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
-------
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
----------
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)