Modal Logic

Aristotle is also the originator of modal logic. In addition to quality (as affirmation or negation) and quantity (as singular, universal, particular, or indefinite), he takes categorical sentences to have a mode; this consists of the fact that the predicate is said to hold of the subject either actually or necessarily or possibly or contingently or impossibly. The latter four are expressed by modal operators that modify the predicate, e.g. ‘It is possible for A to hold of some B’; ‘Anecessarily holds of every B’.
In De Interpretatione 12–13, Aristotle (i) concludes that modal operators modify the whole predicate (or the copula, as he puts it), not just the predicate term of a sentence. (ii) He states the logical relations that hold between modal operators, such as that ‘it is not possible for A not to hold of B’ implies ‘it is necessary for A to hold of B’. (iii) He investigates what the contradictories of modalized sentences are, and decides that they are obtained by placing the negator in front of the modal operator. (iv) He equates the expressions ‘possible’ and ‘contingent’, but wavers between a one-sided interpretation (where necessity implies possibility) and a two-sided interpretation (where possibility implies non-necessity).
Aristotle develops his modal syllogistic in Prior Analytics 1.8–22. He settles on two-sided possibility (contingency) and tests for syllogismhood all possible combinations of premise pairs of sentences with necessity (N), contingency (C) or no (U) modal operator: NN, CC, NU/UN, CU/UC and NC/CN. Syllogisms with the last three types of premise combinations are called mixed modal syllogisms. Apart from the NN category, which mirrors unmodalized syllogisms, all categories contain dubious cases. For instance, Aristotle accepts:

اقتباس :: A necessarily holds of all B.
B holds of all C.
Therefore A necessarily holds of all C.

This and other problematic cases were already disputed in antiquity, and more recently have sparked a host of complex formalized reconstructions of Aristotle's modal syllogistic. As Aristotle's theory is conceivably internally inconsistent, the formal models that have been suggested may all be unsuccessful.

3. The early Peripatetics: Theophrastus and Eudemus

Aristotle's pupil and successor Theophrastus of Eresus (c. 371–c. 287 BCE) wrote more logical treatises than his teacher, with a large overlap in topics. Eudemus of Rhodes (later 4th cent. BCE) wrote books entitled Categories, Analytics and On Speech. Of all these works only a number of fragments and later testimonies survive, mostly in commentators on Aristotle. Theophrastus and Eudemus simplified some aspects of Aristotle's logic, and developed others where Aristotle left us only hints.

3.1 Improvements on and Modifications of Aristotle's Logic

The two Peripatetics seem to have redefined Aristotle's first figure, so that it includes every syllogism in which the middle term is subject of one premise and predicate of the other. In this way, five types of non-modal syllogisms only intimated by Aristotle later in his Prior Analytics(Baralipton, Celantes, Dabitis, Fapesmo and Frisesomorum) are included, but Aristotle's criterion that first figure syllogisms are evident is given up (Theophrastus fr. 91, Fortenbaugh). Theophrastus and Eudemus also improved Aristotle's modal theory. Theophrastus replaced Aristotle's two-sided contingency with one-sided possibility, so that possibility no longer entails non-necessity. Both recognized that the problematic universal negative (‘A possibly holds of noB’) is simply convertible (Theophrastus fr. 102A Fortenbaugh). Moreover, they introduced the principle that in mixed modal syllogisms the conclusion always has the same modal character as the weaker of the premises (Theophrastus frs. 106 and 107 Fortenbaugh), where possibility is weaker than actuality, and actuality than necessity. In this way Aristotle's modal syllogistic is notably simplified and many unsatisfactory theses, like the one mentioned above (that from ‘Necessarily AaB’ and ‘BaC’ one can infer ‘Necessarily AaC’) disappear.

3.2 Prosleptic Syllogisms

Theophrastus introduced the so-called prosleptic premises and syllogisms (Theophrastus fr. 110 Fortenbaugh). A prosleptic premise is of the form:

اقتباس :: For all X, if Φ(X), then Ψ(X)

where Φ(X) and Ψ(X) stand for categorical sentences in which the variable X occurs in place of one of the terms. For example:

اقتباس :: (1) A [holds] of all of that of all of which B [holds].

اقتباس :: (2) A [holds] of none of that which [holds] of all B.

Theophrastus considered such premises to contain three terms, two of which are definite (A, B), one indefinite (‘that’, or the bound variable X). We can represent (1) and (2) as

اقتباس :: ∀X (BaX → AaX)

اقتباس :: ∀X (XaB → AeX)

Prosleptic syllogisms then come about as follows: They are composed of a prosleptic premise and the categorical premise obtained by instantiating a term (C) in the antecedent ‘open categorical sentence’ as premises, and the categorical sentences one obtains by putting in the same term (C) in the consequent ‘open categorical sentence’ as conclusion. For example:

اقتباس :: A [holds] of all of that of all of which B [holds].
B holds of all C.
Therefore, A holds of all C.

Theophrastus distinguished three figures of these syllogisms, depending on the position of the indefinite term (also called ‘middle term’) in the prosleptic premise; for example (1) produces a third figure syllogism, (2) a first figure syllogism. The number of prosleptic syllogisms was presumably equal to that of types of prosleptic sentences: with Theophrastus' concept of the first figure these would be sixty-four (i.e. 32 + 16 + 16). Theophrastus held that certain prosleptic premises were equivalent to certain categorical sentences, e.g. (1) to ‘A is predicated of all B’. However, for many, including (2), no such equivalent can be found, and prosleptic syllogisms thus increased the inferential power of Peripatetic logic.

3.3 Forerunners of Modus Ponens and Modus Tollens

Theophrastus and Eudemus considered complex premises which they called ‘hypothetical premises’ and which had one of the following two (or similar) forms:

اقتباس :: If something is F, it is G

اقتباس :: Either something is F or it is G (with exclusive ‘or’)

They developed arguments with them which they called ‘mixed from a hypothetical premise and a probative premise’ (Theophrastus fr. 112A Fortenbaugh). These arguments were inspired by Aristotle's syllogisms ‘from a hypothesis’ (An. Pr. 1.44); they were forerunners of modus ponens and modus tollens and had the following forms (Theophrastus frs. 111 and 112 Fortenbaugh), employing the exclusive ‘or’:

If something is F, it is G. a is F. Therefore, a is G.	If something is F, it is G. a is not G. Therefore, a is not F.
Either something is F or it is G. a is F. Therefore, a is not G.	Either something is F or it is G. a is not F. Therefore, a is G.

Theophrastus also recognized that the connective particle ‘or’ can be inclusive (Theophrastus fr. 82A Fortenbaugh); and he considered relative quantified sentences such as those containing ‘more’, ‘fewer’, and ‘the same’ (Theophrastus fr. 89 Fortenbaugh), and seems to have discussed syllogisms built from such sentences, again following up upon what Aristotle said about syllogisms from a hypothesis (Theophrastus fr. 111E Fortenbaugh).

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Modal Logic

3. The early Peripatetics: Theophrastus and Eudemus

3.1 Improvements on and Modifications of Aristotle's Logic

3.2 Prosleptic Syllogisms

3.3 Forerunners of Modus Ponens and Modus Tollens

Modal Logic :: تعاليق

Modal Logic

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