Logical form and validity of argument

A reasoning with premises P1, P2, P3, etc. and conclusion C is valid iff its corresponding conditional is valid ( = is aa tautology).

By " corresponding conditional" I mean : " P1 & P2 & P3... --> C"

( For this kind of problem, this definition is perfectly OK).

So build the corresponding conditional of your reasoning by putting an "&" between the premises, adding an arrrow and finally your conclusion.

Build a truth table for this conditional.

Note : here you need a 2 to the 4th power = 16 lines truth table.

In case this conditional has truth value " true" on all lines of the truth table, the reasoning is valid.

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Note : apparently, premises 2 and 3 are useless, since the conclusion can be proved using only premises 1 and 4

(1) P --> ~ Q

(2) Q & S

(3) Since Q and S is true, Q is true.

(4) Since P --> ~Q is true , Q --> ~P is true ( by contraposition)

(5) Since Q --> ~P and Q are true, ~ P is true ( as desired).

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Logic

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