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


Two basic logical truths:


[T1]  P or not-P



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]



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



Therefore Q


Modus Tollens:


If P then 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




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:


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