Propositions Involving Quantifiers

"Express each of these statements using logical operators, predicates, and quantifiers.

c) The disjunction of two contingencies can be a tautology."

The answer to this question is, $\displaystyle \exists x \exists y(\neg T(x) \wedge \neg C(x) \wedge T(y) \wedge \neg C(y) \wedge T(x \vee y)) $

I figured that the answer would involve an implication. Could someone possibly explain how this is the answer?

Re: Propositions Involving Quantifiers

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Originally Posted by

**Bashyboy** "Express each of these statements using logical operators, predicates, and quantifiers.

c) The disjunction of two contingencies can be a tautology."

The answer to this question is, $\displaystyle \exists x \exists y(\neg T(x) \wedge \neg C(x) \wedge T(y) \wedge \neg C(y) \wedge T(x \vee y)) $

Should T(y) be negated as well?

Quote:

Originally Posted by

**Bashyboy** I figured that the answer would involve an implication.

Why do you think so?

Quote:

Originally Posted by

**Bashyboy** Could someone possibly explain how this is the answer?

This statement says that there exist two formulas that are neither tautologies nor contradictions (i.e., they are contingencies) and such that their disjunction is a tautology.

P.S. Courtesy requires defining used notations. It took me a while to figure out that C(x) means that x is a contradiction and not a contingency.