True on Tuesday

Logic Level 2

If it's Tuesday, then this sentence is true.

Suppose it's Wednesday; is the sentence above true?

Yes No Undetermined

3 solutions

Agnishom Chattopadhyay
Dec 5, 2016

In Classical Logic, one has to give every statement a truth value of either True or False, and there can be no statement, which are neither or both.

Now, suppose we have two statements $p$ and $q$ , and want to judge the truth value of "IF $p$ THEN $q$ ". When do we know that this conditional is False? The only way it could be False is when $p$ is True but $q$ is False. Unless this happens, we should default to calling the conditional True.

So, on a Wednesday, the precedent $p$ is False, and hence there is no way for us to tell that the statement is False. Thus, the conditional is true.

This treatment of the conditional is called the material conditional . Because of the strange semantics, it is problematic in epistemology and several other frameworks have been developed to interpret it in a different way.

A slightly different variant where the proposition in hand is

If this statement is True, then it is Wednesday.

This is the celebrated Curry's Paradox

really, i don't agree... Suppose I say "if you give me 1 million dollar i will give you 2 millions" and say we're in a situation where you didnt give me the money, hence in your logic i said truth... For me it's Undertermined

Benjamin Habié - 4 years, 4 months ago

The statement you just stated says that if they do that, you will do this, however, it does not mean if they don’t do that, you won’t do this.

Colin Peaslee - 2 years, 1 month ago

It is nonsensical to speak of the truth-value of a conditional. In "If A, then B", A can have truth-value. B may or may not have truth-value depending on whether it is an implication of A or not. The overall sentence should have no truth-value inspected.

Kishan Nous - 4 years, 4 months ago

Please review this question to be removed or offer different answers to choose from. It seems this question really belongs in the same family of the "Liar Paradox" logic statements, which falls more within the domain of analytical philosophy rather than just math or logic.

Like the "Liar Paradox", the statement "If it's Tuesday, then this sentence is true" is 'semantically closed' and inherently self-referential. Whether or not semantically closed statements can even have truth-values is still debated in the philosophical community. Some interpretations hold that it is impossible for statements like this to even have a truth-value because it is an illogical statement to begin with. In other words, this may not be a valid statement. As a reminder, mathematics and logic are both subject to analytical philosophy.

The objection to the "correct" answer here can be cited from "Tarski’s hierarchy of languages", where it is concluded that no language can contain its own truth predicate. In other words, valid statements can not be 'semantically closed'. See 4.3.1 Tarski’s hierarchy of languages cited from the following link:

https://plato.stanford.edu/entries/liar-paradox/#TarsHierLang

According to this, it seems the correct answer should be "No", as the statement can never be true (or false).

This issue of being viciously circular can be easily avoided if the author changes the statement to something like "If it'sTuesday, then sentence 'X' is true".

NOTE: Unlike the "Liar Paradox", this particular statement may not be inherently contradictory, however, that is not the issue here, as both statements share the quality of being semantically-closed.

Also reference:

https://plato.stanford.edu/entries/self-reference/#ImpHieSetThe

3.1 Building Explicit Hierarchies Building hierarchies is a method to circumvent both the set-theoretic, semantic and epistemic paradoxes. Russell's original solution to his paradox — as well as Tarski's original solution to his undefinability of truth problem — was to build hierarchies. In Russell's case, this led to type theory. In Tarski's case, it led to what is now known as Tarski's hierarchy of languages. In both cases, the idea is to stratify the universe of discourse (sets, sentences) into levels. In type theory, these levels are called types. The fundamental idea of type theory is to introduce the constraint that any set of a given type may only contain elements of lower types (that is, may only contain sets which are located lower in the stratification). This effectively blocks Russell's paradox, since no set can then be a member of itself.

Taylor Shobe - 4 years, 3 months ago

I really like this problem it is so cool. I hope you make more like this ! :)

Charlotte Milanese - 4 years, 5 months ago

The problem with your solution is grammatical in nature. "If it IS Tuesday, then this statement is true." Since it's Wednesday, the statement becomes meaningless, or to use a more mathematical term, undefined. Here's what you are missing: a statement that WAS true on Tuesday may be either true or false on Wednesday. For example the statement "Joe is alive" may be true one day, false the next.

Lawrence Rowswell - 3 years, 8 months ago

Hobart Pao
Jan 19, 2017

The statement is a conditional, and whenever the premise of a conditional is false, the conclusion's truth value doesn't matter (because you can't say that the conditional was "violated" when the premise isn't met), and the statement will be true. This is called a vacuously true statement.

i really like this problem ! :)

Charlotte Milanese - 4 years, 4 months ago

Don Weingarten
Feb 1, 2019

By the rules of formal logic, a proposition can be false only if the initial conditional is true and the conclusion is false. If the conditional is false the statement is true automatically.

True on Tuesday

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