Just like the the investigation of judgements, the investigation of syllogisms is much more extensive in Wolff's Latin than in his German logic. While German logic concentrated on the so-called first figure of syllogisms and noted that all other figures could be reduced to the first figure, in Latin logic Wolff goes through even the second and third figures of syllogism. As I have adequately dealt with all these figures, I shall not touch this topic anymore.
This still doesn't mean that I could leave the rest of this text blank. The three figures all have categorical judgements as premisses, thus, syllogisms of these figures could be called categorical. As categorical judgements are the simplest type of judgements, Wolff also calls them simple syllogisms. In addition, we might also have syllogisms with non-categorical premisses, which appropriately are then called complex syllogisms. Wolff goes on to note that complex syllogisms include at least hypothetical syllogisms, with one hypothetical judgement as a premiss, and disjunctive syllogisms, with one disjunctive judgement as a premiss. There is no indication that these two are all the types of complex syllogisms, although Kant will later appear to assume this.
Just like with the three figures, Wolff is eager to show that we can simplify the variety of syllogisms. Now, I failed to mention last time that Wolff appears not to take disjunctive judgements in the form of the current propositional logic, that is, as a combination of propositions (p or q). Instead, he favours the notion of disjunctive judgements as a combination of concepts: A is B or C. Furthermore, while in modern propositional logic ”p or q” is meant to include the possibility that both p and q are true, Wolffian ”A is B orC” is clearly meant to indicate that A cannot be both B and C at the same time. Thus, a disjunctive syllogism is then of the form
A is B or C
A is / is not B
Then A is not / is C
Wolff points out that we could then understand the judgement ”A is B or C” as a combination of hypothetical judgements ”if A is B, then it is not C” and ”if A is not B, then it is C”. Clearly then disjunctive syllogisms can be reduced to hypothetical judgements.
Wolff continues by noting that an important part of hypothetical syllogisms, including all the disjunctive syllogisms, could be reduced to categorical syllogisms, namely, those in which both the antecedent and consequent of the hypothetical judgement have the same subject, in other words, if the syllogism is of the form:
If A is B, then A is C
A is B
Thus A is C
Wolff suggests that we could always read such a syllogism in the following form:
All Bs are Cs
A is B
Thus, A is C
Problem is that in the original syllogism the hypothetical might hold only for As. Consider the following deduction:
If a triangle is a figure with two equally large angles, then it is an equilateral triangle
This triangle is a figure with two equally large angles
Thus, it is an equilateral triangle
Clearly the corresponding categorical premiss ”all figures with two equally large angles are equilateral triangles” is false. A possible solution is to restrict the scope of the middle term in the following manner:
All As that are Bs are Cs
This A is B
Thus, A is C
Still, even if the reduction works with these hypothetical syllogisms, Wolff has to admit that it isn't so easy in those cases, where the hypothetical judgement doesn't have a subject shared by its antecedent and consequent.
Wolff also considers incomplete syllogisms, which Aristotle had in his Rhetoric called enthymemes. In practice, such entyhmemes are nothing but hidden syllogisms, where we leave some premisses implicit. Thus, if I deduce ”I am hungry, therefore I must eat” I am actually assuming the general premiss ”if someone is hungry, he must eat".
An important subdivision of enthymems is formed by so-called immediate syllogisms, which according to the tradition could be used to prove something without full syllogistic trappings: an example includes ”All As are Bs, thus, some As are Bs”. Wolff points out that we could add as a new premiss the tautology ”some As are As” and then we would have a normal categorical syllogism, albeit one with a tautology as a premiss.
Somewhat more surprising is Wolff's view that even inductions are enthymemes or deductions with implicit assumptions. Namely, if we conclude from the fact that certain individuals or species of things have a feature that this feature is shared by whole of their common genus, we have assumed a further premiss that what holds for an individual or a species holds also for superior genera. It appears problematic to suppose that induction could be deduction, but I would like to point out that it is not meant to be valid deduction, because one of the premisses might well be untrue – the inductive principle or possibility to generalise might be wrong either absolutely or under some circumstances.
I might finally note that Wolff describes a possibility to concatenate syllogisms to form chains of deduction or polysyllogisms and that the highest form of deduction or demonstration can use only axioms, definitions, indubitable experiences and previously proven propositions as premisses. Next time we shall see what all this has to do with truth.