# If this sentence is false, then it is true

Is the sentence "If this sentence is false, then it is true." false or true (even tautologically true), or is it a paradoxon? The sentence p claims (= is equivalent to?) that ~p → p which is equivalent to p, so p is true iff p is true. If this was the claim of p, it would be tautologically true, but I think it's not its claim. So in my opinion, the sentence can be either true or false, and is just a self-referential contingent sentence – but not a paradoxon.

Is my argument correct or flawed?

(ChatGPT considers the sentence to be a paradoxon, but I believe ChatGPT errs.)

• The sentence is not equivalent to ~p → p because the formula does not reflect the self-reference in "this", you need something like p = "~p → p". The sentence is not a paradox, but it cannot be either true or false. It has to be true because assuming it false leads to contradiction, and also "if this sentence is true, it is true" is trivially valid, so it follows by cases. A similar sounding "this sentence is true", known as the Truth-teller, does have indeterminate truth value because both options are consistent. Some call it hypodox. Commented Sep 23, 2023 at 14:04
• @Conifold: So my sentence is a hypodox - even though it looks like a paradox at first sight? BTW I considered something like p = "~p → p". Commented Sep 23, 2023 at 14:40
• It looks like my sentence is somehow equivalent to the truth teller's "this sentence is true". (But what does "somehow" mean?) Commented Sep 23, 2023 at 14:42
• No, the truth-teller sentence might be true and it might be false. Both are consistent. Your sentence cannot be false since that leads to a contradiction. All of this assumes we are trying our best to retain bivalence and non-contradiction and a truth-functional understanding of the conditional, and all while ignoring the inherent problem with self-reference. Commented Sep 23, 2023 at 17:38
• Token-based FOL was designed and evolved carefully enough by logicians to be able to avoid liar type paradox. But if you just express and play with natural language sentences like your titular one which is intended merely to be a variant of the standard liar sentence, then ChatGPT is not wrong, since it's also playing with natural languages statistically based on training parameters and hyperparamaters, and like many people could possibly easily spell out paradoxical sentences... Commented Sep 24, 2023 at 5:58

Your formal rendering of this as `~p -> p` is misleading. This looks like a statement in FOL (first order logic). But first order logic specifically forbids self-referential statements. That was necessary in order to create a system where all statements are decidable. So in FOL, p cannot refer to "this sentence." Second order logic is more powerful, but it too is set up to avoid self-reference. In general any logic that admits self-reference is vulnerable to paradox. So this statement cannot be accurately rendered in a formal logic system.

So let's set aside the formal rendering entirely. Is this a natural language paradox (meaning it cannot be unambiguously true or false)? Let's see if can be true. "If this sentence is false, it is true." We're assuming it's true, so the condition is not met, so the consequent can be either true or false. So there's no contradiction or paradox here.

Now, can it be false? If it is false, then the condition is true, which means the consequent is true, which is a contradiction. So that implies this sentence is vacuously or tautologically true, at least as determined by informal logic. We are in a system that allows paradoxes, but this is not one.

(Note: As of October 2023, ChatGPT does not analyze arguments logically, it uses context clues to guess at an appropriate response. Since this "looks like" a paradox, ChatGPT classifies it as one, inaccurately.)

This is rather close to Curry's paradox, at least an instance of it. To get a sure instance, we would probably go more with:

1. If this very sentence is true, then P.

... where P happens to be "this very same conditional as a whole is false," which might be allowable (if it is allowable for P to be any sentence whatsoever).

But also compare/contrast:

1. If this sentence is false, then it is false.
2. If this sentence is true, then it is false.
3. If this sentence is true, then it is true.
4. If this sentence is true, then it is false and true.
5. If this sentence is false, then it is true and false.

... and so on and on.

What you might be intuiting in your argument is that, per explosive logics, P, any such thing no less, can indeed be inferred from some contradiction. In Dante's Paradiso he says that there is an axiom of logic where we see that a contradiction is "both false and true," even though fully(?) untrue on another level. Even if we do not countenance an explosion in full, allow that we can use the weakened disjunctive syllogism:

1. A and not A.
2. A (x)or not A (instead of A (x)or B, for arbitrary other propositions B; here, we advert specifically to "not A" as the standard negation of A).
3. Since A, then not A.
4. Since not A, then A.
5. A (dialogically: that is, we can execute detachment to get just A, and we have "won the argument" if A was our goal all along).