Define a formal tautology as a statement where by the nature of its atomic components there exists no truth-value assignment where it is not true. A contingent statement is a statement that is true by facts of the world. A statement is necessarily true if the statement is true in all possible worlds. A necessarily true statement is not contingent and a contingent statement is not necessarily true. A formal tautology is necessarily true. But a necessarily true statement is not always a formal tautology.

Are there mathematical theorems which are contingent? Are all mathematical theorems necessarily true?

Define a formal tautology as a statement where by the nature of its atomic components there exists no truth-value assignment where it is not true. A contingent statement is a statement that is true by facts of the world. A statement is necessarily true if the statement is true in all possible worlds. A necessarily true statement is not contingent and a contingent statement is not necessarily true. A formal tautology is necessarily true. But a necessarily true statement is not always a formal tautology.

Are there mathematical theorems which are contingent? Are all mathematical theorems necessarily true?

If we define a tautology as proposition that is always true (or necessarily true) are all mathematical theorems tautologies?

Define a formal tautology as a statement where by the nature of its atomic components there exists no truth-value assignment where it is not true. A contingent statement is a statement that is true by facts of the world. A statement is necessarily true if the statement is true in all possible worlds. A necessarily true statement is not contingent and a contingent statement is not necessarily true. A formal tautology is necessarily true. But a necessarily true statement is not always a formal tautology.

Are there mathematical theorems which are contingent? Are all mathematical theorems necessarily true?