Chapter 2 Logic Flashcards (Complex Conditionals + Beginning Derivation)

Contrapositive

S1 is necessary for S2 can be translated to (~S1 → ~S2) When the antecedent and consequent of a conditional are swapped, then negated. Conditionals and their contrapositives are logically equivalent.

Sc unless Sa: Q translation

(~Sa → Sc) = Contrapositive of Sa otherwise Sc, (~Sc → Sa)

Sa otherwise Sc: Q translation

(~Sc → Sa) = Contrapositive of Sc unless Sa, (~Sa → Sc)

If Sa then Sc, otherwise, So : Q translation

Pair: (Sa → Sc), (~Sa → So)

Sc but not/except if Sa

Pair: (Sa → ~Sc), (~Sa → Sc)

Active Line

A line is active at a line if-f it is either 1) within the main derivation or 2) within a subproof that is not closed at that line.

Closed Subproof

A proof is closed at a line if-f it has been discharged on a prior line by use of →I, ~I, ~E, vE, or 3E that cites that subproof.

Work Backwards/Work Forwards technique

Derive the main operator using the corresponding Introduction rule, then work forwards to derive new lines from premises, assumptions, and further lines from working forwards, using steps in the following priority: →E, R, ~E.

Never open a subproof unless you know how you are going to…

…close the subproof.

→ I Schema

→ E Schema

~I Schema

~E Schema

=I Schema

=E Schema