What Logical Evidence Could not be

Abstract

By playing a crucial role in settling open issues in the philosophical debate about logical consequence, logical evidence has become the holy grail of inquirers investigating the domain of logic. However, despite its indispensable role in this endeavor, logical evidence has retained an aura of mystery. Indeed, there seems to be a great disharmony in conceiving the correct nature and scope of logical evidence among philosophers. In this paper, I examine four widespread conceptions of logical evidence to argue that all should be reconsidered. First, I argue that logical apriorists are more tolerant of logical evidence than empiricists. Second, I argue that evidence for logic should not be read out of natural language. Third, I argue that if logical intuitions are to count as logical evidence, then their evidential content must not be propositional. Finally, I argue that the empiricist proposal of treating experts' judgments as evidence suffers from the same problems as the rationalist conception.

1 The Problem of Logical Evidence

Nowadays, the philosophical debate about logic is dominated by all kinds of views: the monist and the pluralist debate as to whether the correct logic is ‘one or many'¹; the exceptionalists and the anti-exceptionalists argue as to whether logic has a privileged status²; finally, the factualists and the non-factualists dispute as to whether logic can have a descriptive or normative status.³ Moreover, as one might expect, most of these views can be intertwined in various ways.

For the sake of argument, let us suppose that an inquirer rationally believes that some logical theory (or more than one) is correct.⁴ Suppose further that our inquirer rationally believes that one (or more than one) of the above philosophical views is correct. For instance, our inquirer might believe that classical logic is the only true logic, and that logic (in general) is a non-exceptional and descriptive endeavor. In this case, we should be able to see that our agent's beliefs must be well-supported to be epistemically warranted.⁵ Therefore, in order to be considered rational, an accurate, logical belief must be formed based on an intelligible epistemic intermediary that adequately represents logical facts.

Evidence can be broadly conceived as the set of experiences and ‘considerations that epistemically count in favor or against you having certain beliefs' (Chudnoff, 2014, 13).⁶⁷ In other words, evidence represents the epistemic basis for our (propositional) knowledge of reality. Hence, everything we allegedly know about logic also depends (partly) on evidence.

Much has been said about logical evidence. Despite this, it is still unclear what its exact nature is. In the hope of providing clarity, in what follows, I will try to convince the reader of what logical evidence could not be.

Before proceeding, however, a disclaimer is necessary. The topic of logical evidence is undoubtedly vast. At least since the emergence of the first logical anti-exceptionalist accounts (or AEL for short), we can witness an increase in articles about logical evidence. Because of this, the present article will not address many things that have been said so far. Therefore, my analysis will be partly negative: I will try to remove some unpalatable accounts of logical evidence from the current picture. In doing so, I hope to delineate an account of logical evidence that can explain how we can form accurate, logical beliefs.

Here is the plan for the rest of the paper. In §2, I will introduce the reader to the debate between apriorists and logical empiricists. Here, I will argue that apriorists are more tolerant than empiricists in evidence matters. In §3, I will argue that linguistic evidence alone cannot account for the epistemology of logic. In so doing, I will rely on some considerations that have recently been presented in the literature. Finally, in §4 and §5, I will argue that the way apriorists and empiricists currently conceive logical evidence cannot account for logical knowledge.

2 Apriorism and Aposteriorism About Evidence

I think it will be useful to put forward the two main epistemic stances in the debate from the beginning. Hence, we will start by looking at what apriorism and logical aposteriorism (empiricism) could not be.

For starters, take Martin (2020). In presenting the two main apriorist stances in the literature – i.e., rationalism and semanticism – he writes:

Take first logical rationalism. According to logical rationalism, we gain justification for our logical beliefs directly from intuitions regarding a particular proposition. We simply see that the proposition p is true or false (...). Indeed, for the logical rationalist there is nothing else the logician can appeal to. If another party disagrees with us, all we can do is suggest that our interlocutor is not having the right kinds of intuitions, and that they should look a bit harder.

Furthermore, on logical semanticism, he writes:

The logical semanticist is in a similar position. According to her, we gain justification for our logical beliefs directly by grasping the meaning of logical propositions. Again, there is nothing other than the meaning of the propositions for the logician to appeal to. If one's interlocutor fails to assent to the same propositions as us, we are committed to saying this is simply because she has misunderstood its content, and is talking about a different matter than we are. (Martin, 2020, §3, emphasis added)

Rationalism and logical semanticism are the two main apriorist stances in the epistemology logic. According to these views, logic is exceptional because its claims can be a priori warranted via non-experiential evidence. However, although these views have the same goal – explaining the apriority of logic – they differ in the way they attempt to accomplish this task. While rationalism postulates the existence of a mental faculty capable of granting us direct access to facts of a non-empirical nature (cf. Bengson, 2015a; Chudnoff, 2013), semanticism is the thesis that logical claims can be warranted based on linguistic (conceptual) understanding alone (cf. Ayer, 1936; Boghossian & Williamson, 2020; Horvath, 2020).⁸ Therefore, due to these non-experiential sources, rationalism and semanticism represent the main alternatives to the empiricist stances in the epistemology of logic.

Vice versa, logical empiricism is the thesis that logical knowledge is a posteriori. Although Quine's works have breathed new life into logical empiricism, his followers would take many of his teachings cautiously today. Indeed, many Neo-Quineans (AEL advocates) have proposed new empiricist theories of logical epistemology in which they attempt to distance themselves from the traditional tenets advanced by their mentor.⁹

Traditionally, empiricism and apriorism have been conceived as opposite foundational projects. However, the debate between these two epistemological stances has been revised due to at least two developments in the way these views are now conceived. First, although empiricism has often preserved the idea that the epistemic status of claims that fall within a domain of inquiry can only be a posteriori, this view has also moved towards a less foundational conception of the structure of warrant (or knowledge) (cf. Haack, 1993; Maddy, 2007; Sher, 2016). On the contrary, while maintaining a (moderate) foundational spirit, apriorism has opened to a conception of the nature of warrant (or knowledge) in which the ‘boundary stones set in an eternal foundation' (Frege, 1983/2013, XVI) can be supported by various kinds of evidence (cf. Casullo, 2003, 2005).

Interestingly, even some proponents of logical empiricism seem to embrace this new hallmark of apriorism (cf. fn. 9). For example, Williamson (2013) has argued against the idea that there is a sharp distinction in the type of knowledge of necessary and contingent truths.

One may know p both a priori and a posteriori, if one knows it in several ways, some a priori, some a posteriori. Tradition excluded that case on the grounds that only necessities [...] are known a priori whereas only contingencies [...] are known a posteriori. But that was a mistake. (Williamson, 2013, 292)

Similarly, Martin & Hjortland (2022) have argued that moderate empiricist stances can embrace the idea that logic can be warranted a priori.

Admittedly, then, one can coherently maintain that logic's method of theory choice is not extraordinary (thereby calling into question its foundationalism), while maintaining that logic has its own peculiar sources of a priori evidence (thereby continuing to maintain logic's apriority). (Martin & Hjortland, 2022, 148)

Martin & Hjortland and Williamson thus seem to admit that logical claims can be warranted both a posteriori and a priori via empirical and non-empirical evidence. Hence, it will be important to consider whether empiricists can account for what Casullo (2005) calls epistemic overdetermination, that is, the phenomenon by which multiple epistemic sources can sufficiently warrant a single claim, including both a priori and a posteriori sources of evidence.¹⁰

The second fundamental revision to the way apriorism and empiricism are conceived nowadays concerns the link between foundational knowledge and certainty. Many authors today reject the idea that foundational knowledge implies certainty. Indeed, this assumption should appear clear to anyone familiar with Quine's teachings about the unlimited scope of rational revisability (cf. Teixeira, 2018). But it should be noted that this assumption is also germane to all those apriorist accounts that allow for a notion of fallible non-experiential warrant (cf. Summerfield, 1991; Bealer, 1998; BonJour, 1998; Casullo, 2003). Therefore, in the following, in discussing the apriorist and empiricist views, I will rely on these updated conceptions of apriorism and empiricism to be consistent with the current debate.

In general, then, one way in which empiricism can be characterized is as follows:

(E) Given a domain of inquiry D and a set S of claims about D, all the evidence e supporting the elements of S is ultimately experiential.

From E, it follows that knowledge of the elements of S can only be a posteriori. Since only experiential evidence is allowed for E, and since this evidence only ensures a posteriori knowledge, knowledge will necessarily be a posteriori for empiricism.

However, earlier, we saw that moderate empiricist accounts – e.g., AEL – have begun to include a priori evidence in their accounts of logical epistemology (cf. fn. 9) (cf. Russell, 2015; Martin, 2020; Martin & Hjortland, 2022). Despite this change of heart, even these moderate views allow the general spirit of E to be preserved. Indeed, as much as moderate empiricists have been open to more liberal epistemological views, they can (and must) concede that when a priori evidence is holistically mixed with empirical evidence, the evidential basis by which logic is warranted turns out to be empirical. In other words, despite the admittance of a priori evidence, knowledge of the elements of S would still be considered empirical since it is held that what is known based on mixed evidence is ultimately a posteriori (cf. Kripke, 1971, 153).

To better appreciate this last point, consider the following. Two points characterize the epistemology of logic of AEL. First, repurposing a famous motto of Williamson (2000), (i) many advocates of AEL would accept a theory-first epistemology of logic. That is, logical evidence does not support single claims taken in isolation but entire logical theories (cf. Russell, 2015, §2; Priest, 2016, §2; Williamson, 2017, §2; Martin & Hjortland, 2022, §4.1; Ferrari, Martin & Sforza Fogliani, 2023, §2.3). Second, for AEL advocates, the warrant (or knowledge) of logical theories is indirect (inferential), that is, (ii) the epistemic support of a theory depends on non-deductive inferences (cf. fn. 11).

Now, suppose that S in E above represents a logical theory. In this case, from (i), it follows that S is warranted before its elements. Therefore, the elements of S receive the same kind of epistemic support that the whole theory receives. Furthermore, suppose the warranting of S is indispensable to warrant the elements of S. In this case, the elements of S cannot be warranted individually in a direct way. Therefore, the elements of S receive only one kind of warrant, namely, the one provided to S.

On the other hand, we saw that AEL admits that the evidence supporting S can be both experiential and non-experiential. However, since the warrant for S is explained using non-deductive inferences whose evidential basis consists of empirical evidence, S's overall warrant (or knowledge) will be a posteriori via (ii). Therefore, according to AEL, there is only one way to warrant (or know) S: an inferential method that is ultimately based on empirical evidence.¹¹

By contrast, apriorism for some domain of inquiry D is usually conceived as negating E.¹²

(A) Given a domain of inquiry D and a set S of claims about D, not all the evidence e supporting the elements of S is ultimately experiential.¹³

According to this view, evidence has a twofold nature. It follows from A that the evidence supporting the elements of S can be either experiential or non-experiential. Therefore, from A, it follows that apriorists can naturally adopt a sort of evidential pluralism. Moreover, since this pluralism can be tolerated, different types of evidence can warrant S's elements. Thus, contrary to E, the elements of S could be warranted heterogeneously.¹⁴ More importantly, it also follows from A that apriorism can account for epistemic overdetermination. Indeed, if A tolerates evidential pluralism, then, in principle, the single elements of S can be warranted in both an a priori and an a posteriori way; if something can be supported by experiential evidence, then it can also be supported by non-experiential evidence.

These considerations are essential for two reasons. First, as it has been conceived above, E cannot account for the phenomenon of epistemic overdetermination. We have seen that it follows from E that no non-experiential evidence can warrant the elements of S. Yet, again above, we saw that (moderate) empiricists seem to embrace the idea that there can be overdetermination for logic. Consequently, for the moderate view to have teeth, the advocate of E must either deny the relevance of overdetermination or argue against the a priori-a posteriori distinction. Attempts have been undertaken to pursue both strategies. Still, the positive outcomes of these efforts remain uncertain. On the one hand, overdetermination seems to be a well-established phenomenon among epistemic agents. On the other hand, attempts to argue against the a priori-a posteriori distinction have been forcefully challenged (cf. fn. 10) (see also Casullo, forthcoming; Melis & Wright, forthcoming; Schechter, forthcoming).¹⁵

Second, the quotations from Martin (2020) we saw earlier give an inaccurate account of apriorism. His conception of apriorism suggests that A can only tolerate non-experimental evidence to warrant the elements of S. However, as we have seen above, this is exactly the opposite of what proponents of apriorism would claim. Since empirical evidence is tolerated in the apriorist framework, apriorists would argue that logical claims can be a posteriori warranted. However, by virtue of the epistemic overdetermination, they would add that a priori evidence can also warrant logical claims in a non-experiential way. In the ensuing picture, logic receives two kinds of epistemic support: experiential support from a posteriori evidence and non-experiential support from a priori evidence. Therefore, logical apriorist are actually more tolerant than moderate empiricists. In E, overdetermination is precluded; thus, logic cannot be a priori warranted.

The first moral that can be drawn from this discussion comes with a new methodological principle that I will call the maxim of evidence maximization: an inquirer should retain as much evidence as possible in every field of inquiry.¹⁶ Let us then take this maxim to evaluate empiricism and logical apriorism. As we have seen earlier, both views accept that a priori evidence is indispensable for warranting logic. However, we also noted that only apriorism can coherently fit non-experimental evidence into its framework. Conversely, empiricism violates the above maxim in that it cannot allow logic to be warranted a priori via non-experimental evidence; thus, in light of this maxim, we should reject E or consider it incomplete.

Besides, considering the maxim proposed above and the considerations made so far, if (moderate) empiricists were to stick to their guns and allow some non-experiential evidence to support the warrant (or knowledge) of logical claims, they would be proposing an epistemology of logic consistent with the apriorist views. Therefore, contrary to their Quinean heritage, their view would be considered a special case of logical apriorism.

A related – though different – question might be: “Can there be fundamental evidence for a certain domain D? Specifically, should we discard evidence when we possess fundamental evidence for D?” I am sure that some members of both stances would be tempted to say that there is fundamental evidence for some domain D, i.e., only a priori evidence or only a posteriori evidence supports the elements of S. To this end, they would argue that the opponent's evidence should be considered superfluous. Therefore, S would be warranted only by one type of evidence.

I think the above evidential stance is not free of problems. One consequence of this view is that the resulting conception of D's epistemology would be excessively narrow. Indeed, the role of epistemology is not to isolate the most limited set of evidential sources that can account for all the epistemic support of the elements of S. In keeping with the updated conceptions of apriorism and empiricism proposed above, the epistemological enterprise is much more of an ecumenical project that addresses the problem of explaining how old and new evidential sources can ensure the epistemic support needed to explain how beliefs are warranted and known. Reason: let us consider the phenomenon of epistemic overdetermination. Suppose we were to admit only one type of (fundamental) evidence to warrant the elements of S. In this case, we would have to admit that there is a normal source that allows us to warrant our beliefs (cf. fn. 10). However, it would follow from this that an epistemic agent would necessarily have to resort to this source to warrant the elements of S since this is now the only source that can ensure warranted beliefs by providing fundamental evidence. Yet this conclusion undermines the feasibility of having overdetermination (cf. Casullo, forthcoming). Since many epistemologists would be reluctant to argue that an agent cannot rely on a good-standing epistemic source to warrant a belief that p only because she must lean on normal sources, giving up on epistemic overdetermination would be too demanding a move. Therefore, if considering the above argument the fundamentalist conception of evidence appears too hasty, we should reject any view that clings to these evidential standards.

Before concluding, there is one final point I would like to address. One might object that the point I have tried to argue in this section is purely verbal since it stems from my understanding of apriorism. Therefore, if one were to reject A, my point would be blocked. However, I believe there are good reasons to stick to A.

First, we can distinguish at least two conceptions of apriorism in the literature: strong and weak A (since weak A coincides with my conception of A, I will refer to weak A as A to avoid misunderstandings). Strong A holds that the warrant (or knowledge) of the elements of S is a priori and cannot be a posteriori (one can think of Wittgenstein's Tractarian conceptions of logical knowledge). In contrast, A holds that the warrant (or knowledge) of the elements of S can be a priori, e.g., ‘Something may belong in the realm of such statements that can be known a priori but still may be known by particular people on the basis of experience' (Kripke, 1980, 5). Both conceptions face substantial problems. While strong A is a robust – yet possibly false – epistemological view (if we allow warrant by testimony, every element of S can be a posteriori warranted), A is, as might be expected, a weak stance. Yet suppose the debate between empiricists and apriorists is to be taken seriously. In that case, we cannot but admit A since a strong A would be a non-starter.

Second, A is an epistemological stance that a naturalist could easily accept. Take, for example, the case of Kitcher (1983). In his account of mathematical knowledge, he writes: ‘A clearheaded apriorist should admit that people can have empirical knowledge of propositions which can be known a priori' (Kitcher, 1983, 22). Now, note how well this naturalistic conception accords with A. Hence, if A is naturalistically adequate, A should be accepted by an advocate of empiricism. Since A is compatible with Kitcher's naturalistic proposal, the empiricist should easily accept A (for a case concerning logic, see also Maddy, 2018, 20).

Finally, since we made the case that A is naturalistically adequate for any D, a fortiori A should be naturalistically adequate for logic. Therefore, I believe the debate between apriorists and empiricists should be framed in these terms. Nevertheless, I am aware that some traditional apriorists would disagree with this conclusion (cf. fn. 13). Still, the arguments I have offered here provide good reasons to support this revised account.

3 Logic and Language, Language and Logic. On the Linguistic Conception of Logical Evidence

The second misconception regarding logical evidence pertains to the relationship between natural language and logic. Numerous distinguished authors in the field of the philosophy of logic contend that logical evidence originates from natural language (cf. Ayer, 1936; Montague, 1970; Beall & van Fraassen, 2003; Priest, 2006). In what follows, we will assess the cogency of this evidential proposal.

However, before proceeding, I would like to dismiss one possible doubt. The following arguments are not intended to establish that natural language can never be used as evidence for logic. At most, in the wake of the arguments I propose in this section, I will argue that natural language must be honed to be considered logical evidence. In other words, I will argue that language per se cannot be regarded as evidence for logic. Furthermore, from the discussion in this section, the need for theoretical resources that can circumscribe natural language will emerge. Although, for reasons of space, I will not be able to dwell on this latter issue, I will suggest what I think is a suitable asset to fulfill this task in a few passages in this section.

I think it is reasonable to say that the linguistic conception of logical evidence is based on the idea held by some philosophers – usually logical monists – regarding the ultimate goal of logical inquiry. As Commandeur (2022, 46) argues, a form of telic monism is often assumed to apply to logical theorizing. That is, ‘there is only one philosophically primary goal or canonical application of logic'. Let us consider some examples.

[I]f logical pluralism is to be a substantial and controversial thesis, something more must be intended. That something more is the notion of logical consequence – that is, a logic is "correct", or "acceptable", etc., if and only if it is a correct [...] codification of logical consequence. The idea that the philosophically primary (but not only) goal of logical theorizing is to provide a formal codification of logical consequence in natural language traces back (at least) to the work of Alfred Tarski, who also provides us with a useful first approximation of what logical consequence amounts to. (Cook, 2010, 495)

And just as with geometries, pure logics have a canonical application: (deductive) reasoning. A logic with its canonical application delivers an account of ordinary reasoning. One should note that ordinary reasoning, even in science and mathematics, is not carried out in a formal language, but in the vernacular; no doubt the vernacular augmented by many technical terms, but the vernacular none the less. […] In other words, a pure logic with its canonical application is a theory of the validity of ordinary arguments: what follows (deductively) from what. (Priest, 2014, 215- 16)

What do logics represent? It is clear from the various uses of applied logic, they can represent many different sorts of phenomena. But for the purpose of traditional logic, though, theories of consequence are frequently taken to represent natural language inference. (Cotnoir, 2018, 302)

The point made in these quotations should not come as a surprise to us. Indeed, we often read in logic textbooks that logic is the discipline that studies correct reasoning or good arguments. Now, since it could be argued that an argument is always presented in some language, it would be natural to assume that the focus of logic concerns the intrinsic features of the (vernacular) language in which these arguments are presented (cf. Iacona, 2018, §4.3).

Despite its intuitive pull, this conception of logical evidence has come under attack in recent literature. In the rest of this section, we will see why we should resist this evidential proposal.

The first reason to resist the linguistic conception of logical evidence comes from Glanzberg (2015, 2021). In his work, Glanzberg challenges the view of the superiority of natural language for theorizing logic. In other words, he challenges the idea that what we find in natural language is somehow akin to a logical consequence relation. Indeed, according to proponents of the linguistic conception, a correct account of logical consequence would be nothing more than the formal regimentation (or a model) of what we find informally present in natural language – ‘A natural language, as a structure with a syntax and semantics, thus determines a relation of logical consequentiality' (Glanzberg, 2015, 75). Against this conception, Glanzberg (2015, 83) argues that we cannot hope to use linguistic evidence for logical theorizing unless we abstract or strongly idealize this evidence.

It is important to stress that Glanzberg (2015, 115) admits that we can talk about a logic in natural language. Nevertheless, he also argues that this is not the same as saying that language contains a logic (Glanzberg, 2015, fn. 2). For these reasons, more recently, he argued that:

I have urged caution in these matters. I have argued for a real but limited role for model theory in semantics of natural language, and I have likewise argued for limited connections between natural language and logic proper. (Glanzberg, 2021, 209)

Despite these clarifications on the necessary precautions, his basic point remains that ‘the two sets of facts are fundamentally autonomous' (Glanzberg, 2015, 72).

For example, he argues that semantic analysis of natural language and logic differ (cf. Glanzberg, 2015, §3.3.1-2). While the former analysis explains how words obtain their meaning, the latter focuses on formal inferential links between logical consequence bearers. Therefore, the second type of analysis is concerned with evaluating inferential connections independently of the meaning of non-logical terms. In this sense, if logical evidence were linguistic, this evidence would provide qualitatively different epistemic support than that required for a formal analysis of logical terms. In light of this, linguistic evidence seems to be a misleading source if our goal is to understand the notion of logical consequence through a semantic analysis of logical terms.¹⁷

Similarly, Dicher (2021, § 6) argues that the threat of logical nihilism (cf. Cotnoir, 2018; Russell, 2018) seems invited by the continuity relation that seems to hold between the generality thesis and the privileged application of logic to natural language.¹⁸ The generality thesis states that a valid logical principle applies without restriction in all circumstances (cf. Williamson, 2017, ms.). Thus, depending on which concept of validity one adopts, an argument is deemed valid if and only if, in every case in which the premises have a designated value, so does the conclusion. Therefore, it does not matter whether one is working in truth theory, differential topology, or ethnography, a valid rule of inference will preserve truth from premises to the conclusion in all circumstances.

Dicher (2021, 7088) argues that given the breadth and nuances of language, we could, in principle, always find a way to invalidate logical principles that are deemed valid. Below are a couple of examples that I think will help to understand.

The first example concerns the counterexamples to MP proposed by McGee (1985) (fn. 14). His argument proceeds by showing that by implementing the meta-variables of MP with some peculiar claims, one would be warranted to hold the premises without thereby holding the conclusion.¹⁹ According to McGee, this (linguistic) evidence is sufficient to prove that MP is not valid.

As mentioned in the penultimate sentence above, one interpretation of this logical failure can be attributed to the peculiar features of the sentences McGee uses to instantiate MP. Indeed, if we consider the conditional premise of McGee's arguments, we can notice that the consequent of this premise is also a conditional claim. Still, there is reason to doubt another hidden assumption of this argument: the "if… then…" expression of natural language always behaves as a material conditional. In other words, if we assume that validity constitutively depends on the intrinsic features of natural language, it follows that any instantiation of the meta-variables of MP should be evidence of the logicality of this principle.

But as a matter of fact, there seem to be good reasons to question both these assumptions. First, studies on indicative conditionals have shown that these expressions do not always behave in a true functional way (cf. Priest, 2008, §1.6; Florio, Shapiro, & Snyder ms.). Second, the idea that every substitution of the meta-variable can be used as (counter)evidence for the validity of a logical principle is also suspicious. As Dicher argues, this assumption depends on the idea that one cannot have a preliminary "tidying up" of relevant cases that count as (counter)evidence for the validity of a logical principle (cf. Dicher, 2021, 7088). Still, this assumption clashes with logical practice. Citing Frege's metaphor of the eye and the microscope, Dicher contends that:

Details aside, the overarching idea is precisely that of containing the multitude of phenomena that occur in natural language to a sufficient degree to make possible the study of the general laws of thought. It is relative to this purged medium that the generality of logic is predicated. (Dicher, 2021, 7089)

Following Frege, then, Dicher invites us to reconsider the sense of generality with which logical principles apply to language and better consider the evidential relationship between language and logic.²⁰²¹

The second example concerns the relation between natural language and substructural logic. If natural language were the only source of logical evidence, then we would have to conclude that the correct logic is one of the many substructural logics, i.e., a family of logics that denies the validity of structural principles (principles concerning reflexivity, transitivity, idempotence, commutativity, monotonicity and associativity of logical consequences). To see why this conclusion holds, just think of monotonicity and idempotency. From linguistic evidence, we should be able to see that logical consequence is non-monotonic since adding a new premise to an argument does not always assure us that the previously inferred conclusion remains true (cf. Priest, 2016 §2.5). Likewise, again considering linguistic evidence, we could not save the contraction principle either, since in some cases, we would deny that the conjunction of two instances of the same premise is weaker than a single instance (which, according to widespread opinion, is equivalent to the failure of contraction) (cf. Zardini, 2019, fn. 2).

Now, there is no denying that these logics have assumed a very important role in the logical endeavor. Still, the above conclusions would clash with the idea that logical inquiry models general language phenomena. Indeed, rather than modeling general features of the language, substructural logics generally lend themselves to regiment niche phenomena of language, such as paradoxes. Of course, paradoxes are also a very important phenomenon in our language. However, I do not believe that everyone would be inclined to base the choice of the correct logical theory only on the glitches present in the vernacular language (especially if they do not allow us to save the generality of logic) (cf. Williamson, 2017).

Given this, if natural language is to be used as evidence for logic, we must remember two things. On the one hand, as Glanzberg argues, language must be idealized and abstracted in order not to compromise our investigation into logical consequence. On the other hand, as Dicher maintains, linguistic evidence must be tidied up to reduce cases of erroneous (counter)evidence. Ultimately, I believe that the work of these authors supports the thesis I proposed at the beginning of this section: language per se is not sufficient, from an evidential point of view, to determine which is the correct notion of logical consequence.

Finally, the idea that logic has a privileged (canonical) application in natural language appears to be faltering.²² Several authors have argued that we may have a genuine plurality of purposes in mind when it comes to logical theorizing (cf. Eklund, 2020; Hlobil, 2020; Commandeur, 2022). For instance, Eklund argues that at least three independent linguistic projects could motivate logical theorizing in a pluralistic setting. The first project is the mapping project, which involves identifying the expressive limitations of various possible languages. Hence, in the case of logic, this project would determine how the notion of validity would behave in different languages. The second project is the actual language project, which involves figuring out how our natural language behaves with respect to what we are investigating. In our case, proponents of different logics would embark on the challenge of proposing a logical theory that correctly captures the notion of logical consequence in natural language. Finally, the last project is the normative project, which aims to determine the best language. Hence, applying this latter project again to our case would mean we should figure out what the best notion of logical consequence is.

Eklund argues that all three of these projects are relevant for logical theorizing. Therefore, if we focused only on linguistic evidence and argued that logic somehow flows from it, we would have to agree that all three of these projects are equally interesting from a theoretical point of view. Yet, we should also recognize that the projects are also independent of each other. Therefore, if we cannot explain why one of these projects is more relevant, then appealing to linguistic evidence would not be enough to explain knowledge of logical facts.²³

In this section, I have argued that depending solely on one of the traditional sources of logical evidence is insufficient to elucidate our understanding of logical consequences. This conclusion holds significant importance, as it encourages us to scrutinize the means upon which we rely for refining natural language (cf. fn. 20). In the forthcoming sections, I will evaluate other conventional sources of logical evidence closely associated with natural language.

4 Intuiting (Propositional) Evidence

Another type of misconception regarding logical evidence concerns proposals such as Chudnoff's (2011).²⁴ Therefore, the focus of our analysis in this section will be the role of intuitions in logic.

In presenting his account of intuitions, Chudnoff argues that:

Here are some propositions one might unreflectively intuit: if 2>1, then 2>1; 2>1 Here, now, is a worry. When we have an intuition experience whose content is one of these propositions it gives us prima facie justification for believing it. But our intuition experience seems too minimal to have presentational phenomenology. If it does not have a presentational phenomenology, then something else must account for its epistemic properties [...] All you have to do is entertain the proposition and hold it in mind – whereupon it will intuitively strike you as being true. Still, I claim, your intuition experience has presentational phenomenology. What item do you seem to be intellectually aware of? Plausibly, it is the proposition itself, or one of its intrinsic properties such as its form. (Chudnoff, 2011, §6)

To put this quotation in context, Chudnoff is considering the phenomenon of unreflective intuitions. Unreflective intuitions represent an interesting epistemological phenomenon, as these intuitions seem to be rational experiences that grant us immediate access to facts regarding basic logical principles (cf. Carlson, 2022). For example, if we consider the principle of non-contradiction – $\sim \left(\varphi \&\sim \varphi \right)$ – and its negation – $\left(\varphi \&\sim \varphi \right)$– the former principle, but not the latter, immediately appears true when we consider it (considering this principle, we seem to have an impression of felt veridicality). In this case, Chudnoff would argue that we had a non-reflective intuition, which warrants us to believe $\sim \left(\varphi \&\sim \varphi \right)$. However, according to Chudnoff, unreflective intuitions seem to be problematic because they do not instantiate the phenomenological property that explains how we can have warrant for intuitional beliefs, i.e., presentational phenomenology. Chudnoff argues that intuitions are sui generis epistemic sources because they instantiate the same kind of phenomenological properties as perceptual sources to warrant beliefs. That is, ‘an intuition experience possesses presentational phenomenology when in it you both seem to fact-intuit that p and seem to be intellectually item-aware of an item that makes it the case that p' (Chudnoff, 2011, §4). The problematic nature of unreflective intuitions concerns the second conjunct of this analysis. Epistemic agents seem unaware of anything when subjected to unreflective intuitions, as the transition from these unreflective states to beliefs is very rapid. But then, if unreflective intuitions lack the factor that warrants intuitional beliefs, how can these sources warrant (logical) beliefs? Chudnoff (2011, §6) argues that to satisfy this desideratum, the agent subjected to these intuitions must be aware either of the proposition that the intuited content expresses or of an intrinsic feature of it, e.g., its form.²⁵

Let us now apply this account to logic. Although I am sympathetic to the idea that logical principles can be warranted by appealing to (logical) form, I believe that propositions cannot count as logical evidence. Here are some reasons in favor of this view.

First, let us assume with Chudnoff (2011, §5) that (unreflective) intuitions are non-inferential epistemic sources. In this case, we would expect intuitions to produce non-inferentially warranted (logical) beliefs. Yet, Chudnoff's account fails to account for this desideratum. If, in the case of unreflective intuitions, what warrants us to believe that p is propositional in nature, we should consider this warranting evidence a reason to believe that p. Traditionally, in epistemology, reasons are considered propositions used to ground reasoning (cf. Alston, 1989, 176). For example, Burge (2003a) has examined this point at length in connection with the discussion about perceptual content and its epistemic role in belief formation.

I certainly agree with Sellars that reasons must be propositional. Perceptual beliefs are not normally reason based. The normative transition from perception to belief is not a piece of reasoning. If perceptual representations were reasons for perceptual beliefs, such transitions should count as reasoning. But they do not. (Burge, 2003a, 528)

It follows from this quotation that if we were to endure with Chudnoff's firm analogy with perception, then the intuitive content that warrants the (unreflective) belief that p should be considered a reason. Hence, the transition from (unreflective) intuitions to beliefs should be viewed as a piece of reasoning. Yet this conclusion would clash with our initial assumption: intuitions are non-inferential sources.²⁶

Besides, this conclusion also clashes with the phenomenological features that, in Chudnoff's account, motivate the distinction between reflective and unreflective intuitions. If unreflective intuitions allow us to grasp the truth of a proposition immediately, then we would expect these sources to have non-inferential phenomenology. But since these sources require propositional awareness, we should classify the phenomenology of unreflective intuitions as inferential in light of the above argument.

Second, if unreflective intuitions require that there be awareness of a proposition, one may wonder how this awareness can alone warrant a belief. A natural reaction to this proposal would require that intuitional warrant stems from understanding the proposition in question (cf. Goldman, 2007; Sosa, 2007). Yet among the proponents of the sui generis conception of intuitions, few would admit that intuitive warrant depends on understanding, including Chudnoff himself (see especially Chudnoff, 2012).²⁷ Thus, if understanding is not required for intuitional warrant in Chudnoff's account, a fortiori it should not be required for unreflective intuitional warrant either (cf. fn. 26).

But if understanding is not suitable to have intuitional warrant, what warrants the belief that p via unreflective intuitions? Indeed, the idea that one can rationally believe that p by considering only its (propositional) content is controversial. By assumption, if what we rationally believe depends (partly) on the type of source we use to produce warranted beliefs, then Chudnoff's account cannot distinguish between beliefs produced by an agent with an impaired cognitive system and one with a normal cognitive system. The reason is that, on this account, the evidence produced by unreflective intuitions would not be sensitive to the types of sources of evidence. To wit, suppose an agent's impaired cognitive system produces true (unreflective) intuitions about complex logical-mathematical propositions by sheer luck. Furthermore, suppose the agent considers Fermat's last theorem before discovering Wiles' proof. According to Chudnoff, if the impaired agent were aware of the proposition and intuited its truth by sheer luck, this would warrant her belief. However, this would be ‘a great leap in thought' (cf. Schechter, 2019) since we would normally consider (unreflective) intuitions unable to warrant these kinds of beliefs (cf. Boghossian, 2003). More importantly, in this scenario, the phenomenological properties instantiated by the impaired agent would be qualitatively identical to those of a normal epistemic agent. Therefore, contrary to what one might hope, there would be no qualitative difference in terms of competence and reliability regarding the evidence warranting the intuitive belief that p in this case.

Likewise, in principle, using the same (unreflective) sources, our impaired agent could have non-experiential warrant for beliefs we would normally deem empirical. After all, if it is enough to have propositional awareness of p to be warranted, then by slightly extending the effects of her cognitive condition, any proposition could be warranted non-experientially. Yet, this seems highly unsatisfactory. Therefore, Chudnoff's account must be considered inadequate because it yields results about intuitional warrant that contrast with findings concerning both the normal functioning of epistemic sources and the kind of warrant produced by intuitions (cf. Plantinga, 1993; Graham, 2012).

Finally, consider a more technical point advanced by Zardini (2014). The problem that Zardini highlights concerns identifying the bearers of the logical consequence relation.

First, in certain phenomena of context-dependence like reference failure of demonstratives, it is doubtful that any proposition is expressed, but it is not doubtful that there could still be valid or not valid arguments to be made, like, taking for example a case in which one is hallucinating a dagger, the argument from ‘That dagger is on the table' to ‘That dagger is on something'. Second, even in those cases in which propositions are expressed, they seem anyways ill-suited to tracking entailments generated by context-dependence like the one from ‘Actually, snow is white' to ‘Snow is white': it is implausible to think that the proposition that, in @, snow is white entails the proposition that snow is white, since things could have been such that the former proposition is true while the latter false. Third, on many views of propositions [...], the proposition expressed in c^FE by ‘Tomorrow, it'll be sunny' is the same as the proposition expressed in c^FE by ‘Today, it's sunny', and so, on such views, even if the Fregean Entailment were somehow tracked at the level of propositions taking propositions as logical-consequence bearers would implausibly turn that entailment into the trivial entailment from a proposition to itself. (Zardini, 2014, 3478-9)

The purpose of this extended quotation concerns a point already mentioned at the beginning (cf. fn. 6). Although evidence is normally considered propositional, logical evidence might be non-propositional. As we saw in the quotation above, if someone intuiting that p had awareness of p, her evidence for believing that p would not be fine-grained enough to capture basic features about logical consequence like reflexiveness. To wit, if intuiting propositions were enough to be warranted in believing facts about validity, we would not be able to grasp that "that dagger is on the table" logically follows from "that dagger is on the table" in that reference failure would not allow the agent to have awareness of any proposition (mutatis mutandis, similar considerations could also be made for other structural features of the logical consequence). Yet, as we saw above, we would consider this argument valid even though we have no awareness of any (complete) proposition.

Given all of this, one of these two following considerations should hold. Either intuitions do not count as sources of logical evidence because of the propositional nature of their content, or they count as evidence because they present something other than propositions to epistemic agents.

Although propositions might not count as evidence for accounting for the truth of logical claims, there is a suggestion in Chudnoff's initial quote that is worth highlighting: ‘Plausibly, it is the proposition itself, or one of its intrinsic properties such as its form'. I think it is worth making two remarks about this passage. First, suppose that intuiting forms or structures does not count as having propositional evidence (cf. fn. 25). In this case, there seems to be a good reason to regard the second disjunct as true. Second, if intuiting forms is a sound option, then intuitions can satisfy the evidential profile we have outlined so far. Indeed, we should now be able to see that intuitional evidence would align well with the idea that the sources of logical evidence are neither dependent on linguistic (conceptual) competence (cf. fn. 27) nor have propositional outputs.

Furthermore, since it has not yet been ruled out that intuiting forms counts as either experiential or non-experiential evidence (or both), the fact that there are reasons to consider intuitions sources of evidence seems consistent with intuitions being a priori and/or a posteriori sources of evidence. Therefore, this conclusion also aligns well with the apriorist proposal we advanced in §2.

5 Judgments as Evidence

The profile of logical evidence I have outlined so far yields an unusual conception of the epistemological grounds of logic. To recap, let me review some crucial passages.

First, logical apriorism is more tolerant than moderate empiricist views – i.e., AEL. Since epistemic overdetermination is precluded for AEL, we cannot employ non-experiential evidence to warrant logical claims. Secondly, many authors have argued that a logical theory is correct only if it captures the notion of logical consequence underlying natural language. However, we have seen that it would be strange, to say the least, to claim that linguistic evidence can, per se, provide an account of logical knowledge. Third, logical evidence can be non-propositional. There are two reasons for this latter point. On the one hand, propositional evidence is too coarse to capture facts about logical consequence in cases lacking complete propositions. On the other hand, for a piece of evidence to be considered epistemically adequate to support a belief, the source producing this evidence must be truth conducive (the agent must use the source competently) (cf. Sosa, 2007). We have seen that accounts such as Chudnoff's neglect this epistemological feature. Yet by leaving out this detail, we risk admitting evidence that we would otherwise consider unacceptable (cf. fn. 16). Let me pause for a moment on this last remark.

Given the importance of truth conducive evidence, some might argue that we should appeal to experts to ensure this kind of evidence. For example, some proponents of AEL argue that the evidential basis by which we construe and test the correctness of a logical theory consists of experts' judgments regarding matters of logical (in)validity (cf. Martin & Hjortland, 2021, 2022). According to these authors, using only this kind of evidence, we would have a way of warranting our logical beliefs.²⁸²⁹

However, I have some reasons to doubt this latter point. To put it in Frege's terms, the act of judging is the consequence of grasping a thought (cf. May, 2018, §3). Slightly readapting this idea, we could say that judging something as true presupposes that the content of what we are judging has a conceptual nature; judgment presupposes predication. However, to my knowledge, the only way to judge something as true without employing epistemic sources is to assume the controversial notion of self-evidence (cf. fn. 28) (for a discussion about self-evidence, see Jeshion, 2001). For example, we could judge a claim as true if its content had elements in some obvious relation.³⁰

However, two details must be carefully considered when buying into this account. First, although it is reasonable to assume that we can judge something that has obvious content to be self-evidently true, we must still allow sources of evidence to come into play. Indeed, to grasp the obvious relations between the elements that constitute the content of what we are judging, we must first understand these relations. Now, a problem with the epistemic concept of understanding is that the literature on this topic is unclear on whether understanding should be considered an epistemic source (cf. Horvath, 2020) or an epistemic good (cf. Burge, 2020). Be that as it may, suppose we consider understanding in the first sense. In this case, epistemic sources would blatantly come into play when we judge an obvious claim as true. Conversely, suppose we consider understanding as an epistemic good. In this case, sources would be needed to produce this epistemic good, as has been argued by several authors (cf. Bengson, 2015b; Sliwa, 2015). Therefore, it should already seem unclear how it is possible to separate epistemic sources from judgments coherently.³¹³²

Second, for the sake of argument, consider the following borderline case. Even if someone were to argue that judgments concerning self-evident truths can be made independently of epistemic sources (perhaps by recognizing that understanding is necessary but not sufficient for making them), this would still require that something else – over and above understanding – should act as an explanans for making such judgments. However, one might rightly ask: "What could this missing component be?" Now, if something can be self-evident in the way described above, then the only way to make sense of this view would be to hold that the content is somehow special (cf. Teixeira, 2023). In particular, it should be assumed that the semantic features of the content are structured to prevent what is being judged from being false. Yet, the attentive reader should have noticed that this view presupposes that something near metaphysical analyticity should hold in this case. Since we have allowed for the existence of claims that can be judged true regardless of the sources we employ to make such judgments, the truth of such claims must depend on some intrinsic feature of their content. However, this conclusion does not seem to fit well with the empiricist tenets of AEL. Since many AEL advocates would be skeptical towards metaphysical analyticity, we should agree that this strategy is unpalatable. Therefore, it seems that even empiricists who admit that evidence for logic consists of experts' judgments must commit themselves to sources of evidence.

Let me make the same point, but from a different angle. Suppose that logical evidence consists of experts' judgments and that these judgments are independent of sources (cf. fn. 28). In this case, at least three complications would follow from this account. First, even if we restricted logical evidence to experts' judgments, nothing would assure us that such evidence is truth conducive (competently produced). Since, in such cases, what matters for warranting logical claims are only experts' outputs (judgments), nothing can prevent these outputs from being produced deviantly or through epistemically uninteresting methods. Yet, we would not want this kind of evidence to support a theory (logical or whatever).

Second, if we dispensed sources of evidence from the epistemological picture, we could not explain why we rely on experts' judgments in the first place. Since what makes experts such is their competence, and since competence is explained in terms of reliability (competent use of reliable sources), how could we say that an expert is competent (makes reliable judgments) without considering the sources she uses to make such judgments? In the ensuing picture, I think we would have to give away both experts' judgments and their logical competence.

Third, the idea that experts' judgments can count as logical evidence is rather problematic on its own. For instance, wouldn't an expert be inclined to judge as valid only what her preferred logical theory deems to be so (cf. MacFarlane, 2004, 2)? In other words, leaving aside the sources by which she makes such judgments, wouldn't this evidence be cognitively penetrated by her own logical convictions? I strongly believe that an inevitable problem with this empiricist view is that experts' judgments are constitutively biased. Still, we may wonder whether we should rest the warrant for logic on more neutral grounds if such evidence were available.³³ Therefore, I think an account that places so much value on expert judgments should be seriously concerned about this biased effect on evidence if its goal is to provide a neutral method for warranting (or knowing) logical theories (cf. Hlobil, 2020).

I have argued in this section that the problems of apriorists and empiricists are very much alike. Both views provide evidential accounts that do not consider the way evidence is formed. This omission makes these strategies vulnerable to the same kind of criticism. Indeed, both views preclude themselves from explaining why warrant can (partially) ensure, via evidence, that the beliefs we form are reliable and thus can amount to knowledge.

6 Why Nothing Works

I started this paper by assuming that what we allegedly know about logic depends (partly) on facts regarding evidence. Yet, from the discussion presented in the previous sections, nothing seems to work. However, I believe we should not give up. Although we are left with few options, I still believe that an account of logical evidence consistent with the above results can be fully developed. Still, initially, I made it clear that my intention in this article was to offer an account of what logical evidence could not be. Therefore, I have refrained from offering a positive account of what logical evidence is.

Nevertheless, I think it is fair to draw some conclusions that might help offer insights into the debate on the epistemology of logic. In the above discussion, I argued that logical evidence should not be conceived as merely linguistic or propositional. The main reason for this conclusion is that this type of evidence is not subtle enough to capture facts about logical consequence. Furthermore, since I argued that experiential and non-experiential evidence can support logical claims, logical epistemology is based on an evidential duality, i.e., evidential pluralism. In this regard, I argued that we should be apriorists regarding logical evidence since, in most cases, moderate empiricists cannot account for epistemic overdetermination. Although the results presented in this paper introduce some novelties into the notion of logical evidence, I think my account is fit enough for resolving most issues surrounding discussions of logical consequence. Therefore, in order to draw up a logical account that agrees with the available data, I believe we should proceed with our inquiry with this narrow, workable starting point.