M. Kagan, Reductio (Some deductive logic), modified 01.26.2016

Two basic logical truths:

[T1]  P or not-P

Example:

This identification has been altered or it is not the case that this identification has been altered.

[T2]  Not (Q and not-Q)

[T3]  (Not not P) if and only if P [double negation]

Example:

It is not the case that Alethea is a professional basketball  player and that Alethea is not a professional basketball player.

Two basic logical definitions:

(i)  A deductive argument is valid if the truth of the premisses guarantees the truth of the conclusion; i.e., one who accepts the premisses and denies the conclusion is inconsistent.

Example of a valid deductive argument:

P & Q

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Therefore P

(ii)  A deductive argument is sound if it is valid and HAS true premises.

Example of a sound deductive argument:

P or not-P

-----

Therefore not -(Q & not-Q)

A few valid argument forms:

Modus Ponens:

If P then Q

P

--------

Therefore Q

Modus Tollens:

If P then Q

Not-Q

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Therefore not-P

Conditional proof:

Show that if we suppose P, we can infer Q; this allows us to infer:

If P then Q

Reductio:

Show that if we suppose -P, we can infer (Q & not-Q);  this allows us to claim (by conditional proof):

(a)  If not-P then (Q & not-Q)

Which allows us to infer:

P

(when we apply T2 to (a) with modus tollens)