((P => Q) ^ not Q) => not P named Modus tollens meaning the way that denies by denying – denying the consequent Q.
If the compound statement (P => Q) ^ not Q) => not P is given and is true (it is always true) and P => Q is true then not Q must be true – therefore not P must be true (see truth table to follow the reasoning).
P => Q (P implies Q true)
^ not Q (true)
then not P is True
P = an intruder is detected
Q = the alarm goes off
(P=>Q) If an intruder is detected, then the alarm goes off.
(^ not Q) and The alarm does not go off.
(=> not P) then, no intruder is detected
said a little more coherently:
If an intruder is detected, the alarm will go off, and when the alarm does not go off, then, no intruder is detected.
3 no namers – logical implications
Hypothetical Syllogism – a logical implication
Disjunctive Syllogism – a logical implication
Simplification – a logical implication
Addition – a logical implication
Modus tollens – a logical implication
Modus ponens – a logical implication