Thursday, 1 October 2015

An Account of Necessity as an Attribute of Propositions

I hope to say more about this in future, making the account more perspicuous and better defending it, but it is high time I made a blog post about it. My view that this account is correct has been stable since 2012. It fits with my account of propositions and could also be adapted to various other conceptions of propositions and meaning (but not all).

UPDATE 3/10/15: I have made some additions regarding the meaning of 'counterfactual scenario description'.

Some related posts:

A proposition is necessary iff it is, or is implied by, a proposition which is both inherently counterfactually invariant and true.

A proposition is inherently counterfactually invariant iff, if it is held true, it is held fixed across counterfactual scenario descriptions, i.e. its negation does not appear in any counterfactual scenario descriptions.

(I say 'its negation does not appear in any' rather than 'it appears in all' because counterfactual scenario descriptions don't normally deal with everything - are not normally maximal.)

Whether or not a proposition is inherently counterfactually invariant is a matter of the internal meaning of that proposition.

When I speak of 'counterfactual scenario descriptions', I mean not just those actually produced, but those which can be produced. Thus there is an unreduced modal element in my analysis of de dicto necessity.

Not everything which we could call a description of a scenario, where the scenario in question does not in fact obtain, counts as a counterfactual scenario description. If we say, for example, 'Perhaps things are actually such that ...', what goes in the blank is not to be counted as a counterfactual scenario description, even if it is a description of a scenario which does not obtain. Rather, I am talking about descriptions which, so to speak, describe a scenario as counterfactual - e.g. 'Things could have been such that ...'. This is like the distinction two-dimensional semanticists emphasize, between considering a scenario as actual vs. considering it as counterfactual. When a description is made of a scenario which is being considered as counterfactual, then that is a counterfactual scenario description.

We must distinguish between genuine and non-genuine counterfactual scenario descriptions. In the case of the latter, we may always say that the meaning of at least one of the expressions involved is being violated or departed from. (For example, if we suppose that 'Cats are animals' is necessarily true, and yet speak loosely of a counterfactual scenario in which there are robot (and thus non-animal) cats. Here we can either say that we are using 'cat' in a different meaning entirely, in which case the counterfactual scenario description may be genuine, or are beginning with the primary meaning but, as it were, stretching it, in which case we have a non-genuine counterfactual scenario description. Another example: if Euler had squared the circle, he would have been famous for it. The description in the antecedent of what Euler does should be regarded as a non-genuine counterfactual scenario description. I take this notion as primitive, and think it is vague.

Note that it is not the case that a counterfactual scenario description is genuine iff the scenario it describes is metaphysically possible. For instance, if I believe that Hesperus is not Phosphorus, then if I talk about a case in which 'Hesperus had been Phosphorus', this will be a non-genuine counterfactual scenario description, even though it is metaphysically possible, indeed actual, for Hesperus to be Phosphorus. Likewise, if I - again, believing Hesperus not to be Phosphorus - talk about a situation in which Hesperus and Phosphorus are distinct (as I believe them to be), this may be a genuine counterfactual scenario description, even though it is metaphysically impossible for Hesperus to be distinct from Phosphorus.

It cannot be denied that the notion of a genuine counterfactual scenario description, and in turn that of inherent counterfactual invariance, have much of the character of the notion of necessity de dicto. Still, as we have just seen, they behave quite differently. So it is anything but trivial to see that we can put these notions together with those of truth and implication to yield a statement of the conditions under which a proposition is necessary de dicto.

We may also delineate the inherently counterfactual scenarios in another way: they are those which are such that it is a priori that they are necessary if true. I do not think we should think of this as giving the content of the notion, however. It is another way to get a handle on the relevant class of propositions, which may help us to get the notion.

To see why closure under implication is required, consider any disjunction of a necessary truth with a contingently true or false proposition. Such a disjunction will of course be necessary, but it will not be inherently counterfactually invariant, since it can be held true by holding the contingent proposition true and the necessary one false.

My analysis gets the right answer on such a case, since the proposition will be implied by a proposition which is both inherently counterfactually invariant and true - in the simple disjunction case, the necessary disjunct. However, note that the relevant implier will not always be a part of the proposition in question: consider 'Everything is either such that it is either not a cat or is an animal, or such that it is either less than 100 kilograms in weight or not in my room'. This is in fact necessarily true, since all cats are animals and that is a necessary truth. But you might hold it true if you disbelieve that all cats are animals, by believing that nothing in the speaker's room weighs more than 100 kilograms. If that is how you held it true, you would let its negation appear in counterfactual scenario descriptions - namely, descriptions of scenarios in which I have something heavy in my room.

It is very important to see that counterfactual scenario descriptions do not act as "possible worlds" in my account. One easy way to see this is to consider someone who falsely believes a proposition whose negation is necessarily true a posteriori. Examples: 'Hesperus is not Phosphorus', 'Cats are robots', 'Hesperus is Mars'. Such a person will be in a position to produce counterfactual scenario descriptions involving these propositions, despite them being not only false but impossible. My account filters these out from being classed as necessary by means of the truth requirement.

To get a better grip on the role the notion of inherent counterfactual invariance plays in my account, compare Sider’s account on which there is something list-like and arbitrary at the core of the notion of necessity de dicto. The account is given in Writing the Book of the World, but is also rehearsed in ‘Symposia of Writing the Book of the World’ (which has the benefit of being freely available at

According to this account, for a proposition to be necessary is roughly for it to be a logical consequence of a certain class of propositions, the “modal axioms”. Modal axioms come in different sorts, including mathematical truths, analytic truths (under a certain conception of analyticity), “laws of metaphysics”, and “axioms of a metaphysical semantics”. The account is a highly “defl ationary” one in that no metaphysically deep condition is given to unite all the modal axioms. They are given by a mere list (mathematical truths, analytic truths, …), which is selected, so to speak, “by us rather than by the world”—perhaps by linguistic convention.
Once you realize that the “modal axioms” Sider is talking about are all truths, you can see that his account shares a structure with mine: a proposition is necessary if it is or is a consequence of a true proposition fulfilling come condition C. Realizing that something of that form is correct is an important step. And I think my account is preferable to Sider’s because I have something more substantive to say about what the condition C is, such that it is in not in any relevant sense arbitrary whether a proposition has it or not: it is inherent counterfactual invariance. (Of course, if the notion of necessity de dicto seems arbitrary or conventional to you, you might prefer Sider’s account.)

One of the attractive things about my account, I think, is that it does not try to reduce necessity de dicto to non-modal notions. For one thing, that may not be possible. For another, it seems it isn't necessary for an informative analysis of necessity de dicto. By not trying to reduce the modal to the non-modal, my account has more of a chance of being true and insightful. Also, it seems to me to be attractively simple and elegant, while still having enough structure, and involving the hitherto unfamiliar but natural notion inherent counterfactual invariance, so that it is understandable why it was not immediately obvious once the notion of necessity de dicto was clearly isolated by Kripke.

Saturday, 19 September 2015

Two Opposite Types of Granularity Difference

This is another post in my series on semantic granularity. The others so far, in chronological order, are:

John and Mary both use the word 'happiness', and understand each other perfectly in most conversations. They use it in propositions in such a way that we would say that they both attach the same meaning to the propositions used, are on the same page, etc. But in certain relatively peripheral regions of application, they differ systematically.

We want to be able to tie John's peripheral uses of 'happiness' together with his non-peripheral uses - we want to say he's using the word in the same sense in both cases, but we also want to bundle his non-peripheral uses with Mary's. However, we also want to be able to make a semantic distinction between Mary's peripheral use and John's.

The solution is two operate at two granularities. A relatively coarse-grained bundling can tie all John and Mary's uses together - roughly, by ignoring peripheral use features. But we can use a more fine-grained bundling to describe the linguistic difference between Mary and John - at this granularity, we say they mean slightly different things by 'happiness'.

But there is another sort of descriptive problem which we solve with granularity shifts. For example, consider the normal uses made of the word 'hard' in the phrases 'hard man' (meaning something like 'tough guy'), 'hard wood' and 'hard test' (meaning a difficult test). We can distinguish three different senses here - very roughly, (i) toughness, (ii) solidity and resistance, and (iii) difficulty. Or two, a literal sense ('hard wood') and a "metaphorical" sense ('hard man', 'hard test'). Or we can bundle all these together.

Visually, we can think of the first kind of fine-graining as a kind of broadening of considerations which go into bundling, and the second kind of fine-graining as a kind of narrowing.

This is perhaps one of the things which have made granularity considerations, although quite natural, seem difficult - or not even arise as a serious possibility - from the point of view of analytic philosophy. That two different, in a sense opposing, things can be going on in shifts toward finer granularity, can make the matter confusing. But once we see what is happening and master it, it just reveals the richness and power of the approach.

From these examples, it may look like the first, 'broadening' type of fine graining is concerned with intersystematic distinctions – distinctions between different sign systems (in the example above, John's and Mary's idiolects) – and that the 'narrowing' type is concerned with intrasystematic distinctions – between elements and uses of some given sign system.

But this doesn't generally hold. In the broadening case, for example, we may have spoken about two very similar but subtly distinguishable meanings of quite different words in, say, John's idiolect. 'Envy' and 'jealousy', perhaps. Or 'rage' and 'fury'. Likewise, in the narrowing case, where we, at finer grain, distinguish the meaning of 'hard wood' from 'hard test', we could instead make that distinction between these words as used by two different people, or two different words used by two different people.

One thing we can say, perhaps, is that the 'broadening' type of fine graining is about considering how more ground is covered, factoring in more stuff about 'the lay of the land', where the 'narrowing' type is more about marking off different regions. The first involves making more distincions among expression-uses based on how the expressions cover the ground they cover, the second involves making more distinctions among expression-uses based on what ground they are covering in that use.

Tuesday, 8 September 2015

Forthcoming in Logos & Episteme

My paper 'Two New Counterexamples to the Truth-Tracking Theory of Knowledge' is forthcoming in Logos & Episteme. It derives from this blog post. The final draft is available at PhilPapers.

An interesting point about its origin: I was originally playing with what I thought might be a type of counterexample to the truth-tracking account involving weird self-referential propositions. After investigating for a stretch I concluded that the approach was no good, at which point the counterexamples in the present paper (which have nothing to do with self-reference) came into my head. Something about the disappointment at the weird self-referential approach failing, together with the fact that I had during the investigation started to get used to the idea that I was able to refute the truth-tracking theory, caused me to think of the actual counterexamples.

For another recent counterexample to the truth-tracking theory (which also works against some other theories) see Neil Sinhababu and John Williams's paper 'The Backward Clock, Truth-Tracking and Safety' and Sinhababu's blog post about it.

Saturday, 15 August 2015

The Principle of Compositionality and Semantic Granularity

This is another post in my series on semantic granularity. The others so far, in chronological order, are:

If meanings properly get carved up at different granularities, as I maintain, what are the implications for the 'the principle of compositionality'? I believe that granularity considerations can shed light on the status and application of this principle, and clear up much of the confusion surrounding it.

This confusion appears to be considerable. Witness Daniel Cohnitz ('Is Compositionality an A Priori Principle?'):

A superficial look at the literature on the principle of compositionality [...] could suggest that the discussion is as confused as a discussion can be.

I will now quote some classic and some typical formulations of the principle, and then indicate what we can say about it in light of granularity.

From the Tractatus:

I conceive the proposition—like Frege and Russell—as a function of the expressions contained in it. (3.318)

To understand a proposition means to know what is the case, if it is true.
(One can therefore understand it without knowing whether it is true or not.)
One understands it if one understands its constituent parts. (4.024)

It is essential to propositions, that they can communicate a new sense to us. (4.027)

Frege, in a letter to P.F. Jourdain, probably written in 1914:

The possibility of our understanding [my emphasis] propositions which we have never heard before rests evidently on this, that we construct the sense of a proposition out of the parts that correspond to the words.

Theo Janssen, 'Compositionality':

The principle of compositionality reads, in its best known formulation:
The meaning of a compound expression is a function of the meanings of its parts.

But this omits something. The way the parts are put together, not just their meaning, goes into determining the meaning of the whole. ('John loves Mary' means something different from 'Mary loves John'.)

This is the point made by 3.141 of the Tractatus:
The proposition is not a mixture of words (just as the musical theme is not a mixture of tones). 
The proposition is articulate.

The following instances do not omit this:

B.H. Partee, handout for Ling 310 The Structure of Meaning, Lecture 1, February 20, 2006 p.1:

The Principle of Compositionality: The meaning of an expression is a function of the meanings of its parts and of the way they are syntactically combined.

'Compositionality', The Stanford Encyclopedia of Philosophy:

[T]he meaning of a complex expression is fully determined by its structure and the meanings of its constituents.

We have now seen the principle of compositionality stated in various ways.

One idea I want to suggest is that, at a maximally fine granularity, the principle of compositionality can be thought of as guaranteed to hold – a blanket, a priori principle.

However, once we relax the granularity, compositionality can begin to fail in many cases.

This would give us an avenue by which to approach the confusion about whether the principle of compositionality is just a conceptual truth about how all languages – everything we would call a language – must work, or an ideal which is sometimes met and sometimes not, or just something which occurs sometimes (i.e. for some languages, or parts of languages) and not others, without necessarily being an ideal: from a fine-grained point of view, it can be construed as something which applies without exception, but from courser-grained points of view, we can make distinctions between cases where it holds and cases where it doesn't. (The stuff about being an ideal versus not being an ideal can then be considered using this as a basis.)

The proposal I just indicated – that at a maximally fine granularity, compositionality can be thought of as holding a priori and across the board – while it has a certain theoretical appeal, is problematic: it might seem straightforwardly false (in light of the existence of idioms, for example), but of course replies can be made to that (for example, that semantically, idiomatic phrases have less, or just different, parts or constituents than they would appear to have considered literalistically). I do not want to be dogmatic about this, and I suspect there is room for “having it both ways” by means of careful disambiguation.

Some of the points I want to make here can be made without any such strong principle (that is, without the proposal above – call it 'the blanket proposal'). I will therefore do that first, and then come back and reconsider the blanket proposal.

Compositionality Can Depend on Granularity

The proposal summed up in this heading is weaker than the blanket proposal (the proposal that granularity holds a priori across the board at maximally fine granularity but can fail in cases as the granularity is coarsened): all that is claimed is that some complex expressions are such that compositionality may be said to hold of them, given a certain granularity of individuation of their components' meanings, but also such that compositionality then fails to hold at a coarser granularity.

Consider this bit of dialogue from season 2 of Flight of the Conchords:

Murray: Now, we've known each other for quite some time in the professional realm. I'd like to push things forward in the friendship realm.

Jemaine: What's the friendship realm?

Murray: Well, you've heard of a realm?

Bret: Mm.

Murray: Yep?

Jemaine: Yes.

Murray: Well this is like a friendship one.

What makes this sequence is the way Murray's compositional "explanation" of 'the friendship realm' fails drastically. Is this, then, a “counterexample to compositionality”? Not really; or at least, to say that would not be to tell the whole story.

I think we can take two views of this case, and many others like it: a compositional/fine-grained view and a non-compositional/course-grained view.

On the first conception, we think: Murray means something here, he has put together a meaningful proposition. 'Friendship' and 'realm' here are clearly functioning in a way that is continuous with and related to a great many other occurrences of these words. Therefore, whatever this proposition means, that 'friendship' and 'realm' contribute in the way they do, or that they appear here in the way they do, in a proposition with this meaning, is part of – or an aspect of – the meaning, or the use, of these words (given fineness of grain).

Taking this viewpoint does not commit us to saying that these fine-grained component meanings were fully present before the phrase 'friendship realm' was ever used: depending on the details of the case, we could say that some people attached those meanings to the component terms already and some attached others (but that the difference never showed up, or slightly, but not in such a way as to prevent mutual understanding), or that no one used the words with exactly those meanings before the phrase was used, but that they were spontaneously arrived at when the phrase was coined (and understood, if we change the case and suppose it was understood by Jemaine and Brett, as would be more likely in real life) – that is, the meanings of 'friendship' and 'realm' were spontaneously and slightly extended.

On the second conception, by contrast, we think: 'friendship' and 'realm' are familiar words, and they were both presumably around for a long time before 'friendship realm' was ever used. Putting these two meaningful words together, however, doesn't all by itself yield one and only one possible meaning. There is room for going different ways here. ('The friendship realm', for instance, might be used differently from how Murray uses it, to mean some mythical realm in which everyone is friends.) Of course, when 'friendship realm' does end up meaning some particular thing, this will not be unrelated to 'friendship' and 'realm' – they guide the meaning, without determining it. The gap between what they provide and the meaning of 'friendship realm' must be filled by semantic developments pertaining to 'friendship realm'. But we needn't say that these developments affect the meanings of 'friendship' and 'realm' taken by themselves – we can maintain that they haven't taken on new meanings.

(It may be that compositionality doesn't necessarily fail according to this second conception: perhaps the gap between what the pre-existing meanings of 'friendship' and 'realm' and a meaning for 'friendship realm' can be thought of as being filled by a structure or mode of composition, if we construe this as going beyond surface syntax.)

Now, what does this get us, i.e. what does it get us to see that there is all this room for manoeuvring here?

One thing it gets us is a way of explaining a large class of putative counterexamples to the principle of compositionality, without denying any natural point of view. In fact, we have before us two ways of accounting for the idea that this is a counterexample to compositionality. Rather than two equally good ways of accounting for exactly the same thing, I think they cover different sorts of cases, different instances of the idea that this is a counterexample to compositionality; compositionality can be said to fail here in two senses:

(1) While compositionality may be said to hold at a very fine granularity, it can be said not to hold at a coarser one. So, one thing that might be going on when we think we have a counterexample to compositionality is that we are operating at a granularity at which it is a counterexample – and if someone disagrees, we might be talking at crossed granularities.

(2) A dynamic, temporal sense. We make no bones about the fact that now we have some new complex expression, we can say that the meanings of its parts, plus its structure, determines the meaning of the whole. (Thus we are operating at the finer granularity.) But we insist that when it was introduced, its meaning wasn't so determined: rather, it induced spontaneous semantic development. So, we can call a use of language non-compositional in this sense if it involved such development – and we may relativize this to a thinker or speaker, a listener, etc. Something compositional for me might be non-compositional for you at first, but my saying it induces spontaneous semantic development on your side so that it becomes compositional after the fact. We might use 'dynamic compositionality' for this idea; when spontaneous semantic innovations occur involving complex expressions, we can say that those expression uses were dynamically non-compositional.

Neither sense conflicts with the claim that, at fine-grain and timelessly, 'friendship realm' and similar cases are perfectly compositional.

These considerations may also shed some light on the issue of whether, or in what circumstances, compositionality is to be regarded as an ideal to be striven towards, if not attained. In some cases, we may have as an ideal that no spontaneous semantic innovation – no guesswork, we might say, from an interpreter's perspective – be required for a certain language-game (use of language), for example in parts of science, certain personal encounters, or in parts of legislature. That is, dynamic compositionality may be desired. But it is just as clear that sometimes we want dynamic non-compositionality.

Or we might want a language, or a part of language, to be so simple that there is no room for granularity shifting: each expression has a couple of clear rules attached to it, and if you change any of them, it's just not natural to say that the meanings stayed the same. We might be able to ensure this by laying it down that compositionality must not fail. (If we're aware of granularity considerations, we might say 'must not fail at any reasonable granularity', but we need not be aware of them to pull off the trick.)

So, the above considerations get us a few things. What remains?

We have seen that many cases where compositionality seems to fail – such as Murray's use of 'friendship realm' – may be explained away by saying: if you individuate the components' meanings at a finer granularity, this is no longer a counterexample. But the question remains: can that be done in every case? That is, does the blanket proposal hold?

By getting clearer about the blanket proposal, we will get clearer about the nature or meaning of the principle of compositionality. In this connection, we should consider idioms, metaphors and similes, sarcasm and the like, and semantic “outgrowths” like 'Wednesday is fat' and 'The letter a is yellow'.

Such linguistic phenomena also suggest that there may be more to say about the notion of compositionality as an ideal.

The Blanket Proposal

A proponent of the blanket proposal might say that, when compositionality fails, we have decided to bundle together as having one meaning expressions, or possible occurrences of expressions, which on a more fine-grained conception would be regarded as having (perhaps only slightly) different meanings. But is this really plausible in every case?

Here are some difficult types of cases:

Idioms: Consider phrases like 'spitting image', 'dead ringer', 'nest egg', 'piece of cake', 'funny farm', 'loose cannon', 'no dice', 'from scratch', 'kick the bucket'.

Propositions like 'He is a loose cannon'

'Pull strings' a bit more flexible. Frozen metaphor.

Sarcasm and the like: A sarcastic utterance of 'That's just great' may seem to be a kind of counterexample to compositionality: what is meant is that something is terrible, but this depends not just on the meanings of the components and how they are put together, but also on the context: it might be meant non-sarcastically.

But we can invoke an intensional (internal semantic) analogue of Kripke's distinction between speaker's reference and semantic reference here, and insist that what this expression means, even in the sarcastic use, is that the thing in question is very good, even though what the speaker means by it is something else.

From the point of view of expression-meaning, then, sarcasm and the like do not threaten compositionality in the least. But there is nothing stopping us talking about compositionality in connection with speaker-meaning, and saying that it fails in this case ('That's just great').

Or, we may say that what the speaker means by the whole is determined by what the speaker means by the parts, together with their mode of composition, by maintaining that by 'great' the speaker means 'terrible'.

But other cases seem different. Suppose someone takes something to be very obvious, but, by way of parody of some other group who may doubt it, might sarcastically say, 'Of course, empirical studies may prove me wrong' (let's suppose they're quite nerdy). Here, the meaning is something like 'Come on, we know this!', but the sarcasm cannot be located, so to speak, in any particular phrase. The whole construction is bound up with the sarcasm – what is literally meant cannot aptly be stated using the same syntactic form, only negated (e.g. 'Of course, it's not the case that empirical studies may prove me wrong', or some other placement of negation).

For now I remain agnostic.

A Final Observation about Wholes and Parts and Granularity

Observation: Two propositions - or more generally, complex expressions - can be identical in meaning at one granularity, while none of their parts have the same meaning at any reasonable granularity. Or less extremely, while few of their parts, or none of their "key words", have the same meaning at any reasonable granularity. Or again, while their overall structure and arrangement of parts is quite different.

For example: 'Get out!' and 'Leave at once!'.

Sunday, 12 July 2015

An Account of the Analytic/Synthetic Distinction

Some of the intuitive characterisations given, in the last post, of the notion of internality of truth-value – such as 'internal meaning determines truth-value' - sound a lot like a common post-Kantian way of characterising or defining analyticity, namely as 'truth in virtue of meaning'. This raises the question of whether the class of a priori truths is the class of analytic truths, and the question of whether there are, or should be, distinct notions here at all. My answers to these questions will be No and Yes respectively.

The aim here will be to try to clarify an interesting notion of analyticity which is conceptually and extensionally distinct from all the notions of truth a priori identified in the last post (internality of truth, non-Twin-Earthability of truth, Chalmers' epistemic two-dimensionalist account, and traditional conceptions). It is distinct from, but builds on, our internality conception of the a priori.

The account I will give of this notion is inspired by Kant's account of the analytic-synthetic distinction in the Critique of Pure Reason, as well as Wittgenstein's remarks on the synthetic a priori and concept-formation in the Remarks on the Foundations of Mathematics.

It is well known that Kant's definition, or principal explication, of 'analytic' and 'synthetic' is given in terms of subject and predicate:

In all judgments wherein the relation of a subject to the predicate is cogitated (I mention affirmative judgments only here; the application to negative will be very easy), this relation is possible in two different ways. Either the predicate B belongs to the subject A, as somewhat which is contained (though covertly) in the conception A; or the predicate B lies completely out of the conception A, although it stands in connection with it. In the first instance, I term the judgment analytical, in the second, synthetical.

Since modern logic and philosophy of language has taught us not to regard every proposition as being composed of a subject and a predicate, this definition can't be adequate for us. But it is suggestive, and even moreso are some of the other things Kant says about the analytic-synthetic distinction. He says of analytic and synthetic propositions respectively that 'the former may be called explicative, the latter augmentative'. And consider this elaborated version he gives of his main question, that of how synthetic a priori knowledge is possible: 'If I go out of and beyond the conception A, in order to recognize another B as connected with it, what foundation have I to rest on, whereby to render the synthesis possible?'.

The idea that synthetical judgments are 'augmentative', that they 'go out and beyond' 'conceptions', can, I think, be generalized or abstracted from Kant's discussion in such a way that it does not depend on construing all propositions as being of the subject-predicate form. And we get a hint of how to do this from the following passage about the syntheticity of the proposition '7 + 5 = 12':

We might, indeed, at first suppose that the proposition 7 + 5 = 12 is a merely analytical proposition, following (according to the principle of contradiction) from the conception of a sum of seven and five. But if we regard it more narrowly [my emphasis], we find that our conception of the sum of seven and five contains nothing more than the uniting of both sums into one, whereby it cannot at all be cogitated what this single number is which embraces both. The conception of twelve is by no means obtained by merely cogitating the union of seven and five; and we may analyse our conception of such a possible sum as long as we will, still we shall never discover in it the notion of twelve. We must go beyond these conceptions, […]

This regarding-more-narrowly will be the key for us. We said above that a proposition is a priori iff it contains its truth value, i.e. iff its internal meaning determines its truth-value. Our idea now is that a proposition is analytic iff its internal meaning regarded more narrowly in a certain way – or iff a certain sort of part or fragment of its internal meaning – determines its truth-value. And so the next task is to try to clarify what characterises the aspects of internal meaning we are restricting our attention to here.

To do this, we will use the notion of concept- or conceptual-structure-possession, and the notion of understanding. We will not need to involve considerations of knowledge, judgement, being-in-a-position-to-see-that, or anything like that. (Later, we will consider how what we say may shed light on accounts which do involve such considerations.)

As a first approximation, we will say that a proposition is analytic iff the bits of conceptual structure – the part of its internal meaning - one must possess in order to understand it, determines its truth-value. (This involves a terminological departure from the possibly more common procedure of regarding analyticity as implying truth – we say that an analytic proposition can be true or false, just as we say an a priori proposition can be true or false. This has the nice feature of giving us a simple division among propositions in general, not just truths, so that we can say that for propositions in general, being analytic is just not being synthetic, and vice versa.)

We can make this definition easier to handle and more memorable by giving it in two parts:

The meaning-radical of a meaningful expression consists in the bits of conceptual structure, i.e. the part of its internal meaning, one must possess in order to understand it.

A proposition is analytic iff its meaning-radical determines its truth-value.

(As an added bonus, we now have the general concept of a meaning-radical, which we can apply to sub-propositional expressions as well as propositions, and perhaps also to super-propositional expressions such as arguments.)

Consider the fact that we can come to believe false arithmetical propositions - for example on the basis of miscalculation, or misremembering, or false testimony - and that we can apply them.

Contrast the case of a paradigm analytic proposition, such as 'All bachelors are unmarried'. (To get around the irrelevant problem that in English 'bachelor' very arguably doesn't mean 'unmarried man', let us just suppose that it does mean exactly that.) To be sure, someone can assent to the sentence 'Not all bachelors are unmarried', and dissent from 'All bachelors are unmarried', but in such a case we would say that they don't understand this latter as we do – they don't understand our proposition 'All bachelors are unmarried'. So they don't believe – and here we are using words with our meanings kept intact – that not all bachelors are unmarried.

Kant says that we can become 'more clearly convinced' of the syntheticity of arithmetical propositions 'by trying large numbers'. Let us now, therefore, try to illustrate the notion of a meaning-radical, and in turn that of analyticity, by considering an example of a false arithmetical proposition involving numbers larger than 7, 5 and 12. Say '25 x 25 = 600'.

Despite being false a priori, the proposition '25 x 25 = 600' is something we can mistakenly believe and apply while still understanding it correctly (in some suitably minimal, and natural, sense of 'understand'). We have – wrongly – made a connection between our conception of the product of 25 and 25, and our concept of 600.

Why do we say that we understand the proposition in its ordinary sense and are wrong, rather than saying that we are operating in a different system, in which the sentence '25 x 25 = 600' is true, and that we (therefore) don't understand the proposition in its ordinary sense? It is not hard to see what sorts of things make it that way. If we worked it out on a calculator, or calculated it again ourselves, we would unmake the connection. Such developments would show that we did understand the proposition correctly (i.e. in its ordinary sense).

(Suppose an illegal move is made in chess, say that someone moves their king into check (so that it need not be immediately obvious that it is an illegal move). If the maker of this move can easily be brought to accept that their move was illegal, we can maintain that they understand how to play chess and were playing it according to the ordinary rules, but playing wrongly. If they cannot, then either they simply do not understand chess, or are insisting on playing according to deviant rules.)

The fact that in the case of the false belief that '25 x 25 = 600', there is this other option here, if I may put it that way, of saying that we are operating in a different system – an option which we will have to reject because of many things about how things are, so in that sense not an option, but still something which makes sense – shows that there is a possible system, compatible with the meaning-radical of '25 x 25 = 600', in which that sentence holds. That is, the meaning-radical of '25 x 25 = 600' – that bit of conceptual structure – can be incorporated into a larger structure wherein the concept of the product of 25 and 25 (although we might not want to call it that anymore) is connected to that of 600 in such a way that the sentence '25 x 25 = 600' is true. It will have a different internal meaning from our proposition '25 x 25 = 600', despite the system it belongs to incorporating the meaning-radical of our proposition. This is what makes '25 x 25 = 600' synthetic.

On this picture, the full internal meaning of a concept or proposition-meaning may involve connections which do not have to be made in order to understand it.

So, a proposition is analytic iff it has its truth-value in virtue of the bits of conceptual structure someone has to possess in order to understand it. That is, iff the bits of conceptual structure one must have in order to understand it cannot be embedded in a context such that the proposition-radical of that proposition gets a completion such that the resulting proposition has a different truth-value from the proposition in question.

More briefly, a proposition is analytic iff its meaning-radical determines its truth-value.

All a priori propositions, then, on the account I am giving here, will be such that their internal meanings determine their truth-values. But analytic propositions have the further property that their radicals determine their truth-values, whereas the radicals of synthetic a priori propositions can be incorporated into both true propositions and false ones.

A Complication

One complication: perhaps there is a mistaken assumption of uniqueness built into my talk above of the meaning-radical of a proposition. The bit of conceptual structure one must possess in order to understand it. Perhaps one and the same proposition can be understood from more than one angle, as it were, in which case it may be better to talk about multiple meaning-radicals – distinct bits of conceptual structure all of which individually and minimally suffice for understanding.

This gives rise to a choice: if that's how things are, should we call the analytic propositions those which are such that all their meaning-radicals determine their truth-values? Or those which have at least one meaning-radical which determines their truth-value?

I do not want to try to settle the issue of whether we should recognize a possibility of multiple radicals. Furthermore, I have no opinion about which use of terminology is best, in case we should – perhaps it just doesn't matter. If we use 'analytic' for the first, we may say 'weakly analytic' for the second. Or, if we use 'analytic' for the second, we may say 'strongly analytic' for the first. Or we might make 'analytic' mean 'either strongly or weakly analytic'. Or drop it entirely, and always specify 'strong' or 'weak'.

Sunday, 14 June 2015

On a Semantic Account of the A Priori

This post is an attempt at stating and evaluating an approach to analyzing the concept of apriority as it applies to propositions. 

Follow-up: An Account of the Analytic/Synthetic Distinction

My conception of propositions, which sees them as having an internal nature constituted by their place in the system of language and thought to which they belong, but sometimes also external projective relations to the world, is highly suggestive of an approach to analysing the a priori-a posteriori distinction.

Using 'a priori' in such a way as not to imply truth, so that a proposition can be a priori true or a priori false, we might give expression to the basic idea by saying:

A proposition is a priori iff either truth or falsity is an internal property of it.

Or, using 'a priori' like 'necessary', in such a way as to imply truth (that is, meaning what 'true a priori' means on the above usage):

A proposition is a priori iff truth is an internal property of it.

(We will stick to the first usage.)

An account of this sort is attractive from my point of view for two related reasons. Firstly, it explains the feeling that definitions or accounts of the a priori which involve considerations of a knowing subject, and what such a subject can (in some sense) do without need (in some sense) of experience (in some sense), fail to get at the heart of the matter. That is, the feeling that a proposition's being a priori or a posteriori is a matter of the nature of the proposition itself, and that the stuff about being able to know an a priori proposition independent of experience (in some elusive sense) holds as a consequence of that proposition's being a priori, rather than constituting that property.

Secondly, many philosophers are skeptical of the notion of the a priori, or of the idea that anything genuinely falls under it. But topics at the centre of the present work, such as the existence of the necessary a posteriori and the contingent a priori, involve a notion of the a priori, and furthermore, one which is held not to be vacuous. If I can provide a new account of the notion I intend here, perhaps these skeptics will be able to enter into my discussion further than they would be able or willing to otherwise.

On this account, the concepts 'a priori' and 'a posteriori' are broadly logical. They can also be called epistemological if one wishes, but there is a danger in that, since by itself it leaves one free to overlook the distinction between properties like that of being a priori, which have to do with the nature of the propositions which possess them, and blatantly epistemological properties like that of being known, that of being hard to understand, that of being easy to verify, etc., the very constitution of which involves relationships to knowers.

That said, it is of course open to anyone to stipulate that 'a priori' is to have a meaning given in terms of a knowing subject and what they can do. But I would think of the word in that usage as expressing a property the possession of which is explained by possession of the property expressed by the word as I use it - a broadly logical property. I use 'a priori' that way because I find this latter to be of more fundamental interest, but I'm not concerned to insist on or argue for such a usage. The point about a broadly logical property explaining, or having as a consequence, stuff about what a knowing subject can do is supposed to help motivate the notion or notions I want to propose, but even that is secondary. My primary concern is just to propose them and try to make them clear

What do I mean by saying that a priori propositions' truth-values are internal to them? I do not mean that a priori propositions – individuated the way we are individuating propositions here, such that external projective relations are held fixed – necessarily have the truth-values they have. (In that case, it would be difficult or impossible to allow the class of a priori truths to differ from the class of necessary truths.)

The thought is, rather: with many propositions, their internal meanings – that is, their positions in the language-system to which they belong – do not by themselves determine a truth-value; rather, this depends on their external connections to reality, and what lies on the reality end of the connections. But with the a priori propositions, there is no such dependence; internal meaning determines truth-value. Or we might use the pair of locutions 'in virtue of' and 'irrespective of': a posteriori propositions have the truth-values they have partly in virtue of what (if anything) lies on the reality end of external projective relations borne by them, whereas a priori propositions have the truth-values they have irrespective of that. We might even simply say that their internal meanings, in contrast to a posteriori propositions, have truth-values already, all by themselves.

Is Our Notion of A Priority Explicable in Terms of Twin-Earthability?

I think this idea, as explained in various ways above, and guided by our pre-existing, traditional conception of a priority, can have considerable philosophical value. It will turn out to be worthwhile, however, to see what happens if we try to explicate this notion of internality of truth-value by means of the concept of Twin Earthability. We may say:

A proposition is Twin Earthable iff in some possible situation, a proposition with the same internal meaning has a different truth-value.

And then:

A proposition is a priori iff it has its truth-value internally iff it is not Twin Earthable.

This seems a natural strategy, since if a proposition has its truth-value irrespective of its external connections to reality, then Twin Earthing it shouldn't be able to change that. And that is correct, but for the strategy to succeed, we also need it to be the case that Twin Earthing is unable to change a proposition's truth-value only if it has this truth-value irrespective of its external connections to reality. And certain kinds of propositions might seem like counterexamples to this, such as 'I exist', 'Language exists' (where 'language' means concrete linguistic phenomena) and 'I am uttering a sentence now' (where 'uttering' is taken to mean a spatiotemporal process).

I am indebted to discussions of Chalmers ('The Foundations of Two-Dimensional Semantics', 'Epistemic Two-Dimensional Semantics') for these examples, and for the notion of Twin Earthability (although he defines it over sentence-tokens), which Chalmers derived in turn from Putnam's famous Twin Earth thought experiment.

In Chalmers' discussion, the examples are given as counterexamples to a different sort of account of a priority, and for different reasons. Chalmers' target is a particular class of interpretations of what he calls 'The Core Thesis': 'For any sentence S, S is a priori iff S has a necessary 1-intension'. Namely, those interpretations on which intensions are regarded are 'any sort of linguistic or semantic contextual intensions'. (Chalmers goes on to argue for what he calls 'epistemic intensions'.) Chalmers is testing this account against a notion of a priority understood along traditional lines, in terms of knowledge and experience. We, on the other hand, are, at least in the first instance, testing non-Twin-Earthability against our notion of a priority as internality of truth-value. (We will eventually come back and consider how these two notions - non-Twin-Earthability and internality – might line up with a more traditional notion involving knowledge and experience.)

First, it might seem that 'Language exists' and 'I am uttering now', construed the way they are supposed to be construed, namely as implying certain spatiotemporal goings-on, do not carry their truth-values inside themselves. It might seem like they must reach out and get their truth from those spatiotemporal goings-on.

They have non-Twin-Earthability, not because they are true irrespective of what lies on the reality end of their external projective relations to reality (that isn't the case), but rather because any Twin-Earthing of them, any proposition with the same internal meaning, is sure to bear external projective relations to reality such that what lies on the reality end of these relations makes the proposition true. (They are non-Twin-Earthable for what we might call transcendental reasons: their instantiation guarantees their truth; their truth conditions are subsumed under their instantiation-conditions.)

Regarding 'I exist', we might think: this cannot be true in virtue of internal meaning, since what makes it true is that I actually exist – this proposition reaches out to me.

There are numerous unclear and discomforting things about all this, however, especially the 'I exist' case. (If we imagine 'I exist' reaching out to us, aren't we thinking of our bodies? And isn't there a way of construing 'I'-propositions such that they don't imply the existence of bodies?). We will try to address them below.

(To anticipate, since this may ease comprehension: I will end up going along with the train of thought above regarding 'Language exists' and 'I am uttering now', but rejecting it for 'I exist', provided 'I' is construed so as not to imply the existence of a body. These are very confusing topics, however, and I don't wish to be dogmatic.)

On the Peculiarity of these Examples

First, a word about the peculiarity – the weird basicness, so to speak – of these examples.

Throughout my investigation of the difficulties arising with these examples, I have been heartened but also troubled by how peculiar they are – all of them, but especially 'I exist'.

Heartened, because since this proposition ('I exist') is so peculiar, the fact that it raises confusing problems in connection with our analysis of a priority should worry us less than if a less peculiar, more worldly sort of example raised these problems. The confusing problems presumably have a lot to do with the peculiarity of 'I exist', and we know that “the first person” raises confusing problems anyway.

It might even look like this is a peculiarly philosophical proposition, what Wittgenstein might have called a pseudo-proposition, especially when we reflect that it may not even concern a body.

But – and this is why it is troubling – that wouldn't absolve us in any clear way of the need to square our account of a priority with it – it is one thing to be dismissive of certain propositions when you aren't putting forward an account of a general notion in propositional typology, but I am doing that: I'm saying that a proposition (any proposition) is a priori if it has its truth-value internally.

That leaves the option of arguing that 'I exist' is no proposition at all, but that's not an inviting prospect. It seems dogmatic and ad hoc.

Secondly, and connected with this last point: it isn't clear that 'I exist' really is a peculiarly philosophical proposition with no practical use. Couldn't people (indeed, don't they sometimes?) assert 'I exist' in order to make someone more mindful and just toward them? And couldn't someone whose existence is in doubt, but whose supposed appearance is well-known, appear to the doubters and say 'I exist'? (Furthermore, aren't these instances of the very same (internal) proposition-meaning as Descartes tried to establish with the cogito? And that's not to say that weird, peculiarly philosophical things weren't happening there. Clearly they were.)

Accordingly, I will face up to these difficulties as best I can. I will now discuss 'I exist' and argue that it can be maintained to have its truth-value internally, when 'I' is construed so as not to imply the existence of a body. Following that, I will consider the other examples, 'Language exists' and 'I am uttering now'. (These also seem peculiar, and this seems to have to do with their being good candidate instances of Wittgenstein's remark in On Certainty (#83): The truth of certain empirical propositions belongs to our frame of reference. We will discuss this in turn.)

'I Exist'

Here is one consideration which throws doubt on the idea that 'I exist' doesn't have its truth-value internally, since it must reach out to me – i.e. a consideration which suggests that perhaps it does have its truth-value internally after all.

Suppose a computer system used by an administrator allows them to enter queries such as 'user23 space?', and the computer will tell them how much disk space user23 has left. The administrator's username is 'admin', so they can find out how much disk space they have left with 'admin space?'. But to save inputting time, a first-personal pronoun is introduced, so that the administrator and other users can enter 'I' place of their username in such queries. Let us suppose that a command involving 'I' is, under the hood, first transformed so that its occurrences of 'I' are replaced with the username of whoever is logged in (this username, we may suppose, is sitting in a particular memory location, ready for this purpose), and then passed on to be executed.

Now suppose there is a query 'loggedin?', so that the administrator can find out if user23 is logged in with 'user23 loggedin?'. When the administrator enters 'I loggedin?', the computer first looks up the username of the current user where it waits in memory (in this case 'admin'), plugs that in in place of 'I', and sends that to be executed. Suppose the system now searches a constantly updated list for the name 'admin', and returns 'Yes' if it reaches that name, 'No' if it reaches the end without finding it.

There is an obvious inefficiency here. Pointless as the 'I loggedin?' query might be in practise, if for some reason it had to be executed a trillion trillion times (say, at the whim of a rich eccentric, or for an art project), it might be worth modifying the software so that queries containing 'I' are first checked for 'loggedin?', in which case 'Yes' is immediately returned. And note that we don't need to first study the results of 'I loggedin?' working in this inefficient way – don't need to study what's on the dynamic list of users – in order to see this. We look into the system and see that it can only go one way.

It seems to me that if we think of an instance of 'I exist' as functioning like 'I loggedin?' in the unmodified computer system, we may be inclined to think that its truth-value is not internal to it, but if we think of it as functioning more like 'I loggedin?' in the modified system, we will judge that its truth-value is internal to it. On that way of thinking of it, we might say there is a tight conceptual connection, an empirically indefeasible connection, between 'I' and 'exist'. (More on conceptual connections in a future post.)

But what about the first option: thinking of 'I exist' as functioning like 'I loggedin?' in the first system? What might this come to? Well, the truly analogous procedure would be something like: when I ask myself whether I exist, I take 'I' as an abbreviation for my name, and then accordingly ask myself 'Does TH exist?', and then I use the same method I use when I consider whether someone else exists – perhaps trying to observe my body, or traces of my activity. That is plainly not how it works. But perhaps there is a half-way reasonable sort of procedure we can imagine, which contrasts with the immediate verification of 'I exist' based on a tight conceptual connection. Descartes' procedure comes to mind: I must exist, since I would have to exist in order to be thinking about whether I exist. (Perhaps there is a tight connection between 'I' and 'think.)

But this is not an empirical procedure, and it doesn't involve looking out into the world. This suggests that 'I exist', despite what we might have been inclined to say (perhaps on the basis of thinking of 'I' as requiring a body), can be maintained to have its truth-value internally, and so be a priori according to the account proposed.

Here I am using notions like 'empirical procedure', and so making appealing to elements of the more traditional, pre-existing conception of a priority. This would be fishy if I were trying to fully explain those existing conceptions by means of other, independently understandable conceptions. But that's not how I think of it: traditional conceptions of a priority help explain internality of truth-value as well as vice versa. To anticipate: we will end up with two mutually supporting conceptions, giving us two angles on a single important division in propositional typology.

'Language Exists' and 'I am Uttering Now'

What about 'Language exists' and 'I am uttering now'?

Remember, it is stipulated that 'language' and 'uttering' here mean physical, spatiotemporal phenomena.

It seems to me that, so interpreted, these cases clearly do not have their truth-values internally. They do reach out and get their truth, not just their meaning, from outside language – that train of thought is one I want to go along with. (Its not applying, despite possible appearances, to 'I exist', turns on the fact that 'I exist' as construed does not imply the existence of a body.)

There are, however, superficial appearances to the contrary. When we ask ourselves 'Does language exist?' and 'Am I uttering now?', we don't actually do any “looking out into the world”, as I have, speaking figuratively, said these propositions themselves do. It seems that we just immediately, or after a moment's reflection, see that the answer is 'Yes' in both cases. Just like with 'Do I exist?'.

And yet I am insisting that there is a fundamental difference of category here. I think we can get clearer here by considering further the category that 'Language exists' and 'I am uttering now' belong to – considering, that is, what is characteristic of them as opposed to the propositions I want to call a priori?

I think this was one of the main topics, if not the main topic, of Wittgenstein's last work On Certainty. Wittgenstein was responding to Moore's work in epistemology – his 'Defence of Common Sense', wherein he claims to know he has a hand. Apparently Normal Malcolm had some role turning Wittgenstein's attention to this late in the latter's life.

The following remarks seem to characterise the peculiar class to which 'Language exists' etc. belong (I will them the 'Moorean' propositions):

83. The truth of certain empirical propositions belongs to our frame of reference.

136. When Moore says he knows such and such, he is really enumerating a lot of empirical propositions which we affirm without special testing; propositions, that is, which have a peculiar logical role in the system of our empirical propositions.

137. ... Moore's assurance that he knows... does not interest us. The propositions, however, which Moore retails as examples of such known truths are indeed interesting. Not because anyone knows their truth, or believes he knows them, but because they all have a similar role in the system of our empirical judgments.

138. We don't, for example, arrive at any of them as a result of investigation.

Note that these Moorean propositions won't generally be non-Twin-Earthable, like 'Language exists' and 'I am uttering now' (which are clearly special in that their truth is guaranteed transcendentally, i.e. by the preconditions of formulating them). It is the combination of non-Twin-Earthability and Mooreanness which makes these cases, and their nature, require special consideration from the point of view of our account of a priority as internality of truth-value.

Wittgenstein calls the peculiar role of Moorean propositions a 'logical' role, and it is not hard to see why: that we have empirical propositions playing this role, i.e. being part of the framework which guides thought and enquiry, is part of how our practise of thought and inquiry works.

But a proposition's playing this role is not an immutable or intrinsic fact about that proposition, and that is the very thing – I want to say – which makes Moorean propositions empirical rather than a priori.

Here is a metaphor I find helpful: imagine a cluster of metal frames, some of which have two feet (I will call them 'biframes'), others of which have three ('triframes'). The biframes need support in order to stand – they are built to be able to stand or fall. The triframes can stand by themselves. At the periphery of the cluster there are both biframes and triframes, and some of the biframes are held up by bits of wood. No biframe could be supported by wood alone; but some are held up by frames and wood, others by frames alone (which may themselves be held up by wood, or by further frames held up by wood, etc.). Any biframe, if you rearranged the cluster sufficiently and removed certain bits of wood, would fall, but some may be very hard or impossible to get at. (On Certainty #255: 'What I hold fast to is not one proposition but a nest of propositions'.)

Frames correspond to propositions. Their standing corresponds to truth or belief. Wood corresponds to experience, empirical confirmation. Triframes correspond to a priori propositions, biframes to empirical propositions.

The biframes which stand without being in contact with wood, i.e. which are supported by other frames, correspond roughly to Moorean propositions: the special category of empirical propositions with which we are concerned, and with which Moore and Wittgenstein were concerned. Or perhaps better: out of these standing biframes not in direct contact with wood, some are more robust than others with respect to the removal of bits of wood. Some might fall right away if you remove a single bit of wood near them, and perhaps we should not count them in this category (depending on how exactly we project the metaphor). Likewise perhaps those which would fall if a small number of easily identifiable bits of wood were removed. Then, the biframes which correspond to propositions of this special category – 'Language exists', 'I have a hand' etc. – are those which don't clearly depend on any particular bits of wood.

These special biframes are no mere epiphenomena, however: they belong to our 'frame of reference'. So in a sense they are foundations:

401. I want to say: propositions of the form of empirical propositions, and not only propositions of logic, form the foundation of all operating with thoughts (with language). ...

But this is then squared with, for example, the metaphor of biframes, with this brilliant remark:

248. I have arrived at the rock bottom of my convictions. And one might almost say that these foundation-walls are carried by the whole house.

In other words: don't be mislead by a “bottom-up” metaphor into thinking that Moorean propositions are not themselves supported, in a diffuse way, by non-Moorean empirical propositions. The image of the cluster of metal frames (or Wittgenstein's 'nest') helps avoid this: we do not imagine the cluster extending high into the air, but rather getting denser and covering a wider area.

And what's the difference between these and what we call a priori propositions? In our frame metaphor, the correlates of a priori propositions – triframes – stand by construction. Here is what Wittgenstein says about the difference:

655. The mathematical proposition has, as it were officially, been given the stamp of incontestability. I. e.: "Dispute about other things; this is immovable - it is a hinge on which your dispute can turn."

656. And one can not say that of the proposition that I am called L. W. Nor of the proposition that such-and-such people have calculated such-and-such a problem correctly.

657. The propositions of mathematics might be said to be fossilized. - The proposition "I am called...." is not. …

Putting this 'fossilization' talk from On Certainty together with the bits of wood from the metal frame metaphor above, we might say: bits of wood supporting biframes can petrify and become part of the frames themselves. (But that is not a plausible story about the actual origin of each a priori proposition taken individually. Perhaps it is plausible, however, that when we enter new regions of the a priori, so to speak, we often do so in this way. Or that, when humankind apprehended a priori propositions initially, we got there in this way. But this is all by the by.)

But in this case, fresh wood may still grow in the same area. And this sheds light on a distinct metaphor from Wittgenstein's Remarks on the Foundations of Mathematics which helped inspire the one about metal frames above: A mathematical proposition stands on four feet; it is over-determined.

This isn't the same metaphor, but I wager that these four feet do not all correspond to the same sort of thing: one of them can be distinguished as the 'fresh wood' alluded to above: for example, nothing can refute '7 + 5 = 12', it is not held hostage to experience (as we can see from the standard, convincing critiques of Mill's view that arithmetical propositions are empirical generalizations), and yet!: experience bears it out in some sense. We put seven and five things together, and we usually find we have twelve. Or better: our experience is such that this proposition is useful, is possible even (i.e. is something we can grasp, use, instantiate).

Another Angle on the Difference Between Moorean and A Priori Propositions

Another angle from which to see the difference between Moorean propositions and a priori propositions, in particular 'I exist' as we construed it in the previous section, is with a thought experiment involving experiences radically different from those most of us have had. Suppose everything went black and all bodily (kinaesthetic) sensation ceased, and a voice, claiming to be a demon, or some kind of scientist, but not belonging to the world of our experience, announced that space, or everything in it, has been destroyed, so that there is no language (construed as spatiotemporal occurrences) and no uttering (construed likewise).

If this happened to me, I would be unable to refute this. Or at least, no immediate knock-down objection would come to mind; the most near-to-hand strategy for refuting it would probably involve appeal to a belief in psycho-physical parallelism, which the demon would gainsay. I wouldn't know what to think.

If, on the other hand, the voice told me that I no longer exist, it would be totally different: I would steadfastly deny that, and nothing the demon could say to me would shake my belief that I exist. We will come back to this in a moment, when we consider how the conceptions of internality of truth-value and non-Twin-Earthability line up with the traditional conception of a priority.

To summarize our conclusions so far:

'I exist' is a priori (i.e. has its truth-value internally) and non-Twin-Earthable.

'Language exists' and 'I am uttering' are a posteriori and non-Twin-Earthable.

This is when we construe 'I' as not implying the existence of a body, and 'language' and 'uttering' as meaning spatiotemporal phenomena. If we construe 'I' bodily, 'I exist' falls in line with the other two. Conversely, if we construe 'language' and 'uttering' as not necessarily being spatiotemporal, 'Language exists' and 'I am uttering now' may be argued to fall in line with 'I exist'.

So, we appear to have two different notions here, with different extensions, both of which can be legitimate parts of our typology of propositions: internality of truth-value, and non-Twin-Earthability. We shall continue to reserve the term 'a priori' for the notion of internality of truth value. All a priori propositions are non-Twin-Earthable, but not the other way around.

Are All Non-Twin-Earthable A Posteriori Propositions Moorean?

The above discussion of propositions which are non-Twin-Earthable yet a posteriori focused on Moorean propositions – propositions it is hard or impossible to doubt, and which play a peculiar logical role in our thought and language which Wittgenstein tried to describe in On Certainty – such as 'Language exists' and 'I am uttering now'. Their being like that makes it important to distinguish them from a priori propositions, since our traffic with them resembles our traffic with certain a priori propositions in striking ways: we don't have to look out into the world of experience to verify them.

The question now arises: are all the propositions which are non-Twin-Earthable yet a posteriori like that? That is, are they all Moorean?

The answer, and this seems quite definite, is no: take, for example, the conjunction 'Language exists and first-order logic is undecidable'. Adding the second conjunct leaves non-Twin-Earthability unaffected, since it is a priori, but destroys Mooreanness. (Or, if you think it's Moorean that first-order logic is undecidable, pick some less basic a priori fact from the formal sciences.)

Comparison with the Traditional Conception

How do these two notions – non-Twin-Earthability, and determination of truth-value by internal meaning alone (which latter is what we have been using 'apriority'/'a priori' for), line up with the traditional conception of a priori truths as those which can be known without recourse to experience?

Our choice of terminology above has probably given the game away: the a priori propositions (those whose truth-values are internal to them) just are those whose truth-values can be known a priori in the traditional sense, i.e. without recourse to experience. And, since the non-Twin-Earthable propositions outrun those whose truth-values are internal to them (the a priori propositions, on our usage), they also outrun the propositions which are traditionally a priori.

I will not embark on an extensive discussion of the traditional idea, but will be quite rough and ready with it.

To repeat: non-Twin-Earthable propositions are not all traditionally-apriori, but the a priori (on our usage) propositions just are the traditionally-a priori ones. I regard the first part of that suggestion as more certain, and more robust with respect to different precisifications of traditional-apriority, than the second; .

The non-Twin-Earthable propositions which are not traditionally-apriori are the 'transcendent' ones: those whose concrete instantiation ensures their truth: as we just saw, these include both Moorean propositions such as 'Language exists' and 'I am uttering now', but also more complicated, non-obvious transcendental propositions. They give information concerning what goes on in space and time, and no propositions traditionally regarded a priori do that (perhaps an exception must be made for the most extreme Liebnizian rationalism, but even he distinguished between truths grounded by the principle of non-contradiction and those grounded by the principle of sufficient reason).

The case of 'I am uttering now' seems particularly clear: when we know that, we are clearly relying on experience: in the most straightforward case, the very experience of uttering. It could conceivably fail to come off.

A more complicated case could be imagined where some antecedent condition is empirically and reliably connected with me uttering 'I am uttering now', I might observe the condition and be said to know the proposition already, i.e. not by means of my experience of uttering it.

But hold on: is that right? It might be thought that there is a mistake here, similar to the mistake that would be made by saying that the way we know our own intentions is by observation and empirical correlation (cf. Wittgenstein's Remarks on the Philosophy of Psychology). We may just know that we intend to utter, not by observing ourselves, and know that our intention will be carried out, and then utter 'I am uttering now' knowingly. But in that case we may say: very well, but you are still drawing on past experience in knowing that your intention will be carried out. Furthermore, we can maintain high standards of knowledge and say: you don't really know you're uttering now until you experience it (hear it, see it, feel it etc.), because something could go wrong with your attempt at uttering – your larynx could disappear, for example.

So, not all non-Twin-Earthable propositions are traditionally a priori: 'I am uttering now' is the former but not the latter.

Does the converse hold? Are all traditionally a priori propositions non-Twin-Earthable? I say Yes, but I will not bother arguing this directly, since it falls out from the other things I am arguing: that the a priori propositions are just the traditionally a priori propositions, and all a priori propositions are non-Twin-Earthable.

One challenge to the idea that the a priori propositions – i.e. those whose truth-values are determined by their internal meanings alone – just are the traditionally a priori propositions is that already troublesome proposition, 'I exist'. Above, we made a case for saying that – provided the 'I' is construed in such as way as not to require the existence of a body – this proposition is a priori: its internal meaning determines its truth-value. And our intuitive considerations in favour of that were not unrelated to the traditional conception of apriority: we spoke of 'looking out into the world', 'looking into the language-system', and reasoning.

I think we should follow through with this, and say that 'I exist' (on our “bodiless” construal of it) is traditionally a priori: you can know it without recourse to experience.

However, there is an opposing line of thought here. We see it, for example, in Chalmers. Chalmers' idea about 'I exist' is that we know it via introspection, which is a kind of experience. But we may want a narrower concept of experience which this doesn't fall under.

It is admitted by Chalmers that it is 'somewhat controversial' that 'I exists' can only be known on the basis of experience. 'I am uttering now', on the other hand, Chalmers regards as a clear case of something that can only be known the basis of experience.

I propose to go along with this, and say that 'Language exists' and 'I am uttering now', construed so that they give information about spatiotemporal goings-on, is neither traditionally a priori nor a priori in our internality sense.

It is clear that 'Language exists' and 'I am uttering now', construed the way they are supposed to be construed, give information about particular contents of space and time – and that makes us feel that in some sense, surely, these can only be known on the basis of experience. But it seems there is a conceptual gap between 'I exist' and 'I occupy space and time' (or anything else which says something about the contents of space and time) – otherwise Descartes' recovery of his old beliefs would have been considerably easier.

(Here we come upon an interesting difference between space and time. Kant's calling time 'the form of the internal sense' and space 'the form of the external sense' is suggestive of it, so perhaps he comments on it somewhere. Roughly: while perhaps I could be taken out of the “time-space” I was in and put into another one, so that there is no fact about whether I am presently earlier or later than my birth in external time (in the sense of Lewis's distinction between personal and external time), but I cannot imagine not being in any time-space at all. If everything went dark and all bodily sensation ceased, and I heard a voice telling me I have been taken out of space, I would be unable to refute this. Or at least, no immediate knock-down objection would come to mind; the most near-to-hand strategy for refuting it would probably involve appeal to a belief in psycho-physical parallelism, which could only be founded on experience. If the voice, on the other hand, told me I had been deprived of all temporal locatedness and relations, I would immediately be able to see that this was in some important sense wrong. We might say that spatial locatedness is a posteriori, temporal locatedness (in some sense, at least – i.e. Lewis's personal time) a priori.)

Another major source of potential difficulty for my thesis that the traditionally a priori propositions just are the a priori ones in our internality sense, perhaps the most fundamental source of difficulty, is the synthetic a priori; substantial a priori truths which don't seem to be “true in virtue of meaning” in the sense that 'A priori propositions are those whose truth-values are internal to them', for example, is in this discussion (since we stipulated that we would use 'a priori' for internality of truth-value).

To begin to meet this difficulty, we must give an account of the analytic-synthetic distinction. I will make a start on this in the next post.