Proving the Aristotelian square of opposition in predicate ...
Proving the Aristotelian square of opposition in predicate logic:
(Symbols in bold denote the application of the relevant rule of inference from the previous row.)
A universal affirmative proposition (“All S are P”) is mutually contradictory with its respective particular negative proposition (“Some S is not P”).
(x)(Sx ( Px) (Universal affirmative proposition: “All S are P”)
(((x)((Sx ( Px) change of quantifier rule
(((x)(((Sx ( Px) material implication
(((x)(((Sx ( (Px) DeMorgan’s rule
(((x)(Sx ( (Px) double negation
And this, of course, is the negation of a particular negative proposition, expressed in predicate logic as “((x)(Sx ( (Px)”.
A universal negative proposition (“No S are P”) is mutually contradictory with its respective particular affirmative proposition (“Some S is P”).
(x)(Sx ( (Px) (Universal negative proposition: “No S are P”)
(((x)((Sx ( (Px) change of quantifier rule
(((x)(((Sx ( (Px) material implication
(((x)(((Sx ( ((Px) DeMorgan’s rule
(((x)(Sx ( Px) double negation (two applications)
And this, of course, is the negation of a particular affirmative proposition, expressed in predicate logic as “((x)(Sx ( Px)”.
A particular negative proposition (“Some S is not P”) is mutually contradictory with its respective universal affirmative proposition (“All S are P”).
((x)(Sx ( (Px) (Particular negative proposition: “Some S is not P”)
((x)((Sx ( (Px) change of quantifier rule
((x)((Sx ( ((Px) DeMorgan’s rule
((x)(Sx ( ((Px) material implication
((x)(Sx ( Px) double negation
And this, of course, is the negation of a universal affirmative proposition, expressed in predicate logic as “(x)(Sx ( Px)”.
A particular affirmative proposition (“Some S is P”) is mutually contradictory with its respective universal negative proposition (“No S are P”).
((x)(Sx ( Px) (Particular affirmative proposition: “Some S is P”)
((x)((Sx ( Px) change of quantifier rule
((x)((Sx ( (Px) DeMorgan’s rule
((x)(((Sx ( (Px) material implication
((x)(Sx ( (Px) double negation
And this, of course, is the negation of a universal negative proposition, expressed in predicate logic as “(x)(Sx ( (Px)”.
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