No book of logic is not complete without some mention of inferences, and Bolzano’s is no exception. At worst, such an account devolves into a tedious listing of disparate specimens of inferences the author has deemed important. Bolzano is in danger of doing just that, but he does admit that such a listing can never be complete and tries at least to provide an inkling of systematicity to the proceedings.
Just to see more closely, what the method of Bolzano is in going through inferences, I shall give more details of the beginning of Bolzano’s account. He starts with what he calls the simplest kind of proposition: “A has b”. Note that by a lower case letter Bolzano always refers to a characteristic, while a corresponding upper case letter refers to all the objects that have such a characteristic. Thus, Bolzano’s simplest proposition means actually “all A have b”, and indeed, “all A are B” – the traditional form of universal proposition. Just like traditionally, the subject term has some objects to refer to, which Bolzano indeed takes as the first conclusion that can be drawn from this simple type of proposition.
After going through some equally simple conclusions that can be drawn from a single proposition of the simplest type, Bolzano moves on to discuss combinations of this type of propositions. To facilitate the discussion, he moves to an equivalent form “what has a, has b”. Bolzano points out that if we start with two propositions, they could share both representations a and b, only one of them or none.
Starting with the first case, Bolzano points out that the propositions could literally taken be only “what has a, has b” and “what has b, has a”, but he suggests adding the possibility of negating either a or b, leading to eight possible propositions:
1) what has a, has b,
2) what has b, has a,
3) what has a, has not b,
4) what has b, has not a,
5) what has not a, has b,
6) what has not b, has a,
7) what has not a, has not b and
8) what has not b, has not a.
The first proposition could always be taken as 1), because the important thing is the relation of the representations in the two propositions, and if needed, we can always say that not-not-a is equivalent to a. This leaves us seven other options for the second proposition, and even from these, we can take the options 3), 4) and 6), which cannot be connected with 1). Thus, we can infer from 1) and 2) that A and B are equivalent, from 1) and 5) that everything has b, and from 1) and 8) that B is not subordinate to A, but is also not equivalent to the representation of something in general (1) and 7) lead to various conclusions, which I’ll leave unmentioned for simplicity’s sake).
Bolzano goes on for quite a while with such a level of detail, although he eventually moves into a more summarised presentation. We do not need to follow him that far, but we can just note that next he goes on, as he promised, to discuss combination of propositions like “what has a, has b” and “what has a, has c”, sharing only one representation, which are, in effect, Aristotelian syllogisms using only universal premisses. He also considers the case where the propositions do not share any representations – “what has a, has b” and “what has c, has d” – the conclusion of which is that the whole combined of objects of A and B is a part of a whole combined of objects in B and D. Bolzano completes the study of inferences using only the simplest type of proposition with a consideration of cases with more than two premisses.
The next move Bolzano takes is to consider negations of type “it is false that A has b” (not to be confused with “A has not-b”, which would still imply that A refers to some objects, which the negation leaves open). Here we see the pattern Bolzano is making: he starts with inferences from a single negation, then proceeds to discuss inferences using both negations and affirmations of the first type and finally moves on to the case of several negations.
Bolzano follows the same systematic approach with propositions stating that a representation refers (or does not refer) to objects: he looks at individual propositions, then its combinations with propositions of the same or one of the previous types. The only interesting new feature is that in addition to propositions “representation A refers (or does not refer) to objects” he also considers propositions where the A has been analysed further, for instance, “representation of something that has characteristics a and b refers to an object” (these form the traditional particular premisses in Aristotelian syllogisms).
The next couple of types of propositions include such that indicate that a representation refers to one, several of a certain number of objects. What is interesting in these cases is that they imply further propositions about the sums of these sets of objects, at least if we know that they do not share any objects.
Bolzano moves on to discuss propositions about relations of representations, such as “representations A and C are compatible”. He notes that when we speak of mere individual representations, many of these propositions are actually equivalent to propositions considered earlier (thus, the just mentioned example says that some A are C), but his habit of speaking about collections of representations makes the discussion more complex. Thus, a proposition about comprehension between collections of representations (“every object that falls under one of the representations A, B, C… falls also under one of representations M, N, O…”) leads to a complex disjunction, where each individual object belonging to some representation of the first collection belongs to some representation of the second collection.
Bolzano has thus far discussed propositions about representations, but moves next to propositions about propositions, for example, the relations of derivability, equivalence and contradiction. What makes his account hard to follow is his use of variable representations to indicate that it is not individual propositions, but propositions of certain form he is discussing here. This feels an unnecessary complication at this point, when he is, for the most part, speaking only of inferences involving relations of propositions in general (e.g. variations of modus ponens and tollens), without indicating what the propositions are like (the only exception consists of such simple cases as premiss “if A is B, then A is D” leading to conclusion “every B is D”).
The final kind of inferences Bolzano considers involve probability, where the conclusion is not certainly true, but only of certain probability. Bolzano does not really dive deep into such probable inferences, but after some simple inferences (e.g. if probability of proposition is m, probability of its negation is 1 – m), he merely notes induction and analogy as important subtypes or probable inferences.
Ei kommentteja:
Lähetä kommentti