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Determine whether the following arguments are tautologies, contingencies, or con...

Mar 16, 2024

Determine whether the following arguments are tautologies, contingencies, or contradictions in system P. a. (A ≡ ~A) ⊃ ~(A ≡ ~A) b. ~[(A ≡ B) ⊃ (~A ⊃ ~B)] c. (~B & ~A) V ~(B V A)

Solution by Steps

step 1

To determine if the argument is a tautology, contingency, or contradiction, we will analyze the logical structure of each statement

step 2

For statement a, we will use the logical equivalence of

A \equiv \neg A

which is always false, as a statement cannot be both true and its negation at the same time

step 3

Therefore,

(A \equiv \neg A)

simplifies to

F

(false), and

\neg (A \equiv \neg A)

simplifies to

T

(true)

step 4

The conditional

(F) \supset (T)

is always true, because a false hypothesis can never make a conditional false

Answer

a. Tautology

Key Concept

Logical Equivalence and Conditional Statements

Explanation

A statement and its negation are logically equivalent to false, and a conditional with a false antecedent and true consequent is always true, making it a tautology.

Solution by Steps

step 1

For statement b, we will analyze the logical structure of the statement using De Morgan's laws and the properties of logical equivalence and implication

step 2

The statement

(A \equiv B) \supset (\neg A \supset \neg B)

is a tautology because it represents the contrapositive of the implication

A \supset B

step 3

Negating a tautology gives us a contradiction, so

\neg [(A \equiv B) \supset (\neg A \supset \neg B)]

is always false

Answer

b. Contradiction

Key Concept

Contrapositive and Negation of Tautologies

Explanation

The negation of a tautology is a contradiction, and the given statement is the negation of a tautological contrapositive.

Solution by Steps

step 1

For statement c, we will use the distributive laws of logic and De Morgan's laws to simplify the expression

step 2

The statement

(\neg B \land \neg A) \lor \neg (B \lor A)

can be simplified by recognizing that

\neg (B \lor A)

is equivalent to

(\neg B \land \neg A)

step 3

This means the statement simplifies to

(\neg B \land \neg A) \lor (\neg B \land \neg A)

, which is equivalent to

\neg B \land \neg A

step 4

Since

\neg B \land \neg A

is not always true or always false without knowing the truth values of A and B, it is a contingency

Answer

c. Contingency

Key Concept

Distributive Laws and De Morgan's Laws

Explanation

The statement simplifies to a form that is not always true or false, making it a contingency as its truth value depends on the truth values of A and B.