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Why You Don’t Need Brain Surgery To Change Logic

Interview by Richard Marshall.

Image result for hartry field

‘It’s frequently noted that when one becomes aware that certain of one’s beliefs are logically inconsistent, the most rational response is often to keep the inconsistency, because one is not sure how best to eliminate it.  The ability to manage known inconsistencies and other tensions in one’s beliefs is one of the many important factors in rationality that stress on “being logical” obscures.

I occasionally hear the objection that it’s ridiculous to talk of change of logic, since logic deals with the platonic realm or whatever, which doesn’t change.  Reminds me of the joke about how many psychiatrists it takes to change a light bulb.  The answer is only one—but the light bulb has to want to change.’

I’ve heard it joked that you could only change your logic by brain surgery.  I think that view is a severe over-reaction!

I’m enough of a Quinean to think that if “what we mean” by a term rules out a sensible response to questions framed using that term, we should reject that meaning as a bad one.

Priest and Williamson don’t just disagree about truth, they disagree about how to regulate reason: Williamson imposes deductive syllogism as a requirement on good reasoning, Priest doesn’t.

Hartry Field is an American philosopher and notable contributor to philosophy of language, mind, and mathematics. Here he talks about the link between logic and rationality, rational change, what Williamson and Priest are disagreeing about, what validity means, objectivity, whether there’s a uniquely correct logic,  semantic paradoxes and vagueness, paradoxes of indeterminacy, truth and deflationism, metaphysical implications, and what to say to those who don’t like analytic philosophy.

RM:  What made you become a philosopher?

HF: It’s hard to see my path into philosophy as a sequence of rational decisions.  It was easy to fall into the idea of going into graduate school: without that I was in serious danger of being drafted, and even if not, I’d have had to get a job.  As an undergrad my courses were about evenly spread between philosophy, math, history and political science, but I knew I’d never do research in the latter two, and I got sick of doing math toward the end of my undergrad years.  (That was temporary.)  I’d started out with interests mainly in math and physics, but physics fell out rather early for reasons too long and boring to go into; I had no idea what philosophy was.  I don’t think I got a real good sense of what it was even in my undergrad classes (the University of Wisconsin department back then had a heavily ordinary language influence).  Nonetheless there was something in it that appealed to me (especially since I read some Quine and Putnam on my own), and many of the topics connected with issues I’d been thinking of since I was a kid.  (Ones that I learned to keep to myself, given the responses I got from parents and friends if I raised them.)  It does seem to me a total miracle that I ever got into grad school in philosophy: it’s lucky that there was no requirement of a writing sample in those days, since I had nothing to submit that wouldn’t have been a total embarrassment.

Saving Truth From Paradox

RM:  Mr Spock in Star Trek is portrayed as the epitome of rationality because he is very committed to deductive logic. Are we right to think of his deductive logic fetish as a sign of rationality?

HF: The role of deductive logic in rationality is often exaggerated.  The notion of rationality isn’t very precise (and indeed doesn’t have fixed descriptive content: ‘rational’ is an approval-term), but logic is one of only many factors.  For instance there could probably be a perfectly logical advocate of the claim that the moon landing was a hoax, but few would regard such a person rational.  It’s frequently noted that when one becomes aware that certain of one’s beliefs are logically inconsistent, the most rational response is often to keep the inconsistency, because one is not sure how best to eliminate it.  The ability to manage known inconsistencies and other tensions in one’s beliefs is one of the many important factors in rationality that stress on “being logical” obscures.

RM: A problem with the idea that deductive logic is firmly tied to rationality is that it suggests we can’t rationally have a change in logic. But we do. How does your idea of thinking about logic as a ‘science of what preserves truth by a certain kind of necessity, or by necessity plus logical form’ help to overcome this problem?

HF: I don’t actually think that there’s much connection between the rational change issue and the idea that logic is the science of what preserves truth by logical necessity.  (I also have become more qualified about that latter idea.)  But let me address the issue of rational change in logic; we can come back to the other issue if you like.

Rational change in logic is certainly a possibility, one I’ve emphasized a lot in recent years. (And it will typically include change in opinion as to what forms of inference preserve truth.)

Some philosophers (e.g. Putnam) have said that rational change in logic raises no deeper issues than does rational change of other deeply held beliefs such as about the geometry of space, but I don’t think that’s right: in the case of the geometry of space we can lay out the alternatives and use standard inductive methods to choose between them, but in the case of logic we can’t do this because the standard inductive methods are themselves infected with logic.  Recognizing this, other philosophers (e.g. Kripke) have denied that rational change in logic is possible; I’ve heard it joked that you could only change your logic by brain surgery.  I think that view is a severe over-reaction!  What I think is that the change of geometry paradigm is too limited to capture the kinds of change of methodology that are possible, in science and elsewhere, and that change in logic is continuous with certain other kinds of change in scientific methodology.  I’ve come to think that what drives the idea that change of logic is impossible is a picture on which all rational change is by the use of “epistemic rules”.  I have some as-yet-unpublished work that tries to undermine that picture.

Incidentally, I occasionally hear the objection that it’s ridiculous to talk of change of logic, since logic deals with the platonic realm or whatever, which doesn’t change.  Reminds me of the joke about how many psychiatrists it takes to change a light bulb.  The answer is only one—but the light bulb has to want to change.

RM: Ha. I like the psychiatrist joke.  So I guess the point is that “changing an X” needn’t mean “causing a change within an X” but can mean “replacing one X by another”.  But can you say more about the connection between logic and truth preservation?  When Graham Priest’s paraconsistent logic confronts Timothy Williamson’s classical logic, is it necessary truth-preservation or something else that they’re disagreeing about?

HF: Well, one thing they disagree about is whether contradictions can ever be true, but this has to be put carefully: while Williamson says ‘no’ and Priest says ‘yes’, Priest also says ‘no’.  (It’s part of dialetheic doctrine to accept all inferences that the classical logician does, so he accepts that there are no true contradictions for exactly the reasons Williamson does.  It’s just that, as a dialetheist, this doesn’t prevent him from also accepting that there are some.)  Similarly, they disagree over whether every instance of disjunctive syllogism (from A or B and not-A, infer B) preserves truth: here, it’s that while Williamson simply says yes, Priest says no as well as yes.  (He says yes because in the cases where he says no, he regards A as not-true as well as true, so though these are cases that violate the truth preservation, they also aren’t cases that violate it.)

I find it natural to say that Priest, unlike Williamson, simply regards disjunctive syllogism as invalid. (Similarly for the explosion rule, which allows us to infer anything from a contradiction.)  If ‘valid’ for a schema just meant ‘all instances preserve truth (or do so by logical necessity)’, one couldn’t simply say this, since in that sense he regards them as valid as well.  In general I think that even on accounts where ‘true’ is a non-classical predicate, it’s best to regard ‘valid’ as a classical one: which implies that no inference can be both valid and invalid.  (Of course we can regard ‘valid’ as ambiguous, so that an inference can be valid in one sense and invalid in another; it’s just that in any good sense of the term, no inference can be both valid and valid in that sense.)  Taking ‘valid’ to be classical seems required to makes sense of validity as a regulator of reason.

So Priest and Williamson don’t just disagree about truth, they disagree about how to regulate reason: Williamson imposes deductive syllogism as a requirement on good reasoning, Priest doesn’t.  While this fact is not totally divorced from issues about truth-preservation, it’s hard to see it as following from issues about truth preservation given that in non-classical logics the notion of ‘preserves truth’ doesn’t behave classically.  (The case for a regulation of reason requirement irreducible to truth-preservation could be made more strongly if we also considered non-classical logics that allow us to reject some instances of excluded middle, instead of or in addition to accepting some contradictions.)

RM: But doesn’t ‘valid’ just mean ‘preserves truth by logical necessity’?

HF: I doubt it, but if it did it wouldn’t bother me: I’m enough of a Quinean to think that if “what we mean” by a term rules out a sensible response to questions framed using that term, we should reject that meaning as a bad one.  In any case, my picture is that we have a primitive notion of validity, governed by principles that in conjunction with classical logic and some natural principles about truth gives a simple argument that validity coincides with necessary truth-preservation.  Unfortunately, those assumptions about truth are inconsistent with classical logic: they are precisely those which, together with classical logic, lead to the Curry paradox.  Because of this, the case for validity coinciding with necessary truth-preservation vanishes.

Of course you can insist that what you mean by ‘valid’ is ‘preserves truth by logical necessity’, but there is positive reason not to do so: doing so has bizarre consequences.

That’s especially so if your logic is non-classical, but even if you keep classical logic, there’s a positive reason not to identify truth with necessary preservation of validity across the board, i.e. in the logic of truth as well as the logic of the first order connectives.  For instance, if your classical theory of truth is Kripke-Feferman, then insisting that validity coincides with necessary preservation of truth will lead you to say that some deductive inferences about truth that you accept as good are invalid, and that some that you reject as bad are valid.  (For other classical truth theories the situation isn’t quite as extreme, but there too there will be a divergence between good reasoning and reasoning that is “valid” in the sense of necessarily truth-preserving.)

There is still some connection of validity to truth-preservation, but exactly what the connection is depends on one’s solution to the paradoxes of truth.  On my own view, the necessary truth preservation account gives the right answer for 0-premise inferences: a schema for sentences is valid precisely when all its instances are true by logical necessity.  (I’m ignoring the logic of indexicals here.)  For inferences with premises, we have a one-way connection: logically necessary truth preservation suffices for validity.  In the reverse direction, there are more qualified things to say: for instance, if an inference is valid, there can’t be any cases where the premises are clearly true and the conclusion clearly not true.  I’d think that these things give all the connection between validity and truth preservation that we ought to want.

Truth and the Absence of Fact

RM:      Given that the disagreement over logic for you is primarily about regulating belief about other matters, can the logics in disagreement be objective?

HF: The idea of objectivity (whether in logic or elsewhere) is tricky.  There are cases where two views differ as to what inferences preserve truth in which it isn’t obvious that the issue between the logics is objective.  For instance, there’s a good case to be made that the issue between “paraconsistent dialetheic” and “paracomplete” approaches to the semantic paradoxes in a language without conditionals or restricted universal quantifiers isn’t objective, despite the former accepting both the Liar sentence and its negation (and thinking them both validities in the logic of truth) and the latter rejecting them.  That’s because the difference seems to be just a matter of optional norms of acceptance: acceptance for one of these people is rejection of the negation for the other, and rejection for one is acceptance of the negation for the other.  (Once you add a conditional or restricted quantification and assume natural rules for them that both parties are likely to accept, the duality disappears.)

There’s also what I see as a rather boring sort of non-objectivity, where one party simply adopts goals for validity that the other rejects: e.g. where one insists on a relevance requirement that the other sees no point to.

I am nonetheless of the opinion that a great many logical disputes are objective, despite the failure of the definition of validity in terms of truth preservation.  (This includes many disputes of central philosophical importance, such as those concerning the semantic paradoxes or the paradoxes of property theory and those concerning the logic of vagueness.)  That they can still be objective shouldn’t be surprising, given that there are close connections between validity and necessary truth-preservation, even though no identity.

RM:   Does it follow that there is a uniquely correct logic?

HF:  I’m a bit hesitant to say that.  A boring reason why not is that there seems no issue between those who regard the laws of identity as part of logic and those who accept those laws but regard them as non-logical.  Another instance of the same sort is that someone might advocate classical logic as always correct (including for the semantic paradoxes, vagueness and so forth) but nonetheless regard a logic that restricts excluded middle as “basic” because she must use it in carrying out debates with those who reject it.  It may or may not be a good idea to regard excluded middle as “non-basic” on such grounds; but be that as it may, taking it to be non-basic seems compatible with accepting all instances of excluded middle as non-logical truths.  Somewhat similarly, an advocate of a logic without excluded middle who rejects some instances of it might concede that in some domains (perhaps number theory or even set theory) excluded middle is unproblematic; this could be construed as adopting classical logic for those domains.

And I’ve mentioned another boring reason for the claim that no logic is uniquely correct: there are different possible goals for logic.  For instance, some might add in relevance requirements, others not, and the “dispute” between the two positions isn’t over facts but over goals.

A perhaps less boring reason for resisting the claim of a uniquely correct logic connects to the discussion above about whether the dispute between the paraconsistent and paracomplete theorist is objective.  There I said: it probably isn’t objective until you take the dispute to include the treatment of conditionals or restricted quantifiers.  But it needn’t be objective even after you do: the duality between their treatments of other connectives could be extended to the treatment of conditionals and restricted quantifiers, it’s just that it’s rather unnatural to do so.  (Similarly, there are possible alternative treatments of conditionals both within a paracomplete framework;  and there are possible alternatives both within a paraconsistent dialetheic framework.  Whether to regard such choices as “objective” is a somewhat delicate matter, depending on the details of the case.)

I regard a lot of the discussion of logical pluralism as frustrating because it seems to me that there are so many distinct issues that are involved; they aren’t always distinguished, and sometimes the focus is on ones that I don’t see as terribly important.

RM:  There are several considerations besides the semantic paradoxes that make it seem like we need to shift logic. Vagueness is an example. In these cases some are tempted to switch or weaken classical  logic to deal with them. So in the case of a vaguely bald chap we might try and say it’s both true and false that he’s bald, or neither true nor false that the chap’s bald. How do you say we should approach the problem of vagueness?

HF: Well, I don’t think it’s a good idea to say that it’s neither true nor false: the non-classical logic I’m tempted toward would have it that we should reject that it’s true and also reject that it isn’t true.  (It has rejection gluts, in contrast to the acceptance gluts of the dialetheist.)  The advantage of such a logic is that it allows you to say that someone with 0 hairs is bald and someone with a million hairs isn’t, while rejecting that there’s a number such that someone with n hairs is bald and someone with n+1 isn’t.  You can reject that, on my preferred logic; but you can’t accept its negation.  (I’m making the usual simplifying assumption that baldness is just a matter of number of hairs.)

I’m tempted by the use of such a non-classical logic for vagueness, but I have to confess both (a) that the argument for revision of logic in the vagueness case is weaker than in the semantic paradox case and (b) that the costs of going non-classical are higher in the vagueness case.  On (a), I don’t claim it to be obviously untenable that one hair makes the difference between bald and non-bald; it does seem uncomfortable, and leads to questions about why it seems irrational for a person who seems borderline bald to obsess about whether he’s over or under the magic threshold; but sometimes it’s best to accept an uncomfortable conclusion (in order to avoid greater discomfort elsewhere).  On (b), the greater cost is because vagueness is pervasive, and non-classical reasoning is more complicated than classical reasoning.  Insofar as we can be pretty sure that semantic paradoxes are irrelevant to a situation, then non-classicality due to the paradoxes can be ignored: in effect we accept relevant instances of excluded middle as non-logical premises, and with them we can in effect reason classically.  But if vagueness itself induces non-classicality, it’s harder to argue in quite this way.

I’m not sure how ultimately serious this worry is.  That’s because I’m inclined to think that much reasoning involving vague terms is somewhat idealized: we ignore vagueness in much the way that we ignore the curvature of space when we use Euclidean geometry in ordinary circumstances.

An appealing feature of treating vagueness non-classically is that many of the same issues about the details of the needed logic arise in both cases.  (E.g. the argument against the use of Łukasiewicz continuum-valued logic is similar in the two cases.)  Also there are paradoxes that seem to have feet in both camps, such as the paradox of uninteresting natural numbers.  (Presumably there are some, so there must be a smallest.  But that seems to be an interesting number!)

RM:   What about revenge problems for the paradoxes?

HF: This is a long story, but part of the answer connects them with a well-known alleged problem about vagueness: that any attempt to avoid a line between the bald and the non-bald simply draws lines at other places.  For both the semantic paradoxes and vagueness, the set-theoretic model theory that we use to convey an understanding of the logic does draw lines: it has to, since it’s a classical theory.  But by being classical, it can’t represent the non-classical reality entirely faithfully (and it doesn’t need to, since its role is to give to the classical theorist a sense of how the logic works, and to provide a consistency proof that the classical logician will accept).  The fundamental issues aren’t at the model-theoretic level, and for these fundamental issues you can’t draw lines.  For instance, although you can introduce a determinacy operator, and iterate it a long way through the transfinite, you can never introduce an ordinal alpha for which everything is either determinately^alpha true or not determinately^alpha true.  (And quantification over “all the alpha for which the iteration makes sense” is illegitimate.)  This is, in Sainsbury’s term, a logic for concepts without boundaries.  And without introducing boundaries, revenge arguments don’t get off the ground.

RM:   Are we to deal with all paradoxes of indeterminacy in the same way as we do with vagueness?

HF: This may depend on how broadly one uses the term ‘indeterminacy’.  I’m inclined to say that certain questions in set theory, e.g. the size of the continuum or the truth of Suslin’s hypothesis, are indeterminate.  This seems a natural way of talking either from a fictionalist perspective (on which they are like questions about what Little Red Riding Hood did the day after the wolf’s last visit) or from a pluralist perspective (in which case they are rather like questions about whether Euclid’s or Lobachevsky’s claims about parallels are correct in mathematical as opposed to physical geometry).  But indeterminacy in this sense is rather unlike the indeterminacy in vagueness or the semantic paradoxes, and in this case I think there’s no reason to weaken classical logic (except perhaps the minimal weakening involved in going supervaluationist).

RM:   Can we draw conclusions about your attitude towards truth from the way you tackle these semantic paradoxes? Are you a deflationist in some sense – and again, what are the stakes in the deflationist vs non-deflationist dispute? Would we get a sense of the issue if we looked at your account of egocentric content, which is about the representational content of sentences and intentional states which takes a deflationary view of reference and truth conditions?

HF: In my work on the paradoxes I’ve tried to avoid getting involved in the issues surrounding deflationism.  I do assume there that the truth schema (and the related principle that intersubstutition of True(<p) with p in normal contexts preserves believability) has a prima facie appeal, but I think that most non-deflationists agree with me about that.

In other work I have mostly advocated a deflationist position in recent years.  One of the reasons is that it seems hard, from a non-deflationist perspective, to explain why we adhere to the instances of the truth schema as strongly as we do: why it seems for instance to border on the incomprehensible to give a different answer to the question of whether there are electrons than to the question of whether ‘There are electrons’ as we currently use it is true, or to suppose that ‘There are dogs’ as we currently use it doesn’t have the truth conditions that there are dogs but instead has the truth conditions that there are cats.

I do though think that people working in the inflationist tradition have come up with many important insights as to the working of language.  A deflationist needs to account for these, within a deflationist framework.  Maybe in the end a deflationist position sophisticated enough to account for them and an inflationist position sophisticated enough to satisfactorily answer the questions in the previous paragraph will merge.  In the meantime, I like to stress the deflationist side of things, though maybe this is just because fewer people are taking that stance so that there’s a higher chance of coming up with something that advances the debate.

I do think it’s an advantage to my view on the paradoxes that it fits well with deflationism: it seems to me that there’s a strong prima facie oddity in a generally deflationist line that exempts paradoxical sentences from its purview.  At the very least, the exemption calls for explanation, and it isn’t obvious what that explanation would be.

Science without Numbers

RM:   Do your views about logic, meaning and truth have metaphysical implications?

HF: It’s hard for me to give a straightforward answer.  In some sense, not assuming excluded middle everywhere seems “metaphysical”.  On the other hand, one can say that the reasons for rejecting excluded middle are due to odd features of our representational system, taken to include not just our language but also our system of mental representation.  As JC Beall has noted, the truth paradoxes seem like spandrels: simply the inevitable byproduct of having a notion of truth with certain useful features.  In the case of vagueness, the reasons for rejecting excluded middle (obviously) depends on a feature in our representational systems that might be regarded as an imperfection: viz. vagueness!  I just don’t know how to answer whether this makes the failure “metaphysical”.  (The question is reminiscent of the question of whether vagueness is “in language and the mind” or “in the world”.  It’s in the world in that it’s vague whether John is bald.  It’s in language and the mind in that it’s linguistic items and beliefs, hopes, etc. that are vague.  If there’s a deeper question here, I don’t get what it is.)

RM:   There’s a view in some quarters that modern philosophy, especially in the analytic tradition, is just technical logic-chopping, and this is not friendly observation but taken to be damning critique. Yet at the same time people – often the same people -are interested in truth and meaning and questions of objectivity and factual defectiveness, the status of scientific and mathematical facts and so forth. So how would you address their concerns and criticisms – and is there a salient example you can present to illustrate your views on this?

HF: There is work in analytic philosophy that I do regard as logic-chopping: making distinctions with little general philosophical point, systematically avoiding deep issues, and so forth.  There’s a lot of work in non-analytic traditions with many of the same defects.  I’m not sure that it’s useful to generalize about work in one tradition vs. another (and anyway, I’m in no position to do so).  I do think that if people are interested in the questions you mention, the analytic tradition has insights to offer which essentially involve the use of logic.  I don’t rule out that on some of these questions non-analytic traditions might also offer insights that the analytic tradition tends to miss.  We all have to approach questions as best we can given the limitations of our training.

RM:   And finally, apart from your own, are there five books you can recommend to the readers here at 3:AM that will take us further into your philosophical world?

HF: This one is tough for me, since in recent years I’ve been more influenced by articles than by books.  On issues of the paradoxes my work started out by reflection on two articles, one widely known even among non-specialists and one well-known among specialists: Saul Kripke’s “Outline of a Theory of Truth” and Harvey Friedman and Michael Sheard’s “An Axiomatic Approach to Self-Referential Truth”.

Axiomatic Theories of Truth

I learned a lot recently from Volker Halbach’s Axiomatic Theories of Truth, which includes a fair amount of material that had I known about it would have enriched my own book on the paradoxes.

Vagueness and Degrees of Truth

On vagueness, probably the published book that I’ve enjoyed most in recent years is Nicholas Smith’s Vagueness and Degrees of Truth, though I can’t say I agree with all that much of it.

Vagueness and Thought

I’ve also been enjoying a book on vagueness by Andrew Bacon called Vagueness and Thought, though again with far less than total agreement.

Reasoning about Uncertainty

There’s plenty outside these areas too, including areas I’d like to work on but am not up to speed on: examples (somewhat randomly chosen) would be Joseph Halpern’s Reasoning about Uncertainty and

Quantum Physics Without Quantum Philosophy

Detlef Dürr, Shelley Goldstein and Nino Zanghi’s Quantum Physics Without Quantum Philosophy.



Richard Marshall is still biding his time.

Buy his new book here or his first book here to keep him biding!

End Times Series: the first 302

First published in 3:AM Magazine: Thursday, May 3rd, 2018.