Leibnizian Intensional Semantics for Syllogistic Reasoning

In the history of logic, negative terms are also known as indefinite or infinite terms.

For some adjectives (like ‘tall’ and ‘heavy’) it seems less unreasonable to propose that their negative counterparts have essential properties, but it is perhaps no accident that in natural language these negative counterparts are expressed positively by their antonyms (like ‘short’ and ‘light’).

The intensional view is also explicitly discussed in Wittgenstein’s Tractatus, 5.1222: if \(p\) follows from \(q\), then the sense of \(p\) is contained in the sense of \(q\).

According to Leibniz, Aristotle, in contrast to a nominalist like Locke, preferred the intensional interpretation:

Philalethes (expressing Locke’s view) [\(\ldots \)] it appeared to me preferable to reverse the order of the premisses of syllogisms, and to say: All A is B, all B is C, so all A is C, rather than saying All B is C, all A is B, so all A is C. [\(\ldots \)]

Theophilus (expressing Leibniz’s view) [\(\ldots \)] Aristotle may have had a special reason for adopting [what is now] the common arrangement. For rather than saying ‘A is B’ he usually says ‘B is in A’ [\(\ldots \)]. And with that way of stating it he achieves, through the accepted arrangements, the very connection which you insist upon. For instead of saying ‘B is C, A is B, so A is C’, Aristotle will express it thus: ‘C is in B, B is in A, so C is in A’. For instance, instead of saying ‘Rectangles are isogons (i.e. have equal angles), squares are rectangles, so squares are isogons’, Aristotle will put the ‘middle term’ in the middle position without changing the order of the propositions, by stating each of them in a manner which reverses the order of terms, thus: ‘Isogon is in rectangle, rectangle is in square, so isogon is in square’. This manner of statement deserves respect; for indeed the predicate is in the subject, or rather the idea of the predicate is included in the idea of the subject. [\(\ldots \)] The common manner of statements concerns individuals, whereas Aristotle’s refers rather to ideas or universals. [\(\ldots \)]

Leibniz, New Essays on Human Understanding, Book 4, Chap. 17, Sect. 8)

Having a non-empty intersection of the intensions of \(S\) and \(T\) is not enough for the sentence SiP to be true: although both gold and silver clearly share a property (e.g. being a metal) this doesn’t mean that there is something both pure gold and pure silver.

Glashoff [10] rightly complains that Sotirov’s solution is not completely in the spirit of Leibniz’s assumptions: Leibniz asumed that the builiding blocks (the prime numbers) can be an infinite set. This is impossible with Sotirov’s solution.

Sommers [25] proposed an alternative numerical way to account for for syllogistic reasoning without making use of prime numbers, and more in the spirit of the medieval distribution theory. Unfortunately, Sommers’ numerical method alone doesn’t quite do the job. He needs an additional non-numerical rule: the requirement that for a syllogism to be valid, the number of particular conclusions must equal the number of particular premises. Friedman [8] improved on Sommers’ method by getting rid of this additional rule. In fact, he showed that there are at least two purely numerical ways to account for syllogistic reasoning. According to the additional method one should replace \(SaP\) by \(-S+P\), \(SiP\) by \(+S+P\), \(SeP\) by \(-S-P-1\), and \(SoP\) by \(+S-P-1\). Let \(\phi '\) be the result of the replacement of sentence \(\phi \). Then one can show that \(\phi _1, \cdots , \phi _n\vdash \psi \) iff \(\phi '_1 + \cdots + \phi '_n = \psi '\). According to the multiplicational method we replace \(SaP\) by \(\frac{P}{S}\), \(SiP\) by \(2SP\), \(SeP\) by \(\frac{-1}{SP}\), and \(SoP\) by \(\frac{-2S}{P}\). If we denote the result of the replacement in this way by \(\phi "\), it follows that \(\phi _1, \cdots , \phi _n\vdash \psi \) iff \(\phi "_1 \times \cdots \times \phi "_n = \psi "\). Both methods validate all and only the valid syllogism, but the multiplicational method has an advantage because it allows for a natural representation of negative terms: \(\overline{P}\) is represented as \(\frac{1}{P}\). Not using prime numbers makes the calculations easier, but note that the resulting systems are anything but a characteristics universalis. In fact, the resulting systems cannot be thought of as semantic systems at all.

Let \(M\) be assigned \(\langle 10, 3\rangle \), \(S\) be \(\langle 8, 11\rangle \), and \(P\) be \(\langle 5,1\rangle \). On this assignment, the syllogism MaP, \(\textit{MoS}/ \textit{SoP}\) is wrongly predicted to valid.

My proposal is thus closer to Sotirov’s [26] approach.

This is very well known, see, for instance [21, 26].

If we think of the extensional counterpart, this means that ‘some bike is red’ is true not because there actually exists a red bike, but rather that it is possible that such a bike exists. And indeed, what Leibniz considers to be the extension of a term (a set of individuals scattered around all worlds) is very much what in possible worlds semantics is its intension (cf. [13] and [12, p. 49]).

The idea behind these constraints is similar to [18] idea to assume that each term intensionally denotes a proper filter.

Proof: \(x \le y\) iff \(x \star y = x\) iff \(\overline{x \star y} = \overline{x}\) iff \(\overline{x} \circ \overline{y} = \overline{x}\) iff \(\overline{y} \le \overline{x}\).

In fact, we end up with something very close to the syllogistic counterpart of a semantics of Anderson and Belnap’s [1] notion of tautological (or relevant) entailment. Indeed, I have based the semantics on some ideas of van Fraassen [6], which is used to gives a semantics for this logic.

See the same paper for a proof why double negation holds in our semantics.