by Elliot Sober concerns curve-fitting problems and can be put in terms of balancing simplicity (smoothing the curve and thus diminishing the risk of overfitting) and goodness-of-fit (accommodating all data points). As Sober asks: “If curve X is simpler than curve Y, but curve Y fits the data better than curve X, how are these two pieces of information to be combined into an overall judgment about the plausibility of the two curves?” (Sober Reference Sober, Zellner, Keuzenkamp and McAleer2002: 20).

This highlights the difficulties of providing an accurate measure of simplicity. However, a notion of simplicity is essential to explain what makes excessively complex hypotheses problematic, especially when they are on the verge of becoming ad hoc or gerrymandered. For this reason simplicity is widely taken to be a theoretical virtue that can be used effectively as a theory choice criterion both in science and logic.

5.2 Paradigmatic Examples of Logical Anti-Exceptionalism

In the rest of this section, we will swiftly explore some of the most prominent anti-exceptionalist viewpoints in the ongoing debate. The degree to which these diverse perspectives consider logic to be in continuity with the natural sciences depends on which of the various criteria mentioned earlier are taken on board. Our goal is not to offer an exhaustive overview of the perspectives of self-proclaimed anti-exceptionalists in logic. Many and variegated are in fact the views that have been proposed as a kind of logical anti-exceptionalism. Recent explorations into various forms of logical anti-exceptionalism, though not exhaustive, include: Newton Da Costa and Jonas Becker Arenhart,Footnote ¹⁰⁶ Ole Hjortland,Footnote ¹⁰⁷ Penelope Maddy,Footnote ¹⁰⁸ Ben Martin,Footnote ¹⁰⁹ Gillman Payette and Nicole Wyatt,Footnote ¹¹⁰ Jaroslav Peregrin and VladimÍr Svoboda,Footnote ¹¹¹ Graham Priest,Footnote ¹¹² Stephen Read,Footnote ¹¹³ Gila Sher,Footnote ¹¹⁴ Gillian Russell,Footnote ¹¹⁵ Timothy Williamson.Footnote ¹¹⁶ Clearly this is not the venue for attempting a comprehensive review of all these works. Instead, we focus on examining key aspects of three paradigmatic anti-exceptionalist views – those of Timothy Williamson, Penelope Maddy, and the joint proposal by Ole Hjortland and Ben Martin – in order to better understand in what sense and to what extent they consider logic to be similar to the sciences.

Timothy Williamson, a staunch defender of classical logic, takes it as the abductively best logical theory in that it offers the best balance of the most important abductive values such as strength, simplicity, non-ad-hocness, and integration with scientific theories.Footnote ¹¹⁷ In recent works,Footnote ¹¹⁸ Williamson has argued that logic is not primarily about validity (or, more generally, about some kind of linguistic or metalinguistic entities), but it fundamentally concerns true law-like generalizations about the mostly non-linguistic world.Footnote ¹¹⁹ More specifically, Williamson takes the laws of (classical) logic, such as the law of excluded middle – which he formulates, allowing quantification in sentence position, as the universally quantified sentence ∀P(P∨¬P) – to represent broad structural laws that govern the predominantly non-linguistic world. Essentially, he views them as fundamental principles underlying both logic and metaphysics. This conception of logical laws emphasizes that the true universal generalizations which correspond to logical truths are of primary importance. Their significance lies in the insights they provide about the world itself, rather than about the nature of logical truth or validity. In Williamson’s view, then, our interest in these universal truths is not metalinguistic but as directly concerned with understanding reality as our engagement with the true statements of physics. In this respect Williamson’s position strictly echoes Bertrand Russell’s conception according to which “logic is concerned with the real world just as truly as zoology, though with its more abstract and general features” (Russell Reference Russell1919: 169). It also shares with Gila Sher the idea that logical theories focus on the entities denoted by the logical constants, which include negation, conjunction, disjunction, identity, universal and other quantifiers, and possibly more, all of which play a role in nearly all forms of rigorous theoretical inquiry.Footnote ¹²⁰ When expressed as universal generalizations, logical theorems do not describe relationships between sentences but they articulate highly general patterns that pertain to the largely non-linguistic world.

Besides arguing for the theory of classical logic to be abductively the best logical theory in virtue of offering the best combination of strength and simplicity, Williamson’s conception of logic fully qualifies as a science to a great extent, according to the criteria adopted here. Williamson sees logic as the most general science in that it involves generalizations that have a wider scope than those of any other science. Moreover, in concerning the mostly non-linguistic world the subject matter of logic is not of a kind totally dissimilar from the subject matter of the recognized sciences (paradigmatically, physics) and our methods for justifying or revising our logical theory are not dissimilar to those employed in justifying or revising our scientific theories.

Moreover, for Williamson logic plays some significant explanatory role. In its background role as an auxiliary formal tool for scientific theories, logic enhances the scientific strength of those theories as well as their explanatory and predictive power, by extracting more relevant consequences from them. Additionally, logic has an internal explanatory role in “subsuming isolated logical observations under illuminating generalizations – for example, in the twentieth-century streamlining of axioms for modal logic” (Williamson Reference Williamson, Ferrari, Brendel and Carraraforthcoming 1).

Relatedly, Williamson takes our logical theory to be fundamentally descriptive of some of the most general aspects of the world. The laws of logic can be considered normative, but this is a derived sense of normativity tied to the undesirability (or impermissibility, according to the chosen normative principles) of holding (or inferring) false beliefs. This is exactly analogous to the normativity of physical laws: it is incorrect to maintain false beliefs about the physical world.

However, it must be stressed that Williamson’s version of logical anti-exceptionalism is not a form of empiricism, since, contrary to what happens in the sciences, it gives no special role to experience to play. Indeed, logic, along with metaphysics and philosophy more broadly, is mostly conceived as an armchair pursuit. This divergence stems from the fact that in interpreted logic, unlike the natural sciences, evidence from experiment, observation, and measurement have usually no privileged role in accepting a theory. Moreover, despite accepting the general idea that logic is formal, he takes it not to offer a neutral tool for assessing competing metaphysical pictures. Taking logic to be such a neutral arbiter would mean to be forced to endorse some form of logical nihilism, which is certainly unwanted. This is because, simply, different philosophers have argued for different metaphysical views each of which puts into question one or more of the basic principles of classical logic. As a consequence, no single principle of classical logic has been immune from criticism on the basis of some metaphysical view.

All this being said, it is clear that for Williamson logic does not deviate from the standard process of theory formation and evaluation. Furthermore, experience is essential both for enabling beliefs in logical propositions and serving as evidence for/against them, in such a manner that neither aspect can be made null and void (contrary to what a sharp a-priori/a-posteriori distinction would demand). Although in Williamson’s picture the role of empirical (dis)confirmation is significantly downplayed, without this meaning that logic does not concern reality, it is evident, from this brief discussion of his conception of logic, that Williamson provides an excellent example of a logical anti-exceptionalist who sees logic in significant continuity with the sciences in terms of methodology, epistemology, and metaphysics.

A different anti-exceptionalist stance for logic has been developed and defended by Penolepe Maddy in several of her works (e.g., Maddy Reference Maddy2007, Reference Maddy2014, forthcoming). Maddy advocates a naturalist and empiricist conception of logic – a conception that is the result of a more general viewpoint on philosophy that she calls Second Philosophy. A second philosopher is someone who is interested in all aspects of the world and our place in it. In Maddy’s own words, such a philosopher, like a scientist

The second philosopher adopts the same attitude for all kinds of problems, including logical problems. Therefore, it’s evident that Maddy perceives a substantial overlap between the methodologies and the epistemology of logic and the sciences. In this overlap, the role of empirical adequacy is relevant. To get an intuitive picture of what she is after, Maddy asks us to consider the following scenario: a subject S holds a concealed object in her hand, which could be either a common dime or an unfamiliar foreign coin (chosen randomly from a container with only these two types while blindfolded). Upon touching it, S discerns that it’s not a dime, leading to the conclusion that it must be the foreign coin. The immediate question we may ask in relation to this scenario is the following: what does underlie this inference? When tackling such questions, the second philosopher approaches them as inquiries into the world. She begins by noting that the reliability of the coin inference and the truth of the corresponding conditional are not contingent on specific details about the physical composition or distinctive features of dimes. Instead, she asserts that only the following most general structural features of the situation really matter (which effectively exemplifies, at the objectual level, a reasoning pattern based on disjunctive syllogism): an object with one of two properties, lacking one, must possess the other. This insight is then systematically organized in the following way: For any object o and properties P and Q, if either o possesses P or o possesses Q and it is the case that o does not possess Q, then we conclude that o possesses P (and thus that the sentence “the object o possesses the property P” is true). Building on this foundation, the Second Philosopher develops a more comprehensive theory of forms that can generate widely applicable truths and reliable inferences. Maddy thus argues: “Suppose the Second Philosopher now codifies these features of her formal structures into a collection of inference patterns; coining a new term, she calls this ‘rudimentary logic’ [RL] (though without any preconceptions about the term ‘logic’). She takes herself to have shown that this rudimentary logic is satisfied in any situation with formal structure” (Maddy Reference Maddy2014: 95).

Maddy contends that her conception of logic is influenced by the ideas of Kant and Wittgenstein who, she argues, take the validity of logical laws to stem from the structure of our physical world (Maddy Reference Maddy2007: 49). More specifically, Maddy’s conception of what she calls “rudimentary logic” integrates two aspects: a transcendental one, of Kantian heritage, which positions logic at the core of our fundamental conceptualizations of the world; and an empirical one, according to which logic reveals the essential structure of the world. The aim of Maddy’s approach is to have a comprehensive explanation that retains both features. For this purpose, Maddy characterizes an abstract Kant-Frege structure – the KF-world – encompassing objects, properties, and relations. In contrast to conventional first-order structures, KF-worlds incorporate ground-consequent dependencies and uncertainties, giving rise to a rudimentary logic, RL. RL describes the macroworld, supported by common sense and science. Human beings accept it due to innate cognitive mechanisms recognizing the world’s KF-structure.Footnote ¹²¹ Moreover, human cognitive structures align with RL because the macroworld follows a KF-world pattern. As evident from this brief description, Maddy’s position in the philosophy of logic sees logic and the sciences to be aligned in both methodological and epistemological terms. Differently from Williamson, Maddy fully embraces a naturalistic perspective that assigns to experience and empirical adequacy a more prominent role.

As a third example of an anti-exceptionalist conception of logic, let us consider the most recent proposal by Ole Hjortland and Ben Martin, which they label “logical predictivism.” Logical predictivism is grounded in two, arguably uncontroversial, facts: (a) scientific theories strive to explain specific phenomena, and (b) scientific theories demonstrate their efficacy, at least in part, by generating accurate predictions. According to logical predictivism, we should take logical theories to be analogous to scientific theories in these crucial respects. In other words, logical theories should be, at least partially, evaluated on their ability to explain a given set of target phenomena and to make successful predictions. In their own words: “Logics then have phenomena they attempt to explain, and use successful predictions as a criterion to judge the fruitfulness of these explanations. […] [I]f our predictivist account is ultimately successful in reflecting how logicians go about supporting their theories, this will go a considerable way to substantiating the methodological anti-exceptionalist’s claims” (Martin & Hjortland Reference Martin and Hjortland2021: 288).

Hjortland and Martin argue that to provide meaningful insights into a target phenomenon and formulate predictions that are amenable to empirical testing, it is crucial to depart from the conventional idea that logical theories are just a set of valid inference rules or theorems. Instead, they propose to consider logical theories as comprehensive frameworks having as essential components definitions (e.g., let ¬φ be Boolean negation), laws (e.g.: For every valuation, all sentences are either true or false, and not both), and representation rules (such as: ⌜not φ⌝ = ⌜¬φ⌝, or ⌜if φ then ψ⌝ = ⌜φ → ψ⌝). All the aforementioned elements not only provide the underlying semantics but also contribute to the syntax of the theory, laying the groundwork for a formalization of natural language expressions.

Moreover, within the predictivist framework – which sees with a friendly eye a practice-based approach to the epistemology of logic – theories find their initial motivation in instances of arguments which are assessed as acceptable by reliable practitioners. This frequently involves informal mathematical proofs, which are assessed as acceptable by mathematicians, or alternatively, natural-language arguments assessed as acceptable by reliable reasoners (Martin and Hjortland: forthcoming). Consider the case of mathematical proofs. In such cases the mathematicians’ assessments of acceptable inferences serve as a reliable (but of course fallible) guide to distinguishing valid from invalid proofs. It is the task of the logician to argue why certain proofs are held valid or invalid. To do this, a logician follows a number of steps. First, she conjectures that inferences across multiple proofs may be (in)valid for similar reasons, sharing some underlying structure. Secondly, she puts forth a specific hypothesis regarding the validity of an argument form observed in “acceptable” informal proofs. This hypothesis, on its own, does not function as an explanation for the validity of the target proofs; it merely represents a generalization that can later be subject to additional confirmation or falsification. They argue that

They propose to consider an adequate explanation, something entailing a detailed description on how the inherent structure of these arguments guarantees their validity. Subsequently, such a theory should be tested to understand if it is preferable to competing theories. A method of accomplishing this is by generating predictions based on the theory’s postulates and then validating these predictions through testing. A way of testing them is to ask the relevant group of subjects, for instance mathematicians, whether they consider the target inferences acceptable or not. The assumption is that mathematicians’ judgments on the acceptability of proofs can be considered as a guide to the validity of the inferences at issue. The logician assesses the theory’s support based on how well mathematicians’ judgments align with the theory’s predictions. If there is agreement between judgments and predictions, the theory gains further support; conversely, if discrepancies arise, the theory encounters challenges that must be addressed or reconciled, analogously to what happens in the natural sciences. Unlike Williamson and Maddy’s approach, Martin and Hjortland’s logical predictivism focuses on validity as the primary concern of logical theories. They consider the most relevant data for assessing a theory’s adequacy to be considered judgments about instances of logical rules of inference. Their testing methodology is in line with standard methodology within the practice-based philosophies of science and mathematics.