# Paradoxes and inconsistent set of sentences

Let's define logic as:
A statement and it's opposite, obtained by applying the "not" operator, both cannot be true. The not operator is applied to the part of the sentence describing action or attribute.
Example:
a) The apple is red. The apple is not red.
b) Swamy crossed over the fence. Swamy did not cross over the fence.

If in a set of statements, there exist two statements which are both true, and are opposites, then the set is inconsistent.

Do all paradoxes fall under the set of inconsistent statements or set of statements?
(Assuming of course that I have reasonably and objectively defined "logic", and other terms).

• The way classical 1st-order logic works, it follows that p and ¬**p** can’t both be true. “The not operator is applied to the part of the sentence describing action or attribute”: compare ∃xPx and ∃x¬Px. Negation is applied to the predicate, but the sentences can both be true. “in a set of statements, there exist two statements which are both true, and are opposites”. If 'opposites' = p and ¬**p**, then they can’t both be true, by your earlier definition. So, I think you need to revise your definitions a little before we can answer the question. – MarkOxford Apr 14 '18 at 9:23
• @MarkOxford : Can every sentence in natural language be converted into first order logic form? Why is my definition wrong? If two statements exists which are both true and opposites, then they are inconsistent. – novice Apr 14 '18 at 9:30
• Some natural-language sentences are easier to translate than other. E.g. ‘John runs’ is easy, while ‘Jack and Jill have a son’ is harder. (It’s not just ‘Jack has a son and Jill has a son’: they have a son together.) Or take: ‘Earth is a small planet’, which is not: ‘Earth is small and a planet’. Some sentence cannot be translated into FOL, such as sentences that quantify over properties and relations: ‘Jack and Jill have something in common’ might be an example. (See the Geach-Kaplan sentence for a (much) more complicated example.) – MarkOxford Apr 14 '18 at 9:45
• As for your definition: are the two statements both true, or are they opposites? As you say yourself, if they are opposites, they can’t both be true. – MarkOxford Apr 14 '18 at 9:46
• @MarkOxford : Thanks. I'm trying to define "Inconsistent". If they are opposites and true then the set breaks logic and is Inconsistent. – novice Apr 14 '18 at 9:55

The answer above said: no paradox is ever an inconsistent set of statements.

I disagree.

Proof:

The next statement is true.

The previous statement is false.

The examples in the author's question are not actually paradoxes but sentences with no relation to each other.

In my example, you have yourself the actual paradox (loop-type).

First, 'logic' is far broader than inconsistency. You are just trying to capture inconsistency, not all of logic.

Second, there are ways to obtain an inconsistent set of statements other than there being a statement and its negation in the set. For example, the set {P, P->Q, ~Q} is inconsistent, but does not explicitly contain a statement and its negation.

This is why inconsistency is typically defined in terms of truth-assignments: if there is no possible truth-assignment (obeying the truth-functional constraints for the truth-functional operators as we typically define them) such that all statements in the set are true, then the set is inconsistent.

A paradox is an argument for which logic does not determine whether it is true or false, but which seems to be properly formed and meaningful.

An inconsistent set of statements taken together contains a contradiction. So it must be false.

Therefore it has already been determined whether it is true or false.

So no paradox is ever an inconsistent set of statements.