Friday, 21 April 2017

Explaining the A Priori in Terms of Meaning and Essence

It wasn't just the positivists who thought there was a tight connection between meaning and truth in the case of a priori propositions:
However, it seems to me that nevertheless one ingredient of this wrong theory of mathematical truth [i.e. conventionalism] is perfectly correct and really discloses the true nature of mathematics. Namely, it is correct that a mathematical proposition says nothing about the physical or psychical reality existing in space and time, because it is true already owing to the meaning of the terms occurring in it, irrespectively of the world of real things. What is wrong, however, is that the meaning of the terms (that is, the concepts they denote) is asserted to be something man-made and consisting merely in semantical conventions. (Gödel (1951/1995), p. 320.)
Perhaps we should try to recover some insight from the idea, nowadays highly unfashionable within philosophy (but alive and well in the broader intellectual culture, I think), that a priori truths like those of mathematics are in some sense true owing to their meanings. Philosophers often used to express this by calling such propositions 'necessarily true', but since Kripke that sort of usage has been crowded out by another.
  
Noteworthy in this connection is that Kripke was not altogether gung ho about his severance of necessity from apriority:
The case of fixing the reference of ‘one meter’ is a very clear example in which someone, just because he fixed the reference in this way, can in some sense know a priori that the length of this stick is a meter without regarding it as a necessary truth. Maybe the thesis about a prioricity implying necessity can be modified. It does appear to state some insight which might be important, and true, about epistemology. In a way an example like this may seem like a trivial counterexample which is not really the point of what some people think when they think that only necessary truths can be known a priori. Well, if the thesis that all a priori truth is necessary is to be immune from this sort of counterexample, it needs to be modified in some way. [...] And I myself have no idea it should be modified or restated, or if such a modification or restatement is possible. (Kripke (1980), p. 63.)
This may make it sound like the required modification would consist in somehow ruling out the problematic contingent a priori truths from the class of truths whose epistemic status is to be explained. But Chalmers' idea of the tyranny of the subjunctive suggests another route: try instead to find a different notion of necessity - indicative, as opposed to subjunctive, necessity; truth in all worlds considered as actual, rather than truth in all worlds considered as counterfactual - better suited to the explanation of apriority.

Now, in Chalmers' epistemic two-dimensionalist framework, indicative necessity is itself explained in epistemic terms. But if we try for a more full-bloodedly semantic conception of it, we may get something more explanatory of the special epistemic status of a priori truths. The notion we are after is something like: a proposition is indicatively necessary iff, given its meaning, it cannot but be true. And the modality here is not supposed to be epistemic.

But what aspect of its meaning? Sometimes 'meaning' covers relationships to things out in the world, and even the things out there themselves. What we are interested in is internal meaning. Putnam's Twin Earth thought experiment - though this is not how he used it - lets us see the distinction we need here. We want to talk about meaning in the sense in which Earth/Twin Earth pairs of propositions mean the same. This can be articulated using the middle-Wittgenstein idea of the role an expression plays in the system it belongs to (see Wittgenstein (1974, Part I)).

So, what if we say that a proposition is indicatively necessary iff any proposition with its internal meaning must, in a non-epistemic sense, be true? Can indicative necessity in this sense be used to explain apriority?

Maybe not, since there are indicatively necessary truths which are indicatively necessary only because their instantiation requires their truth. Example: language exists. (Language is here understood as a spatiotemporal phenomenon.) This is indicatively necessary, because any proposition with its internal meaning must be true, if only because the very existence of that proposition requires it to be true. Its truth comes about from the preconditions for its utterance, but - you might think - not from the internal meaning itself. It is interesting to note that it is indicatively necessary, but it lacks the special character of a priori propositions whereby they, in some sense, don't place specific requirements on the world.

This situation pattern-matches with Fine's celebrated (1994) distinction between necessary and essential properties. Socrates is necessarily a member of the set {Socrates}, but that membership is not part of his essence, since it doesn't have enough to do with Socrates as he is in himself. Likewise, he is necessarily distinct from the Eiffel Tower, but this is no part of his essence. So let us throw away the ladder of indicative necessity and instead hone in on the notion of essential truth. A proposition is essentially true iff it is of its internal meaning's essence to be true (i.e. to be the internal meaning of a true proposition).

Thus, with encouragement from Gödel and Kripke, we can develop ideas from Chalmers, Putnam, Wittgenstein, and Fine, to yield:

To say that a proposition is a priori is to say that it can, in some sense, be known independent of experience. (You may need experience to get the concepts you need to understand the proposition, but you don't need any particular further experience to know that the proposition is true.) What is distinctive about these propositions which explains their being knowable in that peculiar way? It is that their internal meanings - their roles in language - are, of their very essence, the internal meanings of true propositions; any proposition with that internal meaning must be true, and not for transcendental reasons relating to the pre-conditions of the instantiation of the proposition, but as a result of that internal meaning in itself.

So we can have an account of apriority which explains it in terms of a tight connection between meaning and truth, freed of its accidental associations with conventionalist and deflationary views about meaning, modality and essence.

This is not to say that a priori propositions' truth is to be explained in a case by case way by considerations about meaning and essence. That would be to crowd out the real mathematical justifications of non-trivial mathematical truths. But explaining apriority in general in this way wards off misunderstandings which come from treating a priori truths too much like empirical truths. And that is what makes it an explanation.

References

Chalmers, David J. (1998). The tyranny of the subjunctive. (unpublished)

Fine, Kit (1994). Essence and modality. Philosophical Perspectives 8:1-16.

Gödel, Kurt (1951/1995). Some basic theorems on the foundations of mathematics and their implications. In Solomon Feferman (ed.), Kurt Gödel, Collected Works. Oxford University Press 290-304. (Originally delivered on 26 December 1951 as the 25th annual Josiah Willard Gibbs Lecture at Brown University.) 

Kripke, Saul A. (1980). Naming and Necessity. Harvard University Press.

Putnam, Hilary (1973). Meaning and reference. Journal of Philosophy 70 (19):699-711. 

Wittgenstein, Ludwig (1974). Philosophical Grammar. University of California Press.

Modern Quantificational Logic Doesn't Subsume Traditional Logic

It seems to be a received view about the relationship of traditional Aristotelian logic to modern quantificational logic that the inferences codified in the old-fashioned syllogisms - All men are mortal, Socrates is a man, etc. - are all, in some sense, subsumed by modern quantificational logic. (I know I have tended to assume this.)

But what about:

P1. All men are mortal.
C. Everything is such that (it is a man  it is mortal)?  

This is a logical inference. It is not of the form 'A therefore A'. It embodies a very clever logical discovery! P1 and C are not the same statement. Talk of 'translating' the former by means of the latter papers over all this.

Modern quantificational logic does not really capture the inferences captured by traditional logic, any more than it captures this link between the two. It does capture inferences which, given logical insight, can be seen to parallel those codified by traditional logic, but that is not the same thing.

Monday, 3 April 2017

Reflections on My Claim that Inherent Counterfactual Invariance is Broadly Semantic

This post presupposes knowledge of my account of subjunctive necessity de dicto as expressed in my thesis and in a paper derived from it which I have been working on. (I hope my self-criticism here doesn't cause any should-have-been-blind referees to reject the paper. A revise-and-resubmit verdict I could live with.) Here I try to take a next step in getting clear about the status and significance of the account.

In my thesis and derived paper, I propose that a proposition is necessary iff it is, or is implied by, a proposition which is both inherently counterfactually invariant (ICI) and true, and explicate this notion of ICI.

I claim that ICI is broadly semantic, and put this forward as a key motivation and virtue of the account. I don’t provide much argument for this claim - the intention, I suppose, was that this would just seem self-evident. But I have become increasingly aware of the importance of the fact that this could be challenged, and the importance of getting clearer about the underlying primitive notion of a genuine counterfactual scenario description (CSD).

I do provide one reason, near the end of my presentation of my account, for thinking that ICI is broadly semantic given my preferred approach to propositions and meaning. But there are two reasons for wanting more. One is that it may be hoped that my claim that ICI is broadly semantic could be justified independently of my particular approach to propositions and meaning, where I advocate understanding what I distinguish as the ‘internal’ component of meaning as role in language system. A second, perhaps more suggestive, reason for wanting more is that, even given my preferred approach, the argument I give is basically this: ICI is explained in terms of how a proposition - its negation, really - behaves in certain contexts - namely CSDs. But here of course I have to single out genuine CSDs.

And here’s the thing. (At least, the following seems to be right.) For my claim that ICI is broadly semantic to hold water, the notion of genuineness of a CSD had better be broadly semantic. For it is not enough for a notion to be broadly semantic that it can be characterized in terms of appearance in certain sorts of linguistic context C, where C-hood is blatantly extra-semantic. For instance, we may say a proposition has G iff it (or its negation, to make this more like the ICI case) doesn’t appear in any description which has the property of being written in some notebook I have in my room. In that case, it is plain that whether or not a proposition has G is not a matter of its meaning or nature.

So, I now think that the little argument I give at the end of my presentation of my account, about how my particular approach to propositions and meaning ‘fits well’ with the notion of ICI as broadly semantic only goes so far, and that as an argument that given my particular approach the notion of ICI is or should be seen as broadly semantic, it is weak, since it gives no reason to think that the all-important notion of genuineness of CSD is broadly semantic.

Further, I think it is clear that I want to put forward my account, and I think the account has theoretical value, independent of whether a case can be made that genuineness of CSD is broadly semantic. And so my whole presentation of why my account is interesting and of its motivation is somewhat crude. As a story about what caused it, and the specific things I was thinking, it may have some interest. But as a way of situating the theory and giving a sense of what its value (within philosophy) consists in, it is crude and not really to the point. I do of course hint at other sources of interest (e.g. that the account clarifies the relationship between the notion of necessity and those of truth and implication), and don’t rest everything on the ‘semantic hunch’, but I do perhaps give it too prominent a place - or at least, an incompletely justified place.

So, is the notion of a ‘genuine counterfactual scenario description’ broadly semantic? And what does it mean to be broadly semantic? I may follow up with a post addressing these questions more thoroughly, but for now a couple of remarks. Whatever it is to be broadly semantic, it is not to be conventional in any sense. The idea is perhaps better gotten at, in some ways, by saying that genuine CSD-hood is a conceptual matter. But really I need to roll up my sleeves and investigate this more closely - it is not merely a question of hitting on some formulation. Finally, I propose that the following passage from §520 of Wittgenstein’s Investigations seems very to-the-point when it comes to the questions and difficulties I find myself coming up against here, and may help me plumb the depths of the matter:
So does it depend wholly on our grammar what will be called (logically) possible and what not,—i.e. what that grammar permits?”—But surely that is arbitrary!—Is it arbitrary?—It is not every sentence-like formation that we know how to do something with, not every technique has an application in our life [...].