Formal Logic: Truth-Value Analysis

Question

I'm working through Quine's Methods of Logic book, and I'm stuck on an example he gave in the book to demonstrate truth-value analysis.

The example is determining the truth value of pq V ¬p¬r →q ↔r where 'p' and 'q' are interpreted as false and 'r' as true. The answer he provides to the statement is as follows:

1) ┴┴ v T┴ → ┴ ↔ T

2) ┴┴ v T┴ → ┴

3) ¬(┴┴ v T┴)

4) ¬(┴ v ┴)

5) T

Just in case it needs clarified, "T" is meaning true, and "┴" meaning false. (Apologies if these or anything else written aren't the correct logical connective or the ones commonly used. This is my first post here and I'm new to formal logic)

He has in the book nine laws of resolution, and one of them is "If a conditional has "┴" as a consequent, reduce the whole to the negation of the antecedent." But in practice I'm confused about this rule and it tripped me up. Going from 2 to 3, why does this make sense to reduce the whole to the negation of the antecedent?

Answer 1

See the truth-table for the conditional : p ⇒ q and consider the two rows where q is FALSE (i.e. ⊥).

We have :

p ⇒ ⊥ is ⊥ when p is T

while :

p ⇒ ⊥ is T when p is ⊥.

Conclusion : p ⇒ ⊥ is equivalent to ¬p.