e.g., ‘If the sun is shining, the plants will grow; the sun is shining; therefore the plants will grow’. Symbolically, this is represented by ‘P P Q, P /, Q’. This argument form was explicitly formulated in ancient times by the Stoics as one of the ‘indemonstrables’ and is now known as modus ponens. Another equally basic form of mixed hypothetical syllogism is ‘P P Q, -Q /, ~P’, known as modus tollens.
The disjunctive syllogism. This is an argument in which the leading premise is a disjunction, the other premise being a denial of one of the alternatives, concluding to the remaining alternative; e.g., ‘It is raining or I will go for a walk; but it is not raining; therefore I will go for a walk’. It is not always clear whether the ‘or’ of the disjunctive premise is inclusive or exclusive. Symbolic logic removes the ambiguity by using two different symbols and thus clearly distinguishes between inclusive or weak disjunction, ‘P 7 Q’, which is true provided not both alternatives are false, and exclusive or strong disjunction, ‘P W Q’, which is true provided exactly one alternative is true and exactly one false. The definition of ‘disjunctive syllogism’ presupposes that the lead premise is an inclusive or weak disjunction, on the basis of which two forms are valid: ‘P 7 Q, -P /, Q’ and ‘P 7 Q, -Q /, P’. If the disjunctive premise is exclusive, we have four valid argument forms, and we should speak here of an exclusive disjunctive syllogism. This is defined as an argument in which either from an exclusive disjunction and the denial of one of its disjuncts we infer the remaining disjunct – ‘P W Q, -P /,Q’, and ‘P W Q, -Q /, P’ (modus tollendo ponens); or else, from an exclusive disjunction and one of its disjuncts we infer the denial of the remaining disjunct – ‘P W Q, P /, -Q’, and ‘P W Q, Q /,-P’ (modus ponendo tollens). I.Bo.