Question

Suppose that φ≡ψ. Show that:
1. if φ|=θ, then ψ|=θ;
2. if θ|=φ, then θ|=ψ

Answer 1

In a college-level logic problem, we've shown that if two statements φ and ψ are logically equivalent, then any statement entailed by φ is also entailed by ψ and vice versa, due to them having identical truth conditions.

Step-by-step explanation:

The subject of this question is Mathematics, specifically within the logic or proof-related problems typically encountered in a college level course. We are asked to show two propositions:

1. If φ⊨θ, then ψ⊨θ given that φ≡ψ.
2. If θ⊨φ, then θ⊨ψ given that φ≡ψ.

In logic, the symbol ∝ represents the phrase 'entails' or 'logically implies', and ≡ represents logical equivalence. To show that if two statements φ and ψ are logically equivalent, we must understand that both statements are true under the same conditions or interpretations. Therefore, if φ logically implies θ, ψ being equivalent to φ will necessarily imply θ as well. This is because φ and ψ share the same truth conditions.

Conversely, if θ entails φ and φ is equivalent to ψ, the implication will also hold from θ to ψ. Since φ and ψ have identical truth values in every possible scenario, any statement entailed by φ is also entailed by ψ and vice versa. Hence, if θ entails φ, it also entails ψ due to their logical equivalence.