the premisses of a pure syllogism are both categorical or both pothetical, the pure syllogism is Categorical or Hypothetical. mixed syllogism has one premiss categorical and the other thetical, or one premiss categorical and the other disjunctive, orstly, one conjunctive and the other disjunctive, it is called (1) pothetical-categorical, (2) Disjunctive-categorical, or (3) Contive-disjunctive. By a conjunctive proposition is here meant roposition of the form 'Neither A nor B is C' (Remotive), or of the form 'A as well as B is C' (Copulative), and it is either egorical or hypothetical. The examples we have given are torical. The hypotheticals are of the following forms :It A is, neither B nor C is D (Remotive). If A is, B as well as C is D (Copulative). The subdivisions may be shown in a tabular view: SALOGISMS. I. Pure... II. Mixed. : (1. Categorical, consisting of two categorical premisses. 2. Hypothetical, consisting of two hypothetical premisses. 1. Hypothetical-categorical, consisting of one premiss hypothetical and the other categorical. 2. Disjunctive-categorical, consisting of one premiss disjunctive and the other categorical. 3. Conjunctive-disjunctive, consisting of one premiss conjunctive and the other disjunctive. § 2. I.-Of Pure Syllogisms. The general syllogistic rules and the special rules which we have given in a previous chapter are applicable to hypothetical, as well as to categorical, syllogisms. Of the latter we have given numerous examples. We shall now give some examples of the former. In applying the general and the special rules to pure hypothetical syllogisms, we must remember (1) that the antecedent of a hypothetical proposition corresponds to the subject, and the consequent to the predicate in the corresponding categorical proposition; (2) that the quantity of a hypothetical proposition is the quantity of its antecedent, and is expressed by such phrases as 'in all cases' and 'in some cases' or 'in one case at least,' the former denoting universal and the latter particular quantity; (3) that the quality of a hypothetical proposition is the quality of its consequent; (4) that the rules for the distribution of terms are the same as in categorical propositions, i.e., the antecedent must be distributed in hypothetical propositions of the form A or E, and the consequent in hypothetical propositions of the form E or O. We shall give the following typical examples of Pure Hypothetical Syllogisms, and change them at the same time into the corresponding Categoricals: Changed into the corresponding categorical: Every case of the existence of B is a case of the existence of C, Every case of the existence of A is a case of the existence of B; ... Every case of the existence of A is a case of the existence of C. II. Celarent: E. In all cases, if B is, C is not A. In all cases, if A is, B is E... In all cases, if A is, C is not ... ... ... Changed into the corresponding categorical: (major premiss), (minor premiss); (conclusion). No case of the existence of B is a case of the existence of C, Every case of the existence of A is a case of the existence of B; .. No case of the existence of A is a case of the existence of C. III-Darii : Changed into the corresponding categorical: Every case of the existence of B is a case of the existence of C, Some cases of the existence of A are cases of the existence of B; .. Some cases of the existence of A are cases of the existence of C. No case of the existence of C is a case of the existence of B, Every case of the existence of A is a case of the existence of B; .. No case of the existence of A is a case of the existence of C. Similar examples may be given of the fourth figure, and also of the other moods of the first three figures. § 3. II. Of Mixed Syllogisms. We have seen that there are at least three subdivisions, namely, (1) Hypothetical-categorical, (2) Disjunctive-categorical, (3) Conjunctive-disjunctive. We shall take these in order— 1. Of Hypothetical-categorical Syllogisms. A syllogism of this subdivision consists of a hypothetical major and a categorical minor premiss, the conclusion being categorical. The rules of inference are as follows: (1) If you affirm the antecedent, you may affirm the consequent of a hypothetical premiss, but not conversely, that is, it is not allowed to affirm the antecedent on affirming the conse quent. This rule is for what has been called a Constructive Hypothetical Syllogism. (2) If you deny the consequent, you may deny the antecedent of a hypothetical premiss, but not conversely, that is, it is not allowed to deny the consequent on denying the antecedent. This rule is for what has been called a Destructive Hypothetical Syllogism. Both these rules follow from the nature of the relation of dependence, expressed by a hypothetical proposition, between its antecedent and consequent. The second part of the first rule follows from the fact that the consequent may depend upon other antecedents as well as upon that antecedent, and that therefore the existence or affirmation of the consequent does not necessarily imply the affirmation of that particular antecedent, but of some one of them, and this one may not be the antecedent in question. The second part of the second rule follows from the same fact, for the consequent depending, as it may, on other antecedents as well, may exist while the particular antecedent is absent; and therefore the denial of the consequent does not follow from the denial of the antecedent. For example, in the proposition "If a person be attacked with cholera, he will die," -assuming this to be true-it does not follow that, if he be not attacked with cholera, he will not die; for he may die of consumption, fever, or some other disease. Nor does it follow that if he dies, he must have been attacked with cholera, for he may die of other diseases. All that is really meant by the proposition in question is that if he gets cholera, he is sure to die; if the antecedent is present, the consequent must be present, and that if he does not die, he has not had cholera, i.e. if the consequent does not occur, the antecedent can not have occurred. We shall give some typical examples of Hypotheticalcategorical syllogisms, and change them at the same time into the corresponding categoricals, in order to show that, when thus changed, they conform to the fundamental rules and axioms of categorical syllogisms : I. Constructive Hypothetical-categorical Syllogisms. 1. In all cases, if A is, B is, A is; .. B is. This mode of drawing an inference is called modus ponendo ponens,- -i.e. the mode which by affirming the antecedent affirms the consequent according to the first rule given above; and the syllogism has been called a constructive hypothetical syllogism. A. A. It may be thus changed into a categorical: Every case of the existence of A is a case of the existence of B, A... This is a case of the existence of B. The syllogism is in the mood Barbara. A Hypothetical-categorical syllogism may be also changed into a pure hypothetical syllogism; for the meaning of the minor proposition 'A is' is, that 'if this case is, A is.' By substituting this hypothetical minor premiss for the categorical, we get a pure hypothetical syllogism in the mood Barbara, thus:— The converse of the first rule does not lead to a valid syllogism- In all cases, if A is, B is, B is; .. A is. inference is not valid; and its invalidity can be shown ring it into the corresponding categorical, when it will at the latter violates some of the syllogistic rules, es of the existence of A is a case of the existence of B, aase of the existence of B. |