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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
  • I spent many thousands of hours on this, Gödel's 1931 Incompleteness Theorem, the Tarski Undefinability Theorem and Turing's Halting Problem proof. All these things are interrelated. They are all semantically ill-formed for two different reasons: (1) Self-contradictory (2) Infinitely recursive. The Liar Paradox is the simplest possible example of pathological self-reference(Olcott 2004). – polcott Apr 19 at 15:44
  • "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." That is very apt analysis and much clearer than my way of saying the same thing. – polcott Apr 20 at 17:56
  • Hmm, @Conifold, I always thought that according to Frege, the reference of a sentence is its truth value, whereas its sense is the thought that it expresses. It has admittedly been decades since my formal training in this stuff, but have the meaning of these terms substantially drifted over that time? – gonzo Jun 20 at 17:44
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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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  • "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." That seems exactly correct to me. – polcott Apr 20 at 17:57
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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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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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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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The Liar's Paradox; Aristotle's Laws of Thought: Laws of Non-Contradiction (LNC), Excluded Middle (LEM); Plus, the Law of Bivalence (LBi)

Let: X := "This statement (X) is false".

QUESTIONS to Consider:

Q1. What is a proposition, i.e., what is the logical definition of a proposition? Is X a proposition?

Q2. Is it possible for a proposition to be both true and false?

Q3. Is it possible for a proposition to be neither true nor false, but some middle option between true and false or some otherwise third option besides true and false?

Q4. Can a proposition only bear one truth value, that (single) truth value being either true or false?

ANALYSIS OF THE LIAR'S PARADOX!

If it is true that 'X is false', then X is false, because X stands for: "This statement (X) is false". Therefore, if it is true that 'this statement (X) is false', then X is false.

If it is false that 'X is false', then X is true, because X states that 'this statement (X) is false'. Therefore, if it is false that 'this statement (X) is false', then X is true.

If it is both true and false that 'X is false', then two analyses exist:

If it is false 'that X is both true and false (together)', then X must be the other options besides being both 'true and false', namely that X is either true or false or neither true nor false.

On the other hand, if it is both true and false that 'X is false' then the following conjunction must hold: it is true that 'X is false' and it is false that 'X is false'. To the extent that X is true, it is true that 'X is false' implies 'X is false', and to the extent that X is false, it is false that 'X is false' implies 'X is true'. Therefore X can be both true and false, as was the very assumption that lead us to this.

Questions to consider:

  • A1. Can X be both "neither true nor false" and "true"?

The joint denial, the neither-nor option, excludes the case that 'X is true' as well as the case that 'X is false', i.e. 'X is not true' and 'X is not false'. So, the conjunction 'X is true' and 'X is not true and X is not false' logically implies that 'X is not false', in which case 'X is true'. Here, another question arises: how can something be both true and not true (where not true is implied by the neither-nor option)?

  • A2. Can X be "neither true nor false" and "false"?

The neither-nor option that states "X is neither true nor false" can be restated 'X is not true and X is not false'. Conjoining this statement with the statement that 'X is false' yields that X is both 'not false' and 'false' (a contradiction arises), which in its turn yields that 'X is not true' (i.e., X is false). Neither true nor false denies that X can be either true or false, therefore X cannot be false (a contradiction arises). How can a proposition X be both 'true' and 'not true' (i.e. false)? How can a proposition X be both 'false' and 'not false'? A contradiction arises yet again!

The Laws of Non-Contradiction, Excluded Middle, and Bivalence (i.e., the very definition of a proposition)

Questions to consider here: Is it logically permissible for a proposition to both be true and false? How about neither be true nor false? Let us examine both of these questions, making use of the rudimentary and foundational laws of classical logic. Let us analyze the definition of a proposition also.

There are four distinct options here for a given proposition X:

Syntactical Expressions:

  • [i]. X
  • [ii]. ~X
  • [iii]. Both X and ~X
  • [iv]. Neither X nor ~X

which translates to...

Semantical Expressions:

  1. X is true
  2. X is false
  3. X is both true and false
  4. X is neither true nor false

where [i] = 1, [ii] = 2, [iii] = 3, [iv] = 4

Options 3 and [iii] are logically impermissible in classical bivalent (two-valued) logic.

The law of non-contradiction (henceforth, LNC) states: "Nothing can both be and not be," (where: "nothing" = no thing = not a thing). This can be restated as follows: "Something cannot both be and not be." That is, a proposition X and its negation ~X cannot both be true together: it cannot be the case that 'both X and ~X are true'. This is the Law of Non-Contradiction.

Therefore, the logical conjunction of a proposition (X) with its negation (~X) is excluded by logic i.e., made logically impermissible by the law of non-contradiction (LNC).

The law of non-contradiction also logically entails that any given proposition X cannot be both true and false (simultaneously, at the same time, in the same sense).

It is entailed by the syntax of the law of non-contradiction: ~[X & ~X], that states that it cannot be the case that the conjunction [X & ~X] is true (i.e., "contradictions cannot be"). The form ~[X & ~X] is a true formula for any X.

The law of non-contradiction thus makes contradictions logically impermissible, where contradiction arises where X and ~X are both true together, and not when X and ~X are both false together! A contradiction is the the joint affirmation of X and ~X, where both X and ~X are taken to be true together. In other words, the option that states X and ~X are both true is excluded by the Law of Non-Contradiction, which makes contradictions logically impermissible.

Options (iv) and 4 are logically impermissible in classical bivalent (T,F) logic! The law of excluded middle (LEM): X V ~X, where V = inclusive disjunction = "or", as opposed to an exclusive disjunction (+) = "xor", which excludes the joint affirmation of X and ~X, whereas the "or" operator includes the contradiction (X & ~X), because an inclusive disjunction takes either of the disjuncts or the conjunction of the disjuncts.

The negation of the inclusive-either-or option is the joint denial of X and ~X, which states neither X is true nor ~X is true; that is, the law of excluded middle logically excludes the option in which both X and ~X are both false together!

Either a proposition X is true or its negation ~X is true or both X and ~X are true together.

The logical complement of the inclusive-or ("or") is the neither-nor option (nor)

LEM excludes the joint denial (the "neither-nor" option). Therefore, LEM can be reformulated as stating that "X and ~X cannot both be false together!" Either X is true or ~X is true, and it cannot be neither, but it can be both! That is, by LEM, at least one of X and ~X has to be true, including the option that X and ~X are true together, but excludes the option where X and ~X are both false together!

LEM permits the contradiction: ('X is true' AND '~X is true'); this contradiction is only restricted by non-contradiction, not by excluded middle!

The law of bivalence logically excludes both the joint affirmation and the joint denial

A proposition is defined to obey the law of bivalence:

Proposition X can bear only one truth value, that truth value being true or false; where "or" is to be understood as an exclusive disjunction. Therefore the law of bivalence can be reformulated as stating "A proposition X is either true or false; where "or" is to be understood as an exclusive disjunction.

The law of bivalence is the logical conjunction of excluded middle (LEM) and non-contradiction (LNC). The law of bivalence states EXACTLY ONE of (X, ~X) is true, and the other false. The law of bivalence logically includes both the option that 'X is true and its negation ~X is false' as well as the option that 'X is false, and its negation ~X true'. The law of bivalence logically excludes both the option that 'both X and ~X are true' as well as the option that 'neither X nor ~X is true'.

Therefore, bivalence (LBi) states: [X (+) ~X] = [X xor ~X], where (+) = xor = exclusive disjunction = exclusive-either-or; whereas, LEM states [X V ~X] = [X or ~X]. Not a trivial distinction.

Therefore LBi states that: "something is not neither or both what it is and what it is not".

BY the law of bivalence: (X xor ~X) yields: either X or ~X is true, and it cannot be both, and it cannot be neither!

Please Note the Important Differences Between the Laws of Excluded Middle and Bivalence!

  • Note that the law of excluded middle (LEM) uses the operator "or" (inclusive disjunction), while the law of bivalence (LBi) uses "xor" (exclusive disjunction).
  • Further note that LEM (excluded middle) contains a negation ("not" = "~") in its formula; whereas, LBi (bivalence) does not include a negation in its expression!
  • Moreover note that LEM is only expressed in terms of the truth value of 'true' and does not include 'false'! Bivalence, on the other hand, is mathematically expressed in terms of 'true' and 'false' and does not include ("not" = "~") as a connective.
  • In LEM, the negation operator ("not" = "~") serves as a logical connective, not as a truth function! Whereas, LBi (bivalence) is a principle about negation ("not" = "~") as a truth function!
  • Therefore, LEM is a syntactical principle of logic, while LBi is a semantical principle of logic.

Bivalence states that a truth variable X, i.e., a proposition ("truth-bearer"), can only carry one truth value at a time, that (single) truth value being either "true" or "false"; where or is to be understood as an exclusive disjunction, which logically excludes the conjunction of the contradictory disjuncts X and its negation ~X.

The Law of Bivalence is the Conjunction of the Laws of Excluded Middle and Non-Contradiction!

LEM makes the joint denial (the "neither-nor" option) logically impermissible.

LNC makes the joint affirmation (the "both-and" option) logically impermissible.

Therefore, LBi - being the conjunction of LEM and LNC - makes both the joint denial and the joint affirmation impermissible!

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The Liar Paradox is simply not a truth bearer

Every expression of language that can be definitely resolved to exactly one Boolean value of True or False is a truth bearer. Conversely expressions of language that can not possibly be resolved to exactly one Boolean value of True or False are not truth bearers.

This is easier to see in natural language. Is the expression: What time is it?" {true or false}?

We can see that there is no need to resort to any three-valued logic to handle that case. Expressions of language that are not truth bearers are simply excluded from the system of logic.

Self-contradictory expressions of language such as the Liar Paradox: "This sentence is false" are simply not truth bearers because they are impossible to resolve to exactly one correct value of True or False. As a truth bearer the Liar Paradox is semantically incorrect.

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