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  1. This statement is not true

2.This statement is true only if true and not true.

(1) and (2) are clearly different sentences, but do they express the same proposition?

If yes, then it becomes clear from (2) that the proposition is clearly contradictory, since if (2) is true and therefore the proposition that (2) expresses is the case then (2) should be true and not true.

therefore the proposition that (2) expresses is a contradiction. if it expresses the same proposition as (1) then the proposition that (1) expresses is a contradiction. but then (1) does also 'say' that it is true about itself or to put it sloppily 'not true=true' according to (1) but then the paradox that we seem to have by conventional analysis evaporates , since if we assign to (1) that it is not true , and therefore that it is not the case what it says namely that (1) is 'not true=true' , then it must be not 'not true=true' , which obviously does not result in (1) being true!

Also taking into consideration that e.g. the sentences :

a. It is not the case that it is true that it is raining

b. it is true that it is raining only if it is true and not true that it is raining.

(a) and (b) do clearly express the same proposition even tho they are different sentences.

So do they express the same proposition (according to conventional definition of a proposition)?

  • Hm this is hard. Could you try putting it in formal logic symbols? Thx very much – Math Bob Nov 17 '19 at 21:22
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    Unfortunately, there is no conventional definition of "proposition". "It is used to refer to some or all of the following: the primary bearers of truth-value, the objects of belief and other “propositional attitudes” (i.e., what is believed, doubted, etc.), the referents of that-clauses, and the meanings of sentences", says SEP. In Frege's theory the sense of a sentence is its truth value, so all false sentences express "the same" proposition. To make the question answerable you'll have to settle on your version of "primary bearers". – Conifold Nov 18 '19 at 0:44
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Your first proposition is that statement 1 = statement 2. However, that is false. This becomes clearer if we negate both propositions.

1) This statement is true 2) This statement is untrue if true and not true

1) is always true, while 2) just says it is untrue if it is contradictory. 1) is always true, while 2) doesn't always have to be true. Negate both sides again and you will recognize 1) does not equal 2). Or you could just make truth tables and 1)'s truth table will be different from 2)'s, this is just the long way. That debunks the rest of the proof, and the liar's paradox, as it has from ancient times, remains unsolved.

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Minimalist formal languages don't bother defining 'False' (nor 'True' for that matter), or even 'not', they start from 'nand' -- the operation 'the first of these propositions contradicts the second' (sometimes represented as the Sheffer Stroke). Then they derive everything else.

Avoiding pedantic symbol-mongering, to get to a normal range of symbols they effectively define False as pretty much (A and not A), and not A as (A implies False) (and then they define True as not False).

So from the point of view of such formal systems, these two are literally the same proposition (not A) is formally equivalent to (A implies A and not A). If A, is circularly, the statement (not A) or (X and not A), it should not make a difference.

Those formalisms are equivalent to 'normal' versions of formal logic that do include False, so 'up to isomorphism', yes, these are the same sentences.

Addition: (You can decide that referring directly to yourself does make a difference. It has less to do with the notion of proposition than with the idea of how referring to things works. The basic idea is called 'ramification'. But it leads to a highly artificial way of looking at language.

It seems more logical to simply admit that negation 'has holes in it', kind of the way division 'has a hole' around zero. There are statements it is not safe to negate. Our notions of language just can't be complete.)

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Sentence (2) is a version of Curry's paradox, while (1) is simple instance of the liar paradox. Both of course are close cousins: They involve self-reference as well as semantical predicates like 'is true'.

However, there are some differences: Curry paradoxes involve principles concerning conditional reasoning, while liars don't. This is quite obvious when it comes to a formalization of the respective paradoxical derivations. Some tie the identity of propositions to properties of their proofs (or proofs of their negations). If you buy this sort of conception, then, no, (1) and (2) do not express the same proposition.

But you had better make clear what your preferred identity condition for propositions is; otherwise chances are that we are talking past each other.

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At the risk of misunderstanding you (my apologies in advance): it's hard to say that liar sentences correspond to propositions. There's no abstract proposition, in the universe of sets (or Forms or whatever), that somehow is referring to itself and self-encoding as false. At least, I doubt there is such an entity. (I have no idea, maybe Zalta's work covers another option, here; been too long since my studies on that score!)

OTOH there is an abstract sentence-type that corresponds to the liar sentence-tokens, i.e. the generic indexical function for "This sentence" there. But the liar index should actually always point "outward" (e.g., you say the liar index while pointing at a different sentence, to which "This" actually refers).

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