# Propositions Involving Quantifiers

• Jun 11th 2013, 12:29 PM
Bashyboy
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?
• Jun 12th 2013, 08:52 AM
emakarov
Re: Propositions Involving Quantifiers
Quote:

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.