**((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

In English

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.

see Wikipedia

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

Logical Implications