Paradoxes

If this sentence is true, linguistics club is the best club ever.

Paradoxes

Simple liar: “I am lying”: If you are lying, you said something false. As this was the only thing said, if you are telling the truth about saying something false, you are lying because you said something true and thus were not lying.

Note: This is actually sort of important because it messes up semantics a bit.

Strengthened liar: “This statement is not true”: Same as the simple liar, except here there is no possible in between state here, as there is for the “I am lying” statement.

Truth-teller: “This statement is true”: This statement is not a paradox, but it’s problematic for semantics. For example, is this a tautology? It’s truth value depends on its truth value, so it isn’t really a tautology.

Liar cycle: Socrates: “What Plato is saying is false.” Plato: “What Socrates is saying is true.”

We’re interested in the liar cycle because it eliminates the self-reference.

Errors: “This sentence has two erors.”

The error is that “error” is spelled wrong, but there’s also the error that the sentence only has one error. But given that other error, the sentence now has two errors, so that is no longer an error.

Curry’s paradox: “If this sentence is true, then Dick Cheney is a lizard.” We’re assuming for the purposes of this example that Dick Cheney is not a lizard, but for the record many people believe otherwise: http://www.debatepolitics.com/archives/8069-dick-cheney-reptile-shapeshifter.html

If this sentence is true, Dick Cheney is a lizard. But Dick Cheney is lizard, so this sentence can’t be true. However, in logic, that makes this sentence vacuously true, because if the statement is false, it doesn’t matter whether Dick Cheney is or isn’t a lizard—the sentence is never proven false. So Dick Cheney is a lizard. But Dick Cheney isn’t a lizard…

Yablo’s paradox: Yablo created a paradox that doesn’t require self-reference

                S1: For all k > 1, Sk is untrue (all subsequent statements are false)

                S2: For all k > 2, Sk is untrue

               

                Sn: For all k > n, Sk is untrue

                If S1 is true, then all of the subsequent statements must be false. But if all the subsequent statements are false, then S2 is true. So S1 can’t be true. If S1 is false, then at least one subsequent statement is true. Let Sn be the true statement. But if Sn is true, we have the same problem as we did when S1 was true for all those subsequent statements, so we have a paradox.

The solution/reason we don’t believe Dick Cheney is a lizard: sentences can be both true and false. The logic that uses this idea of truth values is called paraconsistent logic.

This might sound weird, but if you think about it, it makes sense with people. People aren’t perfectly logical. We believe contradicting things.

These paradoxes may also be classified as a sort of “semantic babble”, or meaningless on the semantic level: Tarski’s solution says that these paradoxical sentences aren’t natural sentences. In natural language, he says, there is a hierarchy of sentences. The basic sentence is one where no sentence contains true. The level above can refer to level 0 sentences only, and so on. Basically, the sentence “This sentence is false” is a contrived sentence that doesn’t actually have natural meaning, so there is no meaning, so there is no paradox.

 

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