The Thesis of Logical IndeterminacyIntroducing logical indeterminacyLogical indeterminacy is the thesis that logic by itself cannot tell us which logic, if any, is the “correct” or One True Logic. Given that there are numerous different candidates for the One True Logic, all of them provably sound and complete, there is no basis within logic itself for determining which of these logics is correct.

As such, this thesis of Logical Indeterminacy differs from that of Logical Pluralism, i.e., it does not imply the denial of the thesis that there is One True Logic. Pluralism claims that thereis more than one acceptable logic. Indeterminacy claims that, whether there is One True Logic or not, the choice of such a logic or logics cannot be made on purely logical grounds, but must be made at least in part on metalogical1 considerations.

This thesis may well seem obvious and trivial. After all, there is a well-established branch of philosophy known as “philosophy of logic.” Not surprisingly, in the philosophy of logic, thinkers attempt to resolve debates over the nature of the One True Logic (or over whether there is One True Logic) by doing

1 Because the term metalogical is used formally to refer to the metalanguage usedto describe a formal object language, I need to clarify immediately that I wish to use “metalogical” in a broader sense to refer to considerations that go beyond a particular logic, but that deal with logics in a more general fashion, particularly in evaluative or comparative contexts. I considered using “extralogical” for this purpose, but this to me suggests considerations that are separate from logic altogether. As will hopefully be obvious, the considerations in play here are “logical” in the broad sense that they relate to questions of the purpose and value of logic in general as well as particular logics. But they are “beyond logic” in the sense that no investigation of any particular logic, by means of, for example, soundness andcompleteness proofs, determination of which theorems are derivable from the axioms or principles of inference of particular logic, proofs of consistency, investigation of models, etc., can settle the kind of questions discussed here. Nor will it help if we begin by defining a formal language with which we can evaluate logics—no doubt one could do this, but it wouldn’t answer the question.

philosophy – as opposed to simply, say, doing soundness and completeness proofs, or constructing an entirely new formal language. If the question of which logic or logics is “correct” could be settled through purely logical methods, then there wouldbe no need for “philosophy of logic.” Logic, pure and simple, would suffice. But unless pretty much our entire literature in philosophy of logic represents a giant intellectual wrong turn, logic by itself does not suffice.

Nonetheless, many writers of papers on logic seem to assume that if they can devise the right formal language or prove the right series of propositions, they can thereby resolve fundamental issues in logic. And there is a line of thought, going back at least to Wittgenstein, that there is no standpoint outside of logic from which we can think about logic, so that these questions have to be resolved within logic. So perhaps there willbe some value in advancing and defending the thesis of logical indeterminism.

There is a weaker, albeit I believe quite correct, version of logical indeterminism which holds that, even if there is one truelogic, there are potentially different and equivalent symbolisms one might use to express it. As Gila Sher puts it:Even if there are "eternal" logical truths, I cannot see why there should be eternal conceptual (or linguistic) carriers of these truths, why the logical structure of human thought (language) should be "fixed once and for aiL" I believe that new logicalstructures can be constructed. Some of the innovations of modern logic appear to me more of the nature of invention than of discovery. Consider, for instance, Frege's construal of number statements. Was this a discovery of the form that, unbeknownst to us, we had always used to express number statements, or was it rather a proposal for anew form that allowed us to express number statements more fruitfully?...Defining the field of our investigation to be language as we currently use it, we can invoke the principle of multiformity of language…Once we accept the multiformity of language, change in the "official" classification of logical terms is in principle licensed.2

This to me seems true so far as it goes, but the thesis of logical indeterminism defended here goes well beyond that. On this view, the indeterminism extends beyond alternate symbolisms of the same principles to the basic principles themselves: given that there are disputes over which “basic” principles of logic, 2 Sher (1991), pp. 6-7.

if any, are in fact “eternal,” by what means do we determine which principles, and ultimately which logic, to accept? According to the thesis of logical indeterminism, if there are any means for determining which logic we should accept, they are metalogical rather than logical in nature.

If there was a logic that possessed all of the qualities that thinkers have historically found desirable in a logic – logic that validated all of the theorems traditionally held to be fundamental to logic, and all of the principles considered essential for valid reasoning, including the naïve theory of truth, then the thesis of logical indeterminism would of course be false. Or perhaps more accurately, it would be beside the point, in the way that a thesis of arithmetical indeterminism would seem beside the point, given that there is no serious rivalto standard (we could call it “classical”) arithmetic. Similarly, if there were no serious debate over which logic we should prefer, there would be nothing motivating a problem of logical indeterminacy.

An ideal logic such as I am imagining would validate the laws of noncontradiction and excluded middle, and rules such as modus ponens, modus tollens, disjunctive syllogism, and contraction. Itwould observe the standard introduction and elimination rules forits connectives. It would validate the T-schema and be perfectlytruth-functional and bivalent. It would have great expressive power, a plausible semantics, and would be able to express all meaningful sentences of the language for which it is a logic. And it would avoid paradoxes such as the Liar and Curry paradoxes, and the paradoxes of the material conditional. If there were such a logic, there would probably not be much seriousdebate that that logic would be the correct logic for reasoning. But as even a cursory review of the literature in symbolic logic makes obvious, no such logic has been devised. After more than acentury of effort, it seems highly unlikely that such a logic willbe devised. This suggests the conclusion that no such logic can be devised. And this, in its most basic form, is the core argument for logical indeterminism.

If by a strictly logical argument we mean a deductively valid argument, then a strictly logical argument must be a deduction from a set of axioms and theorems (themselves derived from axioms), or (in a non-axiomatic system) it must be a derivation from a particular set of premises using introduction and elimination rules. But of course, those deductions or derivations are valid only for the specific logics which employ those axioms or rules. However, when we ask the question “which logic should we prefer?” we are asking the question of precisely which axioms or rules we should prefer. But such axioms or rulesare themselves never deduced or derived, but rather assumed or stipulated for a particular logic. And that in turn assumes a choice of a particular logic, which is precisely at issue. So itwould seem to follow that the question of which logic we should prefer cannot be deduced from a set of axioms or derived using introduction and elimination rules. If such deduction or derivation be taken to constitute “strictly logical” means of resolving a question, then the question of which logic we should use cannot be resolved through strictly logical means. This is the thesis of logical indeterminism.

Even if an ideal logic were indeed to prove impossible, still, iflogicians as a group were to come to general agreement about which logic we should prefer – the One True Logic - then for all practical purposes, logical indeterminism would cease to be a problem. Perhaps we would still worry about the foundations of such a logic, as mathematicians of the 18th century worried aboutthe foundations of mathematics, even though there was no serious question as to “which” mathematics we should use in everyday life, and as theoretical mathematics was continuing to develop inscope and complexity. But this would be a quite different question than that of indeterminism between logics. In any case,that is far from the situation obtaining today3. Instead, rather3 One other difference between the mathematical and the logical case is that, while standard arithmetic and mathematics are employed universally for measurement and calculation, there is no single logic that is employed in a similar manner. Indeed, most ordinary people in their reasoning don’t use anyparticular symbolic logic whatsoever, as opposed to simply applying basic rules of general reasoning and inference. Their reasoning may be consistent with a particular logic, or (more likely in my opinion) any number of logics,

than converging on a single logic universally agreed to be correct—our One True Logic—logicians over the last several decades have devised an ever-increasing proliferation of logics, most of them devised in an attempt to overcome the shortcomings of earlier logics, including the “classical” logic developed by Frege, Russell and Whitehead. One could say that, by devising more and more logics, we increase the chances of finding the One True Logic. But one could also say that, if there was One True Logic, surely we should have stumbled across it by now?

Whether such an ideal logic is possible or not, there’s no question that over a century of research – and the creation of literally hundreds of logics – has failed to find it. Assuming –and it is a far from trivial assumption – that there is One True Logic, the search for it has so far been a difficult and unsuccessful one. Why has this been the case? One can imagine various reasons. The reason might, for instance, turn out to be largely sociological. Certain schools of logical thought, or projects seeking to develop certain types of logic, might attractsupport because of the influence wielded by their founders, or bythe institutions where those schools and projects have developed.4 Thus, the great variety of logics being advocated bydifferent professional logicians might simply be a function of the persuasiveness or influence of certain logicians, rather thanbecause of any substantive dilemmas or disagreements in logic. At the other extreme, it might turn out that, at the deepest levels of logic, there is a fundamental conflict between basic logical values. Consistency and expressive completeness, as we’ll see, might be two such values. Relevance and truth-functionality might be two others. Investigations into the “deepstructure” of logic might clarify such conflicts. It may be thatsuch conflicts in logic are inevitable and irresolvable. Or the problem might lie somewhere in our ordinary language or intuitivereasoning. Perhaps ordinary language is ultimately too logicallyimperfect, at some fundamental level, for any symbolic logic to

but that does not mean that they are using such a logic or logics.4 Think, for instance, of the Brazilian school of paraconsistent logicians ledby Walter Carnielli (and inspired by the work of Newton Da Costa), or the Belgian project in adaptive logic led by Diderik Batens.

adequately model it. Or perhaps our innate “logical faculty,” ifone exists, has some deep flaw that infects our reasoning and hence any attempt to model it in a symbolic system. One explanation that intrigues me involves the conditional—that thereis no adequate account of the conditional that does not involve metaphysical views about, inter alia, causation, counterfactuals, etc. So no logic that employs a conditional can avoid some sort of meta- or extra-logical commitment. This would validate the thesis of logical indeterminism, at least so far as any logic with a conditional is concerned.

Ultimately, finding an explanation for logical indeterminism is beyond the scope of the current essay, however intriguing these speculations may be. Nor do we offer to “prove” (in any formal sense of “proof”) the thesis of logical indeterminism (though we will offer two informal arguments)5. Rather, we seek here to explain the thesis, and develop considerations in its support. In part, this will involve looking at several current debates in the philosophy of logic, and arguing that “purely logical” methods are insufficient for settling them.

The differences among the various leading candidates for the One True Logic are not trivial. Classical logic (CL) rejects all contradictions, accepts excluded middle and disjunctive syllogism, and validates all conditionals with false antecedent or true consequent. The dialetheic logic MiLP – Minimally Inconsistent Logic of Paradox – accepts some (but not most) contradictions, and restricts disjunctive syllogism. Relevant logics do not validate all of the conditionals validated by CL, including some p q for true p and true q. Intuitionist logic rejects excluded middle. These logics cannot all be true, at least if they are meant to range over the same cases. Yet each logic is provably sound and complete. In that sense, there is nothing “wrong” with any of these logics, at least from a 5 A formal proof that an “ideal” logic (thinking of “P” as “perfect,” we couldcall it a “P-logic”) is impossible would certainly constitute such a proof. Iam not sure what form such a proof might take, but I cannot rule out its possibility. (We would then be in the ironic position of having a strictly logical proof of the thesis of logical indeterminism.)

strictly logical point of view. At the same time, all of them, asnoted above, have their shortcomings, at least as compared to an “ideal” logic as described above. It follows that if we are to choose one of these logics over the others, we will have to do soon metalogical grounds. There is no “purely logical” method for making such a choice.

First, however, let us develop further the thesis of logical indeterminism.

Indeterminism, pluralism, and monismThe thesis of logical indeterminism is distinct from both pluralism and monism in logic. Pluralism – at least the sort of pluralism defended by J.C. Beall and Greg Restall6 - holds that different logics, even if they don’t validate the same sentences,can still coexist, since they are used for different kinds of cases. Indeterminism holds that these logics are indeed incompatible – they can’t all be true – and that there is no strictly logical criterion for choosing among them. Beall and Restall hold that the disagreement between, say, classicists and relevantists is merely apparent: the classicist is treating of “worlds”, while the relevantist is treating of “situations.” Each is correct, given their choice of indices by which to evaluate the truth-values of sentences. Another way of expressing this might be to say that the monologist7 thinks that only one kind of index is correct for assessing truth-value – whereas the pluralist thinks we need not restrict ourselves to one choice of index.

Beall and Restall carefully outline a schema in which “worlds,” “situations,” and “constructions” all co-exist in the same logical space – worlds are maximal constructions and situations, with consistent situations also being constructions (there are alsoinconsistent situations, which are not constructions). But not 6 Beall, J.C. and Restall, Greg (2000). 7 The term is introduced by Ben Burgis in his (2011).

everyone will accept this schema: actualists, for example, will reject worlds altogether in favor of situations (all of which areunderstood to occur within the one, actual world). For their part, intuitionists and constuctivists will prefer constructions to worlds. For my part, I am an indeterminist about this matter as well: there seems to me to be no clear method for settling thedebate over whether worlds, situations or constructions – or somecombination thereof – should serve as our indices for evaluating the truth of sentences. Here again, the answer cannot be found within logic. If anything, one’s position on worlds, etc., will be a function of one’s beliefs about ontology – and will thus have to be settled in whatever manner ontological questions are settled (assuming, of course, that there is such a method). Or perhaps there are considerations of “conceptual scheme” that might prove decisive (though it is perhaps just as likely that one conceptual scheme will prove as good as another). But I can think of no reason drawn from logic itself to prefer worlds, situations, or constructions.

Ultimately, in the debate over whether or not we can accept only one, or more than one logic (or index), it’s not clear to me how we might determine an answer. There are strong lines of thought8

on both sides. On the one hand, there is the line of thought that there must be one unitary, fundamental logic which underliesall of the others, in the sense that it’s the logic we employ when we’re deciding which logic to employ at what times. (In other words, the logic we employ in doing metalogic.) And on the other hand is the line of thought that reasoning is too diverse in its purposes and methods to admit of only one correct formalization of reasoning. (Not to mention fear of an infinite regress – if there are more than one candidate for the “logic of metalogic,” what logic do we use to decide among those? And so on.) With two such strong and opposing lines of thought in play, what sort of argument can prove that one and not the other is correct? Thus,

8 I originally wrote “intuitions” here, but then I read some of Herman Cappelen’s work on that subject.

I am also an indeterminist on the question of whether there is One True Logic in the first place.

Nonetheless, the question of whether there is One True Logic, andthe question of whether we can determine which logic that is, aretwo different questions. The thesis of logical indeterminism as I am developing it does not imply the denial of the thesis that there is One True Logic. It is conceivable that, though there isin fact One True Logic, we cannot discover which logic it is. The thesis of logical indeterminism merely asserts that we have no method – within logic at least – for determining which logic or logics are correct. Now to me, a situation in which there is One True Logic, but we are unable to determine which logic it is,would be deeply strange. (Indeed one of the reasons I am skeptical about the existence of One True Logic is the suspicion that, if there were such a logic, it would be fairly obvious – inthe sense that Peano Arithmetic is obviously the One True Arithmetic.) Nonetheless, I know of no way to rule such a possibility out.

Examples of logically indeterminate argumentsTo illustrate the inconclusive nature of arguments over which logics are “true,” consider first the debate between classical logicians and dialetheists over how to handle paradoxes such as the Liar9. Dialetheists are those who claim that there are validcontradictions, that is, valid statements of the form p ᴧ ¬p. Although accepting contradictions goes against centuries of logical tradition – and would seem to undermine one of the primary methods of proving statements (i.e., by deriving a contradiction from the negation of a statement) – dialetheists argue that there are benefits to accepting contradictions that make the costs worth paying. In particular, they argue that 9 The liar is a statement that asserts its own untruth, i.e. “this statement is not true,” or more formally, “‘L’ is true” in which L is “L is not true.” It therefore seems, prima facie, that if it is true, it is not true, and if it is not true, it is true.

dialetheism offers a treatment of paradoxical statements such as the Liar that does not require us either to restrict the T-scheme(“Pa” is true iff Pa), or to posit a hierarchy of languages such that no language can contain its own semantic or truth predicates(but can only make semantic statements or ascriptions of truth about statements in a lower-level language). By accepting certain contradictions – by giving up consistency – we can gain semantic closure. That is, we can have a language within which we can speak of that very language’s semantics, and within which we can give the meaning of all of the terms of that language (including “true”). This is important, according to one prominent advocate of dialetheism, because for philosophical reasons we require a language that is universal: that is, it can speak of language in general, not just of one particular language, as in a hierarchy. To do this, such a language needs tobe able to speak about its own syntax, without requiring ascent to a “higher” level of language in a hierarchy. It has to be, touse the term introduced above, semantically closed. But, argue dialethiests (and paraconsistentists such as J.C. Beall), any semantically closed language can generate paradoxes such as the Liar.10 Since the paradoxes are unavoidable (Bremer refers to them as “antinomies,” since, he argues, valid proofs can be constructed for both the Liar’s truth and non-truth), if we are to have a logical language that is universal and semantically closed, we need a logic that can handle contradiction. That is to say, we need a dialetheic logic.

This universality, naturally, comes at a price. And the price ofpure dialetheism (the price of Graham Priest’s original system LP) indeed appears steep: not only must we accept true contradictions – a serious enough cost in itself – but we also face the loss of disjunctive syllogism as a reasoning tool (if p is a dialetheia, then p v q, ¬p Ⱶ q fails, because we also have p).But disjunctive syllogism is a fairly basic and highly intuitive form of reasoning. If one of two options must be true, and we 10 The argument here follows Bremer (2008a). The classic presentation of dialetheism is Priest, Graham (1979), also his (1987).

discover that one of the options is false, then we want to be able to conclude that the other option is true. If you tell me that a coin toss failed to come up heads, I want to be able to conclude that it therefore came up tails. If my wife knows that I am either at home or at work, and she sees that I am not at home, it seems she ought to be able to conclude that I am at work. Of course, we have to be sure that the two options given really are the only two options for disjunctive syllogism to work. And on reflection, it’s clear that we can be more confident of this in some situations than in others. To take thefirst example given above, it’s pretty hard to imagine a coin toss coming up something other than heads or tails. But in the second example, it’s not hard at all to imagine (knowing me) thatI might in fact have slipped off to a pub and thus am neither at work or at home (but please don’t tell my wife or my boss!). Still, as long as we can rule out the more exotic cases, disjunctive syllogism would seem to be a very useful logical tool.

The thoughtful reader will likely now be asking, “wait a minute –aren’t those considerations involving disjunctive syllogism precisely the sort of logical considerations you say are insufficient for deciding between logics?” But while those considerations are indeed logical in nature, I deny that they aresufficient to enable us to decide between dialetheic logics such as LP and their rivals. We still need to justify a metalogical principle of some sort to accomplish that task. “That disjunctive syllogism is not valid in LP” is a logical statement. “That disjunctive syllogism is not worth giving up to obtain completeness and avoid paradox” is a metalogical statement. As is usual with these types of statements, the first is not debatable,but the second very much so. It is debated by dialetheists such as Priest, who argue that the cost of losing (or restricting – see below) disjunctive syllogism is worth it in order to defuse the paradoxes (d.s. is also implicated in the Curry paradox) while preserving the T-scheme and gaining semantic closure. While dialetheism certainly is a minority view among logicians,

it remains strongly held by some of the most important figures incontemporary symbolic logic. Clearly, simply pointing out that LP fails to validate disjunctive syllogism has not sufficed to settle that argument.

What’s more, dialetheists have a means of reducing the cost of dialetheism: the so-called “classical recapture.” This is the idea that a dialetheic logic can avoid much of the cost of givingup disjunctive syllogism via the path of “minimal inconsistency,”in which contradictions are accepted only in certain highly circumscribed areas of discourse. Following the argument developed by Beall in his Spandrels of Truth, contradictions might only be acceptable in the fragment of language that employs a truth predicate, or even more minimally, to sentences involving self-ascriptions of a truth predicate. By minimizing the domain of acceptable contradictions, we widen the domain of language in which we can continue to employ non-contradiction and disjunctivesyllogism. Accordingly, Priest developed a modification of LP known as MiLP (for “Minimally inconsistent Logic of Paradox”), inwhich disjunctive syllogism is not invalidated, but merely restricted. Priest has claimed that MiLP is the One True Logic. By thus reducing dialetheism’s cost, MiLP makes it a more attractive alternative to CP. What’s more, MiLP blunts the criticism that dialetheism requires giving up disjunctive syllogism, which seems like a perfectly good logical reason to prefer CP to LP.

At this point, it may seem like the thesis of Logical Indeterminism really amounts to a mere technical quibble. Am I not merely defining discussion of the most basic principles of logic as metalogic, and hence outside of logic proper? But surely every field of investigation involving a formal language, such as mathematics or geometry, employs its most basic principles which are in some sense prior to or outside the formallanguage itself. It seems a mere matter of definition or taxonomy whether such statements are “strictly speaking” mathematical, geometrical – or logical. The problem in the case

of logic, however, is that the basic principles themselves are inconflict, or otherwise controversial in ways that the basic principles of mathematics or geometry are not. The debate between dialetheism and classical logic is largely a conflict between the principles of consistency and completeness. Seemingly basic principles such as excluded middle are still debated – and rejected – by intuitionists, among others. There is no comparable debate over, say, the successor relation in arithmetic. Precisely because such conflicts involve fundamentalprinciples, they cannot be resolved within logic, because fundamental “logical” principles can be adduced on either side ofthe argument.

To take another example, logical completeness is as fundamental alogical principle as we can imagine. Soundness and completeness proofs are a fundamental litmus test, certainly for any logic proposed as a candidate for the One True Logic. A failure of deductive completeness is generally taken to indicate some flaw or other insufficiency in a logic. Sometimes, logics that have been proven incomplete are derided as not being “real” logic. Logical purists in the tradition of Quine have long regarded second-order logic as, in Quine’s famous phrase, “set theory in sheep’s clothing.” Furthermore, it has long been known that second-order logic is in fact incomplete. That is to say, there is at least one formula which we can know to be universally true that second-order logic does not validate. A Quinean might therefore argue that this fact constitutes a decisive logical argument against counting SOL as “real logic.” Yet plenty of logicians both past and present, have been perfectly willing to use second-order logic, and many philosophers have found second-order notation far more useful in expressing a number of conceptsthan first-order notation.11 Indeed, that second-order logic is more expressive than first-order logic is arguably a “logical” fact as much as is the incompleteness of SOL. So again, we have a conflict of basic logical principles: deductive completeness 11 Among such concepts, in addition to that of “set” and “number,” are “universal,” “property,” and “predicate.”

versus expressive power. “Second-order logic is incomplete” is alogical fact, but “second-order logic isn’t real logic” is a thesis of metalogic – and a highly debatable one, at that. The first statement can be adduced in support of the second, but doesnot by itself necessitate it. A further statement, such as a definition of “real logic” (if one can be had), is needed for us to conclude that second-order logic isn’t real logic.

One could insist, I suppose, that completeness is part of the definition of “real logic,” and insist that one has ruled out SOLon “strictly logical” grounds. Indeed, at least as far as FOL’s are concerned, it is generally agreed that failure of completeness indicates that there is something wrong with a logic.12 But there is no principle of classical or any other logic I know that holds that only complete logics are “real” logics – in part because the question “what is a ‘real’ logic” isnot the sort of question that basic logical principles (for any logic) typically address. Rather, that question arises at the level of metalogic. Using strictly “logical” principles, I can easily tell you which logics are modal logics, which are relevantor “fuzzy” logics, etc., but I cannot tell you which logics are the “real” ones. This therefore seems to me more a statement of value than of logic. “Which logics are real logics” is a question that by its very nature seems to come from outside of logic.

Yet another example of a dispute which cannot be settled by strictly logical considerations is that between classical and intuitionist logic. Someone who is a skeptic or deflationist about truth, or a pragmatist, might prefer intuitionist logic, which replaces the concept of truth with that of provability. Thus, instead of saying “p ᴧ q is true if p is true and q is 12 This is not to say that incomplete logics lack all value whatsoever. Some modal logics, including some of C.L. Lewis’ first formalizations of modal logic, are incomplete (although fortunately S4 and S5, one or the other of which is the preference of the vast majority of modal logicians, have been proven complete). They nevertheless may have some heuristic value, in that they may illuminate some feature either of modality itself or of our attempts to treat it symbolically.

true,” we say “p ᴧ q is provable if there is a proof of p and a proof of q.” The lack of a proof of p, however, does not imply the falseness of p, merely that there is (as yet) no proof of p. It might well be the case that we have no proof of either p or ¬p, in which case p v ¬p fails: intuitionist logic rejects excluded middle.

We have soundness and completeness proofs for intuitionist logic just as we do for classical logic. So intuitionist logic meets at least these minimal criteria for logical respectability. On what logical basis, therefore, can we choose between classical and intuitionist logic as candidates for the One True Logic? It’s true that intuitionism fails to resolve the Liar paradox, due to the fact that “neither true nor false” implies “not true,” which permits a “revenge” version of the Liar to be constructed for intuitionism: “this statement is not true.” But classical logic also fails to solve the Liar!

This is not to say that there are no arguments for choosing between classical and intuitionist logic. Of course, there are metalogical arguments for each position. For instance, a mathematician who wants to reason from the present state (as opposed to some ideal future state) of mathematical knowledge maybe more concerned to reason about what has or hasn’t been proven rather than about what is or isn’t true. For that reason, she may prefer an intuitionist logic. This would be a function, not of her logical views, but of her practice of mathematics. Certainly, if she has settled on a particular philosophy of mathematics, this will drive the choice of a logic – if she is a constructivist, for instance, she will likely prefer intuitionism. (This involves giving up a number of theorems thathave only been proven non-constructively, but if one if philosophically committed to constructivism, that is simply a cost one has to be willing to pay.) But were she to convert to realism, instead, then she would likely choose classical logic. So settling this debate in philosophy of mathematics (assuming that the debate can be settled) may in turn settle the choice of

classical versus intuitionist logic, at least for reasoning aboutmathematics.

There is also in the vicinity an argument for logical pluralism –one may prefer intuitionist logic for reasoning about mathematical provability, and classical logic for reasoning aboutmathematical truth. (For that matter, one might prefer classical logic for reasoning about mathematics, and intuitionist logic forreasoning about physics, where several rival theories are in competition, none of which have yet been proven.) But again, these issues will be resolved, if they will be resolved at all, outside of pure logic.

Another reason (for some) to prefer intuitionist logic is that itrejects excluded middle – so anyone who rejects excluded middle for whatever philosophical or metalogical reason will have a reason to prefer intuitionism. Again, how can logic by itself tell us whether we should accept excluded middle or not? It can give us various pros and cons regarding excluded middle – it can tell us, for instance, that without excluded middle, the denial of p will not imply the truth of ¬p. But that merely gives us a cost of rejecting excluded middle – it doesn’t demonstrate that excluded middle is true, simpliciter. It can also tell us, as mentioned above, that rejecting excluded middle does not itself solve the Liar paradox (because of inevitable revenge problems). But that by itself does not compel us to accept excluded middle: lots of other proposed solutions to the Liar have failed to put the paradox to rest. We might have other reasons to reject excluded middle: we might think, for instance, that value statements in ethics or aesthetics are neither true nor false. Someone who believes in a strict dichotomy between “statements offact” and “statements of value”, or someone who is a moral expressivist might take this view. Granted, they could also holdthat such value statements are simply false – indeed, most expressivists probably do hold this. But I can imagine a sort ofagnosticism about such statements, such that one may be as reluctant to endorse the denial of a value statement as to

endorse its assertion. One might not want to assert, for instance, that Beethoven is better than Ozzy Osbourne. But nonetheless, one might not want to go as far as to deny that Beethoven is better than Ozzy Osbourne. Thus, such a person may regard “Beethoven is better than Ozzy Osbourne,” intuitively, as neither true nor false.13 For that reason, she may prefer an intuitionist logic. Thus, there are perfectly respectable reasons to favor either classical logic or intuitionist logic. But these are philosophical or metalogical, rather than logical reasons. Thus, they are consistent with the thesis of logical indeterminism.

The argument from metalanguageAn argument that could be made against this thesis is that, in fact, classical logic can be preferred on strictly logical grounds – because, no matter how exotic and non-classical a logicmay be, the logic of the metalanguage is typically classical,

13 I recognize that the standard approach to such statements of aesthetic value is to relativize their interpretation or truth-value assessment to a particular context (or index) – that of the speaker of the statement, or of the person to whom the statement is directed, or perhaps even to an eavesdropper. Thus, “Beethoven is better than Ozzy Osbourne,” when spoken by your crusty old music teacher, is true iff Beethoven is better than Ozzy Osbourne in the judgement of your crusty old music teacher. But, to lay my cards on the table, I am a dissenter from this view. For me, your crusty old music teacher does not take herself to be saying “Beethoven is better than Ozzy Osburne in my judgement,” but “Beethoven is better than Ozzy Osbourne” simpliciter. She does not mean for her statement to be understood indexically, or “in a context.” She is so convinced of the obvious superiority of Ludwig Van to Ozzy that she regards “Beethoven is better than Ozzy Osbourne” as an objective truth. So the relativist, or any other indexical or contextual account does not accurately describe the statement the music teacher takes herself to be making. This forces the relativist to the claim that the music teacher does not correctly understand her own statement. On my view, the semantics should be understood homophonically: “Beethoven is better than Ozzy Osbourne,”when spoken by your crusty old music teacher – or anyone for that matter – is true iff Beethoven is better than Ozzy Osbourne. But there is no objective fact of the matter as to whether Beethoven is better than Ozzy. So, “Beethoven is better than Ozzy Osbourne” is in my view neither true nor false.

first-order, and non-modal. If one regards considerations relating to a logic’s metalanguage as “strictly logical” for purposes of this discussion, as I am quite willing to do, then this would seem to be a powerful “logical” argument in favor of classical logic – and against the thesis of logical indeterminism. What’s wrong with it?

First, as Timothy Williamson in his (2013) points out, the choiceof classical logic as metalogic is not inevitable for all logics.For instance, committed philosophical intuitionists have developed semantics for intuitionistic object languages that employ intuitionistic logic in the metalanguage as well as the object language. The problem is, as he explains, that this semantics comes at a cost: not all formulas validated using classical logic in the metalanguage are validated when we switch to intuitionistic logic in the metalanguage. Worse, the first-order completeness theorem for intuitionistic logic does not go through in the “intuitionistic” metalanguage, whereas it is validon the “classical” metalanguage. Indeed, for all the examples Williamson considers – fuzzy logic, modal logic, and logical pluralism in addition to intuitionism – he regards the attempts to use non-classical logic in the metalanguage problematic. Yet he is broadminded enough to allow that the various research projects in non-classical metalanguages are still ongoing, and concludes that “(t)o an extent much greater than is widely realized, unorthodoxy in the object language can be fully explored and fairly assessed only through unorthodoxy in the metalanguage.”14 He thus concludes that classical logic is not necessarily the unique logic for formal metalanguages.

Williamson’s conclusion seems somewhat tentative, however, particularly considering that there seems to be no current program in non-classical metalanguages that he is actually willing to endorse. (One is tempted to submit, tongue-in-cheek, that Williamson is actually employing an intuitionistic logic to assess the situation, since it seems that for him, the statement “there is a viable metalanguage that employs a non-classical 14 Williamson (2013), p. 230.

logic” has neither been proven nor disproven.) Can one make a stronger case for non-classical logic?

In the case of intuitionistic logic, it seems to me a dedicated intuitionist could counter that the philosophical virtues of intuitionism outweigh its lack of bivalence – in other words, that the benefits of intuitionism make the cost of losing bivalence worth paying. And this argument cannot be settled on purely logical considerations: we have to do some philosophy. Onecould, for instance, adduce philosophical considerations against completeness. If one thinks, for example, that the world is in aconstant state of becoming, such that new truths are constantly coming into being, then one will never be able to enumerate all of the truths. In set theory, Patrick Grim has shown that there can be no set of all facts15. If there were any such set, one could derive that set’s powerset, which by Cantor’s power set axiom would be larger than the set of all facts. That this set islarger than the original set of all facts would constitute a further fact not in the set – thus contradicting the premise thatthe original set contained all the facts. Similar reasoning would seem to apply to any putative “set of all true statements.”The powerset axiom (PSA) implies that a still larger set can be generated from any such set by the power set operation. One could say that the statement of this fact16 is a logical truth (guaranteed by PSA), and that this statement was not in the originalset. So there is a logical truth that is not in the original set. But that set was supposed to be the set of all logical truths. So the idea of a “set of all logically true statements” leads to contradiction17. This would seem to cut against the 15 In Grim (1984).16 This blocks the objection that the set generated by the powerset operation contains no additional logical truths to the original set, but is merely a “repackaging” of the logical truths in the original set. Even granting that, the fact that additional sets are generated by the powerset operation makes true an additional statement (“additional sets are generated by the powerset operation on any putative set of all logical truths”), that is itself a logical truth courtesy of the Powerset Theorem. 17 At least, it does if classical logic is the logic of our metalanguage (in this case ordinary, if philosophical, English)(!) But if LP is taken as the

claim that verifying “all of the logically true statements” of a formal language – i.e. completeness – should be part of the logical gold standard. (It also suggests that perhaps set theoryshould not be employed so indiscriminately within symbolic logic,but that is another matter.) This makes the failure of completeness for an intuitionistic logic with intuitionistic metalanguage less of a mark against intuitionistic logic. Particularly if one feels that intuitionistic logic is a better fit for such an incomplete, still evolving world, then one might decide that intuitionistic logic is to be preferred despite its lack of completeness. Indeed, its lack of completeness might even be viewed as part of what makes it such an appropriate logicfor our world – if we see our world as incomplete in some important way.

Even if it turns out that classical logic is the “best” logic to employ as the logic of the metalanguage of most, if not all otherlogics, we still need not abandon the thesis of logical indeterminism. A further premise is required: that classical logic’s unique suitability as a logic for metalanguages makes it the best choice for an object language. Remember that the real question at issue is which logic we should prefer for the object language, not the metalanguage. If classical logic in the metalanguage is just as compatible with intuitionistic logic or paraconsistent logic in the object language as it is with classical logic in the object language, why should the choice of classical logic for the object language be forced by its utility as the logic for the metalanguage? It might just be that classical logic is the best logic for dealing with the domain of logical truths, since that domain is necessary and eternal – unlike the domain of contingent, empirical truths, which are not necessary and may not be eternal. That would not make it the

logic of our metalanguage, it could be argued that “there is a set of all logically true statements for language L” might be true as well as false, in which case there can still be a set of all logically true statements - if we are willing to countenance dialetheism.

best logic for those of us who inhabit a non-abstract realm to use.

Indeed, the friend of logical indeterminism can turn the argumentaround. She can argue that the fact that metalanguages employingclassical logic are compatible with such a wide range of non-classical logics as object languages shows that the choice between rival logics cannot be made using logic by itself. Even should classical logic become universally accepted as the best logic for the metalanguages of all other logics, this could just as easily be seen as showing the equality of all those other logics to serve as object languages, rather than showing the superiority of classical logic itself in that capacity.

What Role Do Logical Considerations Play?None of this is to say that logical considerations are irrelevantto deciding between rival logics. Arguably, there is a core of basic logical principles – such as those I used above to define an “ideal” logic, or some subset thereof – such that a failure ofone of those principles in a logic is a reason, all things being equal, to reject it in favor of a logic which validates that principle. But all things are not equal. One might have a metalogical reason – such as desire for a logically and semantically complete language, or a desire to avoid paradox – toprefer a logic in which a basic principle fails to one in which it doesn’t.

Such metalogical reasons had better be good ones – one shouldn’t give up a base logical principle willy-nilly. But if one can mitigate the effects of giving up a base principle, it might be made more palatable. Thus the “classical recaptures” described above, which preserve classical logic for important fragments of the language while allowing departures from classical logic for arestricted class of statements. Not just dialetheists, by the way: advocates of solving the paradoxes by restricting excluded middle, such as Hartry Field, also limit their restriction of

excluded middle to a certain class of statements – while allowingexcluded middle to apply to mathematical and scientific statements. Thus, they also attempt the “classical recapture.” These moves reflect how base logical considerations can certainlyplay critical roles in our metalogical judgments.

For another example of the interplay between logical and what I call metalogical principles, consider the debate over which quantifiers should be accepted in symbolic logic. Traditionally,logicians have employed just a handful of quantifiers: the universal and existential quantifiers, and perhaps also the “indexical” quantifier when we want to refer to specific objects.Linguists, on the other hand, use a wide variety of quantifiers to analyze the structure of natural language, such as “a few,” “most,” “some,” and others which, while seemingly more vague thanthe existential and universal quantifiers, taken together offer amore fine-grained and accurate representation of how we quantify things in everyday speech. But these “generalized” quantifiers have been given logically precise semantics by a tradition of logicians beginning with the Polish mathematician and logician Andrzej Mostowski, a student of Tarski. For instance, one contemporary friend of generalized quantifiers has defined a generalized quantifier as “a function on cardinal numbers (sizes of universes) assigning to each cardinal number x another function I that says how many objects are allowed to fall under a set B and its complement in a universe of size x in order for Q(B) to be "true."” One such quantifier is the “most” quantifier Mx,which defines a function that goes to T (understood as “true”) ifthe objects that fall in B outnumber the objects that fall outside of B.18 The fact that such quantifiers can be given logically precise definitions is clearly a fact that can be demonstrated “within” logic – either through the semantic method of giving truth conditions described in terms of a model structure, or the proof-theoretic method of giving introduction and elimination rules. So doesn’t this count as a purely logicalresolution of the question of whether generalized quantifiers maybe used properly in logic?

18 The quote and example are from Sher (1991), pp. 11-13.

Well, no. As it happens, the functions described above as defining the generalized quantifiers are generally understood as functions over sets (representing models in a standard Tarskian semantics) of varying sizes.19 The traditional existential and universal quantifiers, on the other hand, are naturally read in amanner that makes no reference whatsoever to sets (“there is a x…” and “For all x…”) And whether or not sets may be legitimatelyinvoked for that or any other purpose is a major controversy in philosophy of logic, going back to the earliest days of symbolic logic.20 On the one hand, it has traditionally been a truism among logicians that logic presupposes nothing about the nature of the world or what manner of entities exist in it. Logic, on this view, is neutral as to metaphysics or ontology. So logic should not presuppose the existence of sets. On the other hand, modern symbolic logic largely has its origin in the project of Frege, Russell, and Whitehead to found mathematics upon logic, which employed set theory as a medium for building up mathematical concepts from logical concepts. Key to this projectwas the idea, still held by most mathematicians, that numbers ultimately are sets—and thus amenable to treatment by set theory. If set theory could be shown in turn to be derivable from logic, then logic would thereby also provide a foundation for mathematics. The “concept script” which Frege devised to expresshis theory of the foundations of arithmetic was developed to makeassertions about sets, such as the Comprehension Axiom ∀x(x ∈ F ≡ Fx), which employs the symbol of set membership along with the universal quantifier and the biconditional. If this project had succeeded, then set theory would be derivable from and reducible to logic, and there would thus be no problem in employing set-theoretical concepts. But since Russell’s paradox undermined the19 Mostowski, for instance, defines quantifiers as function upon members of I, which refers to “an arbitrary set,” (1957, p. 12) and refers to sets throughout his (1957).20 One can replace talk of sets in the quantifier definitions with talk of numbers (if one takes numbers to be primitive, as opposed to being constructions from sets), but then one is simply trading commitment to one type of abstract entity for commitment to another type of abstract entity.

Comprehension Axiom, which had made set membership equivalent to being in the extension of a predicate21, this brand of logicism has long been out of fashion (a “neo” version has been put forward by Crispin Wright and Bob Hale, however)22. Despite this, most contemporary logicians think nothing of employing set theoretical symbols in their formal logics. But for nominalists and other ontological minimalists in logic, presupposing the existence of sets in order to introduce a whole menagerie of new quantifiers is a bridge too far.

The dispute over the proper relationship between set theory and logic is also reflected in the debates over which type of semantics best characterizes logical consequence – model theoretic (which employs sets), proof-theoretic (which does not),or some other approach. This debate, however, does not seem likethe sort of debate that can be solved within logic, by research in the properties of a given logic or logics. The question of the proper relationship of set theory to logic is a question to be resolved philosophically, if possible – not logically.

Again, this is not to say that logical considerations don’t play a role in these debates. For instance, the fact that model theoretic semantics validates certain intuitively valid logical inferences that proof theory does not validate certainly counts as a point in favor of model theory. For many, this is a decisive consideration. But not for all: it’s clear from the most cursory comparison of model and proof theory that proof theory is ontologically minimalistic: since it is based on rules for the introduction and “elimination” of symbolic expressions,

21 Russell showed that, if we allow “is not a member of itself” as a predicate, we could construct a set of all non-self-membered sets. Is that set (the Russell set) a member of itself, or not? If we presuppose that it isa member of itself, we can easily prove that it isn’t, and vice versa. Thus theprinciple that every predicate defines a set, which is what the Comprehension Axiom expresses, leads to contradiction.22 One could restrict the Comprehension Axiom by excluding such predicates as “is (not) a member of itself,” but then one would of course be sacrificing completeness of the logical language, and thus we are back to the debate between completeness and consistency.

it doesn’t presuppose much of anything in the way of ontology. Model theory on the other hand cannot avoid ontological presupposition of some sort. To begin with, it presupposes the existence of models! Further, in standard formulations of model-theoretic semantics, models are defined as sets of sentences, andlogical validity is typically described in terms of sentences following from sets of premises – so the standard model theory, at least, endorses the existence of sets.

Whether or not this dependence of model theory on entities such as models and sets is problematic depends on one’s philosophical inclinations. If one has no problem thinking of sets as part of the basic furniture of the world, then one is more likely to embrace model theory. If one prefers as austere an ontology as possible, however, one will more likely incline toward proof theory. Again, logical analysis can give us a list of pros and cons, but it doesn’t tell us how to weigh their relative importance. This requires that we do philosophy – if indeed philosophy can find a method for resolving such questions, instead of merely expressing them.

Yet another example of important logical results that must still be supplemented by philosophical considerations is offered by thequestion, discussed above, of which logic should be employed in the metalanguage, as opposed to the object language, of a given logic. In particular, can (or should) we use the same logic in both the object language and metalanguage? Williamson (2013) offers the example of fuzzy logic. He shows how numerous formulas in fuzzy logic are valid when classical logic is employed in the metalanguage, but invalid when fuzzy logic itselfis employed in the metalanguage. Indeed, a fuzzy logic with fuzzy metalanguage is extremely weak – very few formulae can be validated. Does this not suggest that classical logic should be preferred as the logic for expressing the semantics of fuzzy logic – and doesn’t that in turn suggest that classical logic is in some sense fundamental to fuzzy logic?

Perhaps – but only if one presupposes that such “extreme” weakness is disqualifying, or at least severely disadvantaging for a logic. And this is a conclusion that goes beyond purely logical to philosophical considerations. One requires a further premise to the effect that a logic that is not “extremely weak” is to be preferred to a logic that is “extremely weak” – and sucha premise cannot be demonstrated by purely “logical” considerations as commonly understood.

This is not, of course, to say that no considerations whatsoever can demonstrate the superiority of a strong logic to a weak one: for instance, a very weak logic will be too weak to express many of our scientific judgments. This is an excellent reason to prefer a stronger logic—but not a strictly logical one. Rather it relies on an empirical fact about scientific statements: that they require a fairly strong underlying logic. For such a reasonto be decisive, therefore, would count in favor of the thesis of logical indeterminism.

Monism revisitedA pluralist might point out that many of the arguments given above for, in particular, intuitionistic logic may be taken to support pluralism rather than indeterminism. (We briefly suggested one such argument above.) But the thesis of logical indeterminism is not inconsistent with logical monism, as well, implausible though that might seem prima facie. It might conceivably be the case that there is in fact One True Logic – but that we are unable to determine what that logic is. Perhaps there is, as Manuel Bremer has surmised, a “logical faculty” ultimately located in the brain in the way that a “linguistic faculty” is thought by many to exist.23 If so, then this facultymay operate according to a particular logic – and if we could discover that, we will arguably have uncovered the laws by which humans actually reason. Of course, if such a logic were

23 In Bremer (2008).

discovered by neurological investigations, that discovery would be consistent with the thesis of logical indeterminism, which states only that such questions cannot be solved by purely logical means. Clearly, neuroscience is not a branch of logic. It’s possible that a rigorous analysis of our intuitions about validity and meaning might still enable us to uncover, through a sort of psychological investigation, the logic of our “logical faculty.” But there is no guarantee that this will be the case: over a century of investigation and debate in logic have led not to one but to many logics, so it seems premature to assume that conceptual analysis or “experimental” philosophy will enable us to converge on one logic sometime in the future. We would thus have a situation (barring a breakthrough in neuroscience) in which there is a single logic of human reasoning – but we can’t discover which it is. Such a situation would seem strange to me:the logic by which we intuitively operate would be as it were locked away in our subconsciousness. (Bremer does not, it shouldbe noted, believe this is the case: he regards his “universal” logic UL4 as a strong candidate for the One True Logic24.) But I cannot rule out a priori that it is the case. It may be that, ultimately, a formal specification of the logic by which our species reasons is simply beyond us.

Furthermore, even if we could discover, through neurological or other means, the precise logic by which we humans actually reason, how would we know that that logic is the One True Logic? Humans are not, in general, known for their logical infallibility. If we presume that such a “logical faculty” is a product of our evolution, why shouldn’t we think that further evolution would improve that faculty – and utilize a different, better logic than that by which we currently reason? The question of which logic is the One True Logic would thus remain open.

Metalogical indeterminism?24 See Bremer (unpublished).

As outlined here, the thesis of logical indeterminism claims onlythat there is no strictly logical means to settle which of the various rival logics, if any, is the One True Logic. Understood conservatively, the thesis goes no further than that. But could we go further? Could we advance a thesis of metalogical indeterminism?

Conclusively ruling out a metalogical resolution of our debates over these fundamental issues in logic certainly seems more difficult than ruling out a strictly logical resolution – the domain of metalogic appears to be considerably larger and less well defined. It may be that presently unforeseen developments in philosophy or in the special sciences will provide conclusive metalogical arguments that are currently lacking. This seems to preclude conclusively establishing a thesis of metalogical indeterminism.

Still, when we consider such apparently intractable debates like those rehearsed above, where disputants appear to disagree about fundamental logical issues, we might well wonder how such disputes could ever be resolved. Indeed, relatively little contemporary literature in the philosophy of logic seems to address this meta-question directly. Here, however, are three fairly recent developments in the philosophy of logic that I think offer some possibility of progress on meta-logical issues.

Williamson (2014) tackles the question of method head-on by arguing in favor of an abductive approach. He makes an abductive argument in favor of classical logic that takes into account (among other things) the impact of logical choices on other fields of human inquiry. He puts the dilemma in somewhat different term – as between non-contradiction and disquotationalism rather than non-contradiction and completeness.But restricting disquotationalism amounts to a restriction on language, so a restriction on disquotation is a restriction on linguistic completeness. In any case, Williamson points out thatmost of our mathematics and science can be represented in a formal language that makes no reference to truth whatsoever.

These inquiries could be carried out in a language entirely lacking in a truth operator or predicate, and thus would never have to contend with any paradoxes involving truth operators or predicates. But, he argues, restricting contradiction – either directly by restricting the Law of Non-contradiction and Explosion, or indirectly by restricting excluded middle – would be fatal to mathematics and science. Given the central role of mathematics and science in our system of knowledge, he concludes,we should be willing to restrict disquotationalism before we restrict non-contradiction.

Williamson makes a powerful and persuasive argument – the more sosince most participants in the discussion will likely readily agree that mathematics and natural science (particularly physics)indeed sit at the core of human knowledge.25 And is argument hasthe virtue of intellectual modesty – he does not view his argument as true based on a knockdown, deductively valid argument, but rather as an inference to the best explanation based on a number of criteria, included fit to evidence, simplicity, and impact upon other bodies of human knowledge. Still, some objections could be raised.

Most broadly, it can be argued that, even granting his argument that excluded middle is required for core mathematical and scientific reasoning, and that mathematical and scientific theories can be constructed without use of a truth predicate – nonetheless these theories are not constructed in a vacuum. We use these theories, often as a foundation for theories in other fields of knowledge. We build bridges and immunize our kids on the basis of such theories. We are therefore very much concernedto know if they are true. So while a truth predicate may not be needed to construct the theory, it is needed to evaluate it, to tell us whether we need to take the theory seriously or dismiss it as an intellectual curiosity. Thus, it is basic to our

25 We can, however, imagine arguments that ethics, metaphysics, and indeed theology are more central to human concerns.

fundamental epistemic practice across all disciplines, including mathematics and physics.

Furthermore, that classical logic is necessary for mathematical and scientific reasoning is not beyond challenge. In mathematics, constructivists have advocated a method of proof which does not depend upon deriving contradictions, and hence does not require excluded middle. Although historically mathematicians have made much use of proofs based on deriving contradictions, and constructive proofs do not exist for all of the theorems in mathematics that have been proven non-constructively, it is not conclusive that constructive proofs could not have been used instead, had the history of mathematics gone differently, nor is it conclusive that constructive proofs are impossible where we do not have them currently. While it would require considerable work to replace every important proof based on non-contradiction with a constructive proof, if we have strong enough philosophical grounds for rejecting excluded middleand/or embracing constructivism, this may be a price worth paying.

As for scientific reasoning, the biggest problem with classical logic as I see is not its use of excluded middle, but the paradoxes of the material conditional, which we discussed above. As is well known, classical logic validates any material conditional with a false antecedent. This includes conditionals that go against our understanding of natural law, as when I say, “if Milwaukee is the capital of the United States, then the speedof light is 100 m.p.h..” No scientist thinks the speed of light is in any way dependent on which city happens to be the capital of the United States, so a scientist would likely reject such a conditional. The classicist, of course, has a response, namely that the scientist is not rejecting the indicative conditional quoted above, but the counterfactual “if Milwaukee were the capital of the United States, then the speed of light would be 100 m.p.h.,” which is clearly false.

For the classicist, someone who objects to vacuously true conditionals such as “if Milwaukee is the capital of the United States, then the speed of light is 100 m.p.h.” misunderstands logical validity. The purpose of the material conditional is to model neither natural law nor the ordinary English-language conditional. Logical validity is the preservation of truth. In ensuring logical validity, we ensure that one cannot go from truth to falsity. Thus, if one starts off with a true statement p, any disjunction of that statement with any other statement is guaranteed to be true, by the classical truth conditions for disjunction. The other statement q may be as false or outrageousas you can imagine, but the truth of p will guarantee the truth of p v q. So, outrageous though a statement like “if Milwaukee is the capital of the United States, then the speed of light is 100 m.p.h.” may sound, they are harmless, from the standpoint of logical validity, because they do not lead us from truth to falsity. From the standpoint of logical validity, going from falsity to falsity is not a problem. Logical validity is concerned with truth preservation, and where there is no truth tobegin with, there is none to preserve.

For scientists who worry about endorsing bizarre theories of natural law, classical logic can be extended with modal logic anda treatment of counterfactuals. Historically, this was done via the introduction of the possibility operator ◊, so that a lawlikerelationship between antecedent and consequent could be represented as ¬◊¬p (not possibility not p)26. Later, counterfactual operators were introduced to represent “if it werethe case the p it would possibly be the case that…” and “if it were the case that p it would necessarily be the case that…” Thusthe ordinary English language conditional could be treated in symbolic logic as ¬◊¬(p → q). As a result, though one must stillaccept the vacuous truth of “if Milwaukee is the capital of the United States, then the speed of light is 100 m.p.h.”, one can still securely deny that “if Milwaukee were the capital of the 26Subsequently the box (“□”) was introduced to replace the ¬◊¬ construction.

United States, then the speed of light would be 100 m.p.h.” Thus, argues the classicist, the idea of a “paradox of the material conditional” is based on a misunderstanding of logical validity.

I am not so sure, however, that the scientist would rest easy allowing the indicative version to be held true, however vacuously. The indicative conditional still asserts a violation of natural law as its consequent, something to which a scientist may wish to deny even vacuous truth. To put it another way, to ascientist, consistency of natural law may be a greater value thantruth-functionality of a formal propositional calculus. For suchreasons, one might imagine that scientists are actually reasoning, at least implicitly, with some sort of relevant logic.

More broadly, we can say that, though Williamson’s abductive method does seem like the most likely way forward, it is not entirely unproblematic. For one thing, such a method, like “inference to the best explanation” methods in general, is very much an “all things being equal” sort of approach. But all things are not always equal – and it’s not always clear whether all things are equal or not. One can imagine all sorts of philosophical reasons why all things might not be equal. For instance, someone who has a strongly-held intuition that there are statements that are neither true nor false, who is convinced that certain statements (the paradoxes, perhaps, or statements about morality or aesthetics) are simply not “truth-apt,” will insist on rejecting excluded middle, even if the “logical price” of doing so is quite high. Such a person might even be willing to give up important theorems in mathematics, if proving them requires excluded middle. Obviously, such a person is not going to feel compelled by an abductive argument such as Williamson’s.

Even if we’re willing to dismiss such opponents of classical logical principles as monomaniacs or ideologues, there remains the more practical problem of how to weigh the various criteria that may be employed in an abductive argument. Willliamson, for instance, lists fit to evidence, plus “strength, simplicity, elegance, and unifying power” as criteria for evaluating rival

logics.27 Prima facie, these seem like reasonable criteria. But how exactly do we weigh them relative to each other? For example, what if “fit to evidence” seems to require a highly complex theory – because the evidence indicates a highly complicated situation? Such a theory would, however, fare poorly on the criterion of simplicity. Say a we have a rival theory that is considerably simpler – easier to understand, but also quicker to deliver results due to relative ease of computability – but does a poorer job of tracking the logical “evidence.” Which theory should we prefer? Williamson does not give us a clear method of resolving this question.

Depending on how one weighs the various criteria by which logics are judged, one might offer plausible-sounding abductive arguments for different logics. Williamson makes a fairly persuasive abductive argument in favor of classical logic. But Ican just as easily imagine abductive arguments in favor of intuitionist, relevant, and even dialetheic logics. For example,if one has a strong belief that certain statements are neither true nor false, and one gives more weight to “fit to evidence,” say, than to implications for other fields of knowledge, one has the outline of an abductive argument in favor of an intuitionist logic. Analogously, a relevantist could make a similar claim that relevant logics “fit the evidence” that CL endorses irrelevant inferences, and thus derive an abductive argument for relevant logic. So it is not clear that adopting an abductive method is going to lead to a conclusive answer to the question ofwhich logic we should prefer, however plausible the basic approach seems.

For these reasons, I am not convinced that an “abductive” approach is going to resolve our fundamental questions concerninglogic. But I can’t rule out that an abductive argument in favor of a particular logic could be developed someday that would command broad assent among logicians. Such a development doesn’t seem likely to me, but I can’t rule it out.

27 Williamson (2014), p. 14.

Another recent attempt at a conclusive metalogical argument in favor of a particular logic is the recent revival of the “semantic defectiveness” described above. This is the claim thatparadoxical statements like the Liar “semantically defective,” ornot “truth apt” – and therefore should be considered no more valid than a line of gibberish. The obvious problem with this approach is that many of the sorts of statements that semantic defectiveness advocates want to rule out, such as the Liar, seem prima facie to be meaningful. That is, they appear to be grammatically well-formed, and we feel that we understand them – more or less. Thus, a semantic defectiveness approach seems to leave us with a language that contains meaningful expressions that our logic fails to provide a truth value for. But proponents of the semantic defectiveness approach think that our feeling that the Liar is meaningful is illusory. As discussed above, Benjamin Burgis, one of the best contemporary advocates ofthis sort of view28, claims that sentences that fail to “ground out” in sentences that are about something other than truth are semantically defective in this way29. For this to work as a dissolution of the Liar paradox, he has carefully to distinguish having a truth value called “undefined” from not having any truthvalue whatsoever. The Liar, on this view, fails even to qualify for an assessment for truth value – it is simply not “truth apt.”This is critical, because otherwise the value “undefined” can be lumped together with “false” to create the category “not true” – and as we have seen above, a strengthened Liar can then be constructed with “this statement is not true,” or even “this statement is false or undefined.”

Burgis’ “semantic defectiveness” approach represents a serious revival of the meaninglessness strategy, long regarded as a dead end, for resolving (or dissolving) the Liar paradox. Should it become generally accepted, it would be an example of a logical issue being settled as a result of settling a metalogical issue (as noted above, this therefore would not contradict the thesis 28 Another is Bradley Armour-Garb.29 Burgis (2011), p. 128ff.

of logical indeterminism). However, this would require that quite a number of other thinkers either give up or overlook their strongly-held intuition that the Liar sentence is in fact meaningful (and that Burgis’ solution therefore requires an expressive limitation of our language). While I find Burgis’ argument persuasive (or at least, I can think of no obvious refutation of it), it has yet to stand the test of time and inevitable counterattacks. Its success in doing so would certainly count against a thesis of metalogical indeterminism.

Another conceivable route to resolving the completeness-consistency dilemma – and this is highly speculative – is via some sort of “structural” approach that manages to unify rival logics in some way30. For instance, it might be noticed that, for all their differences, dialetheic and classical treatments ofthe Liar paradox, as well as Field’s solution referenced above, have something structurally in common: each identifies a group ofparadoxical statements and then effects a sort of separation between those statements and the larger class of non-paradoxical statements, separating the paradoxical statements to receive different treatment under the logic in question. In classical logic, either their truth-value is assessed at a higher level in a hierarchy of languages, or the paradoxical statements are simply ruled defective in some way. In the case of MiLP, the paradoxical statements are differentiated by allowing them to take {1,0} as a truth value (whereas non-paradoxical statements, of course, can only take {1} or {0} as truth values). Similarly,Field in his solution of the paradoxes “preserves” classical logic for a broad range of types of statements for which excludedmiddle applies, but relaxes it for a more narrow range of

30 Béziau’s “universal logic” program (see Béziau 1994 and later papers) takesa sort of structural approach, but at a higher level of abstraction than suggested here. Interestingly, his approach (as I understand it) is a structuralism based in mathematics (mathematics for him being the “study of structures”) such that logic is, for him, a branch of mathematics (rather than, as in logicism, its foundation). Thus his view involves ontological commitment, at least to structures, if not to sets – and thus is not ontologically innocent, despite its high level abstraction.

statements, such as those employing a truth predicate31. But perhaps the specifics of how these paradoxical statements are to be dealt with are less important, from the viewpoint of deep structure, than the simple fact that, in all three of these approaches, the same group of problematic statements is separatedout for special treatment. Such a structural view would look for other deep commonalities among logics that, at the “surface” level appear quite different.

The challenge for such “structuralist” views of logic, of course,will be to flesh out ideas such as those briefly sketched above, so that they amount to more than simply pointing out that, yes, it’s striking how so many logics have such things in common. Béziau has offered some interesting ideas along these lines, but in broad outline, rather than (as yet) a fully developed program.Still, the conceivability of such a development by Béziau or someone else is a consideration against a thesis of metalogical indeterminacy. For this reason, along with the others adduced above, although I suspect that ultimately such metalogical questions may turn out to be indeterminate, it seems premature toadvance a thesis of metalogical indeterminacy. But there is clearly much more to be said on the question of the method we should use – if there is any such method – to settle the metalogical questions that have to be settled if, in turn, we areto settle the fundamental logical questions.

ConclusionWe have here given an account of the thesis of Logical Indeterminism, and adduced considerations in its favor. As stated above, we offer no formal proof of the thesis, but we can offer two informal arguments, which are summarized below:

The Argument from the Multiplicity of Logics

31 See (Field 2007), p. 14.

There is no ideal logic, that is, no logic can have all of the characteristics that, ideally, we want a logic to have. Classical logic endorses irrelevant inferences and (along with the T-scheme) generates paradox. Relevant logics lack a truth-functional conditional (because considerations other than mere truth and falsity of antecedent and consequent determine whether a given conditional is true or false). Intuitionist logics give up excluded middle. Paraconsistent logics give up explosion (generally, by restricting Disjunctive Syllogism, which is an even worse consequence). Dialetheic logics restrict consistency.It seems that every logic we’ve devised “comes up short” in some way.

As a result, there is not one universally agreed-upon logic, but a variety of rival logics, each of which has virtues and shortcomings. Despite their shortcomings, we have soundness and completeness proofs for the strongest candidate logics. So thereis nothing defective in any of them, from a strictly logical point of view. But they can’t all be correct, as each invalidates the others.

Which logic we should use, then, will depend upon how we weigh these various logical virtues and shortcomings. But how we do that does not depend on purely logical reasons. If we were to seek a purely logical answer to that question, that would raise the question of which logic to use in considering that question. But that is precisely the question we are trying to answer (particularly if we are assuming that there is only One True Logic). So, other than purely logical means must be found.

The Argument from the Nature of Deductive Reasoning

By “logical means,” we mean the methods of deductive reasoning, as symbolized in a formal language.

This can be carried out by deductions from a set of axioms, as inthe case of axiomatic systems, or derivations from a premise set using introduction and elimination rules for logical connectives,as in the case of a proof-theoretic system.

However when we are debating which logic we should use, we are debating, inter alia, about which sets of axioms, or which introduction and elimination rules, we should work from. But axioms and rules are generally given or stipulated for a logic – they are not themselves arrived at through deduction or derivation. So the axioms or rules of the One True Logic cannot be arrived at through deduction or derivation. But by the definition of “logical means” given above, this means that the axioms or rules of the One True Logic cannot be arrived at through logical means.

Thus the question of which logic (or logics) we should use cannotbe arrived at through logical means. This is precisely the thesis of Logical Indeterminism.

As mentioned at the beginning of this paper, in a sense the entire field of philosophy of logic represents attempts to resolve the deepest questions of logic by non-deductive means. This does not mean that logical methods are not also employed. Aphilosopher of logic may, for instance, devise a formal language to demonstrate that something is expressible in language, or to offer a better way of modeling ordinary language expressions. But ultimately such demonstrations must combine with metalogical propositions to yield conclusions about deep questions in logic. So it would seem odd for a philosopher of logic to question the thesis of Logical Indeterminism. A more likely response might beto dismiss it as trivial. Of course we must use metalogical methods to settle metalogical questions, one might say. But in commencing a search for an answer that has proven elusive, perhaps the best way to start is by ruling out looking in places where we can know we won’t find it.

The big question, of course, is what means, if any, we do have for deciding which logic we should use. Prima facie, it seems that some form of inference to the best explanation – perhaps a Williamsonian “abductive” argument – is our most likely means of

answering this question. There is, of course, no guarantee that any particular abductive argument will convince everyone, the waya valid deductive argument will tend to do. But that is a topic for another paper. For now, we conclude that some grounds outside of logic (narrowly understood) must be found for choosingone logic over another. Logic alone will not accomplish this task – we must do some philosophy as well. Paradoxical though itmay sound, the deepest questions in logic cannot be solved with logic alone.

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