Construct the truth-table for the following propositional formulas. In each case, explain whether the formula is...

Construct the truth-table for the following propositional formulas. In each case, explain whether the formula is a tautology, a contradiction, or neither. (Explain how you arrive at this conclusion.) (a) ¬((p → ¬p) → ¬p) (b) (p → (q ∧ r)) → (¬r → ¬p) (c) (p → ¬q) → ¬(¬p → q)

Expert Solution

If the truth table gives only TRUE values then the proposition is called Tautology

If the truth table gives only FALSE values then the proposition is called Contradiction

If the truth table gives neither tautology nor contradiction then it is called as contingency

¬((p → ¬p) → ¬p)

P	Q	¬p	¬q	p → ¬p	(p → ¬p) → ¬p	¬((p → ¬p) → ¬p)
T	T	F	F	F	T	F
T	F	F	T	F	T	F
F	T	T	F	T	T	F
F	F	T	T	T	T	F

So this was a contradiction

(p → (q ∧ r)) → (¬r → ¬p)

P	Q	R	¬p	¬r	(q ∧ r)	(p → (q ∧ r))	(¬r → ¬p)	(p → (q ∧ r)) → (¬r → ¬p)
T	T	T	F	F	T	T	T	T
T	T	F	F	T	T	T	F	F
T	F	T	F	F	T	T	T	T
T	F	F	F	T	F	F	F	T
F	T	T	T	F	T	T	T	T
F	T	F	T	T	T	T	T	T
F	F	T	T	F	T	T	T	T
F	F	F	T	T	F	T	T	T

So this was a contingency

(p → ¬q) → ¬(¬p → q)

P	Q	¬p	¬q	p → ¬q	¬p → q	¬(¬p → q)	(p → ¬q) → ¬(¬p → q)
T	T	F	F	F	T	F	T
T	F	F	T	T	T	F	F
F	T	T	F	T	F	T	T
F	F	T	T	T	F	T	T

So this was a contingency

Colby Messinger answered 1 year ago

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