logicandlanguage.net

I played about with possible T-modalities, trying to capture the idea that T(A) means "bivalently A" in a non-classical logic, and from a slightly whacky motivation to find a proof-theoretic semantic view from which T(A) was plausibly a truth predicate for which disquotation was non-truth-preserving. I didn't find anything that worked (hey, surprise!), but I was playing around with the kind of side conditions you see in the introduction rules for modalities (ie. all side assumption must have a certain structure).

What you were up to in the above reminded me of these long abandoned ideas: isn't the kind of analysis you are doing sensitive to what undischarged assumptions you have?

Hi Charles!

isn't the kind of analysis you are doing sensitive to what undischarged assumptions you have?

I'm not certain I understand the question, but this is what I think you are asking about: the rewriting method I showed only obviously works for standard (including classical) logics where we can think of the premises as members of a set. But if we want to think about non-classical logics (linear, relevant and affine) which are fussier about assumption discharging, then we'll have to take the premises to form multisets, and local reduction of the truth predicate requires that I show, for a proof π from X to A containing a non-normal pair, that it can be converted into a proof π' from X' to A without that non-normal pair, such that:

• If π is linear, so is π' and X=X'.
• If π is affine, so is π' and X' is a submultiset of X.
• If π is relevant, then so is π', and X' is a modest super-multiset of X containing more repeats.
• If π is standard, then so is π', and X' is a variant of X found by possibly deleting some elements from X and duplicating others.

In the (unusually simple) case of 'T' I can do this by pointing out that neither [TI] nor [TE] allow the discharging of assumptions, and applying them does not involve making new assumptions. The multiset of assumptions in force after the [TE] step to get B is the same multiset of assumptions that is in force right before [TI] is applied to B. This multiset will also be the multiset in force at the point where we derive B in the second proof. π2 will have the same effect on that set in the second proof as it does in the first. So the assumptions in force at the end of both proofs - the original, and the new proof - will be the same, regardless of the discharge policy of the logic.

In fact, though I have only been trying to show that non-normal pairs can be removed, it would be easy to add a 'T' predicate case to the proof of a a full-blown normalisation theorem (showing that ALL the non-normal pairs can be removed.) Whenever a non-normal pair is removed like this, the proof must get two steps shorter. (This does not always happen when we remove non-normal pairs involving the conditional in standard and relevant logics.) Proofs have a finite length, so eventually all the non-normal 'T'-pairs will be gone, which is to say that normalisation will come to an end: we will have a normal proof.

But maybe I've completely misunderstood your question, in which case please get back to me and I'll try again.

I was thinking more along the lines that maybe not all propositions might be considered "disquotation neutral", and so you might want to have side conditions ensuring that the introduction rule is only applied when the set of side assumptions is safe, ie. contains no "disquotation unsafe" formulae.

Er, no - this is definitely meant to be the unrestricted, paradox-inducing, niave truth-schema. There IS trouble ahead, and that is what suggests the analogy with tonk.

...whilst my analysis is trying to axiomatise a theory of truth for which Tarski's T-schema does not go through.

Attempt #2: Turn the T-schema into an S5 modality, [truly], which is dual to -{apparently}- (HTML doesn't like angle brakets, so possiblyish modalities I give -{ ... }-). The usual theory of S5 will give you the T-schemas:

• [truly] (A&B) iff ([truly]A & [truly] B)
  
  
• [truly] (A => B) iff ([truly]A => [truly] B),
  
  

• [truly] (A \/ B) iff ([truly]A \/ [truly] B)
  

If you add either direction, I believe that the S5 modality will collapse onto the trivial modality, as they will with the scheme for negation, which tells you that the inference rules for the [truly]/-{apparently}- modality behave together well with those for & and =>, but interfere with those for \/ and negation.

Time for another question. I don't understand what "A" means that is different from "T(A)". Surely this is a syntax question, not a meaning question? Some syntaxes will have a syntax which allows the atom "A", whereas others will take a propositional form like "there exists a" or "a is true".

But if this is a meaning question, maybe you can't get from A to T(A). Let's say "A" refers to "Santa Claus". True, in its natural language sense, means that something corresponds properly to the external world. T(A), then, is maybe a claim that "Santa Claus" is not just some syntax, but refers properly to something in the external world. That is, the proposition T(A) is synthetic - that is true contingently on the external world.

One is then left with the hoary problem of whether you can prove that any analytic statements are necessarily true :)

I can certainly imagine a logic with no conception of "false", where all you had was true things, and claims about what is and is not mutually exclusive. In the real world, you find objects, not false anti-objects :).

I have a feeling that I'm out of my depth however...

Hey Tennessee,

1. There is a syntactic thing going on here and it does have to do with "A". I'm using capital letters at the beginning of the alphabet in a slightly unusual way here (though it is not unusual when talking about truth), as schematic sentence letters. You form an instance of a schema with "A" in it by replacing every instance of "A" with a sentence. When I say it's a rule that you can go from A to T(A), what I've really given you is a schema for forming the rules. Anything which is an instance of the schema is a rule. E.g.

Snow is white
--------------
T(Snow is white)

is a rule, and

Snow is white &-(grass is green)
-----------------------------------
T(Snow is white & -(grass is green))

is a rule too.

2. We have to do things in this slightly odd way here because T is basically an implication-rule version of Tarski's T-schema: True(S) iff S. We can't just use a variable over both occurrances of 'S' here because '(S)' does't contain an instance of 'S' at all, it's just a name for the sentence S.

(2b. Incidentally I agree with Matt Wiener that 'true' is, in the first place, a property of propositions. I just don't think that means it is not a property of sentences; sentences get to be true by expressing true propositions. He's also right that my version of the rule wasn't exactly wearing its disquotational nature on its sleave, but, nonetheless, that was the intended interpretation of "T(A)". I think it would be good if I were more careful about these things in this forum in future. I'll see what I can do.)

3. Tennessee - another thing about the schema is that it only talks about the case where A is SENTENCE. It doesn't say anything about True(Santa Claus). And that is as is should be - Santa Claus isn't the kind of thing that can be true (in the sense I'm interested in), so I don't particularly mind that my rule doesn't tell you what follows if he is true...

4. Doesn't T(A) mean the same as A? That's actually kind of a cool question. Lots of smart people have thought something like that. For example, Frege wrote that "it's true that I smell the scent of violets" has the same content as "I smell the scent of violets" and the very brilliant P.F. Ramsey flirted with a similar "redundancy theory" (that's what people call the view) of truth. The problem with this kind of view is that it hasn't taken into account other kinds of context in which we use "true". For example, I can say "The fourth sentence of Greg's new book is true", or "every sentence Tennessee has ever uttered is true." But surely "the forth sentence of Greg's new book is true" does not MEAN the same as (er, let me check) "the book is designed to serve a number of different purposes, and it can be used a number of different ways."

5. Just because a sentence is about a concrete object, that doesn't mean it isn't analytic. "All bachelors are unmarried" is analytic, but bachelors are real, worldy people. "Snow is white or it is not the case that snow is white" is analytic, but that doesn't mean snow is abstract.

6. I don't think all analytic sentences are necessary. The most famous example of a sentence which is true in virtue of meaning, but which expresses fact that could have been otherwise is "I am here now." Something about the meaning of the words guarantees the truth of that sentence, but I still could have been somewhere else. But I'm not sure I understand why it is a "hoary old problem" to prove that *any* analytic sentences are true; doesn't the rule of necessitation in modal logics make it easy to prove the necessity of the axioms?

Well, I can respond, but unfortunately it involves including some personal opinions about things, rather than just arguing about what you've said :)

"Tennessee - another thing about the schema is that it only talks about the case where A is SENTENCE. It doesn't say anything about True(Santa Claus). And that is as is should be - Santa Claus isn't the kind of thing that can be true (in the sense I'm interested in), so I don't particularly mind that my rule doesn't tell you what follows if he is true..."

Okay, this, to me, makes no sense. An example of a sentence might be "Roses are red". Why is this a sentence, but "Roses" is not? One might say that all sentences express propositions, and that's consistent with your suggestion that truth is a property of propositions. Or, maybe less strongly, express a natural conjecture. But plenty of *linguists* will argue about what is, or is not a sentence. But it seems to me that one cannot establish the truth of *any* sentence without understanding its subject and object.

If you're me, then you also believe that atoms like "Santa Claus" are either true or false, or both.

"But surely "the forth sentence of Greg's new book is true" does not MEAN the same as (er, let me check) "the book is designed to serve a number of different purposes, and it can be used a number of different ways."

One man's modus ponons is another's modus tollens... :) Once again, I apologise for talking here about linguistics rather than on your own logical terms. We first, I would say, need to agree on what meaning is. For me, it's the interpretation given by an intelligence. One can interpret the first proposition (that the fourth sentence of Greg's book is true) *without knowing* what the actual sentence of Greg's book is, especially if you are willing to take it on authority rather than understand it yourself.

So here's a question for you - is it possible to understand what the truth of a proposition means if you don't also understand the content of that proposition?

Certainly, the truth of that statement *implies* that Greg's book is "... designed to serve a number of different purposes, and it can be used a number of different ways." If you buy into particular theories of reference and meaning, you could most certainly argue that this means the same thing as "the fourth sentence of Greg's book is true".

Okay, so am I just arguing semantics on an issue we both understand? I don't think so. Because if we could agree, we might be able to say once and for all what is going on here.

"Just because a sentence is about a concrete object, that doesn't mean it isn't analytic. "All bachelors are unmarried" is analytic, but bachelors are real, worldy people. "Snow is white or it is not the case that snow is white" is analytic, but that doesn't mean snow is abstract."

I think you missed my point. If you buy into a causal theory of reference - i.e. that sentences are only *meaningful* if some intelligence understands what in-the-world the symbols of the sentence represent, then the truth of a proposition about the world is synthetic.

You point about (Snow is white or snow is not white) being a truism is interesting, but I would argue a self-defeating one. It is *impossible* to make any claim about snow with that sentence. In fact, I would argue that because the truth of that sentence is not dependant on any property of snow, it does not in fact *refer* to any property of snow.

"But I'm not sure I understand why it is a "hoary old problem" to prove that *any* analytic sentences are true; doesn't the rule of necessitation in modal logics make it easy to prove the necessity of the axioms?"

Something like 2+2=4 is necessarily true, because it's part of a consistent system which is defined in just that special way.

What's interesting, is that it appears to refer properly to how the world *really* is. For example, if I have two tiles, and add another two tiles, I get four tiles. Is this a contingent accident, or not?

If 2 + 2 = 4, and 2 tiles and 2 tiles gives 4 tiles, are all other things which are necessarily true about analytic statements, also necessarily true in reference to the world around us?

The "hoary old problem" is precisely that boundary. We believe that even the laws of physics are the kinds of things which can change - our best science describes *different* physical laws for different places, yet we still believe that in the real world, *anything* we are able to notice as being made of distinct parts *must* be broken down in according with these analytic views.

Is it perhaps the *nature of synthesis* rather than the nature of reality that makes this true? Can we in fact refer to things which are not merely logically necessary, but also true by reference to the real world, inside some analytic statements?

Anyway, I'm sorry if I'm just rambling like a first year high on crack. Maybe one day I'll publish a dense and incomprehensible book that will turn out to be mostly wrong, but containing some interesting paragraphs for use in philosophy essays yet to come...

If you're me, then you also believe that atoms like "Santa Claus" are either true or false, or both.

Hmm, puzzling. Can you help me understand this? Assuming the name "Santa Claus" can be true or false, which is it? And why? (If you happen to know.)