I have been working on a paper about the notion of logical truth as a bit of a side project and it’s now at the state where I could use some input from people who know their logic better than I do (which is not much to ask). The draft is available here. I had a chat about this with Oswaldo Chateaubriand at the ENFA 4 conference in Portugal in September, and I have been reading parts of his massive book on logic since then. The book is nearly impossible to get now, but apparently there should be reprints coming out soon. Anyway, I am very sympathetic towards Chateaubriand’s conception of logic, which takes logic to be a part of metaphysics, and the key topic here is no doubt the notion of logical truth.

The purpose of the paper is to defend a metaphysical interpretation of logical truth as opposed to a linguistic one. I take my cue from Davidson’s classic paper ‘In Defense of Convention T’, where he distinguishes between absolute truth and truth in a model or relative truth. These two senses of truth can be seen to represent a metaphysical and a linguistic interpretation of logical truth and it is in my interest to see whether they can be reconciled. The notion of absolute truth is the one familiar from Tarski’s T-schema: ‘Snow is white’ is true if and only if snow is white. Truth in a model, or relative truth, instead of being a property of sentences as absolute truth would appear to be, is evaluated in terms of the relation between sentences and models, where ‘models’ is interpreted in a wide sense.

Now, drawing on the work of John Etchemendy, I propose that we can reduce truth in a model to absolute truth, when ‘truth in a model’ is defined as follows:

Once we have specified the class of models, our definition of truth in a model is guided by straightforward semantic intuitions, intuitions about the influence of the world on truth values of sentences in our language. Our criterion here is simple: a sentence is to be true in a model if and only if it would have been true had the model been accurate –- that is, had the world actually been as depicted by that model. (J. Etchemendy, The Concept of Logical Consequence, Cambridge, Mass.: Harvard University Press. (1990), p. 24.)

It seems indeed that once we have specified the class of models, we can define truth in a model according to Etchemendy’s suggestion, but now the problem concerns the delimitation of the class of models. A crucial question is what does the introduction of modality to this picture imply: it must be the case that the model could have been true, that is, it must be the case that the world could have turned out to be like the model depicts. Etchemendy talks about genuinely possible configurations of the world (ibid., 25), but he does not say much about the nature of the modality involved. My suggestion is that the modality in question must be metaphysical modality, as that will be the only type of modality that guarantees the needed connection to the world. I talk about this in much more detail in the actual paper.

What ties this to contemporary discussion is the question concerning the implications of the metaphysical interpretation of logical truth for the recent discussion about logical pluralism and the normativity of logic. I briefly discuss Hartry Field’s two papers about these matters, namely his ‘Pluralism in Logic’ and ‘What is the Normative Role of Logic’. I think that once the metaphysical interpretation of logical truth is adopted, there cannot be any very interesting sense of logical pluralism since the view effectively implies monism about logic, that there is a One True Logic — here I agree with Field — but according to Field, there is another question that we must consider, namely whether pluralism about logic could be seen as pluralism about the normative content of logical consequence, in the sense of epistemic normativity.

Well, epistemic normativity is something that I do not want to go into in any detail, but instead of the expressivist account that Field defends, I try to make a case for the normativity of logic which is compatible with the metaphysical interpretation of logical truth. I think that we can cash out the normative role of logic by asking how could it not be better in a normative sense to reason according to the constraints of reality? A more technical case for this can be constructed by analysing Field’s notion of genuine validity; this is something that, according to Field, a classical logician and an intuitionist, for instance, will have to share to have a debate in the first place. But if this is indeed the case, then genuine validity will have to be very close to something like a metaphysical notion of truth.

Indeed, Per Martin-Löf, the developer of intuitionist type theory, takes validity to be ‘nothing but the notion of truth or reality applied to the particular acts and objects with which we are concerned in logic’ (‘Truth of a Proposition, Evidence of a Judgment, Validity of a Proof’, Synthese 73, 1987, p. 419). So the notion of validity is fundamental, but not necessarily in the sense that Field suggests. In fact, Martin-Löf suggests that this fundamental notion of validity connects directly with what he calls the metaphysical notion of truth, as opposed to the truth of a proposition, so the case for connecting the normativity of logic and the metaphysical interpretation of logical truth is straight forward.

See my paper for further discussion.