Abstract
We proceed to develop logical systems intended to provide formal interpretations for the different layers of the modal syllogistic. In this Chapter, we construct an axiomatic system and formal semantics for an apodeictic system which includes ecthetic proofs of Baroco XLL and Bocardo LXL. In our system, Aristotle’s restrictions on Substitution and Cut are dropped. And the problematic moods are integrated with the apodeictic. A related system is formulated in which only assertoric forms appear, with or without modalised terms. We then give a classification of alternative approaches. The axiomatic system of McCall, with Johnson’s semantics — which rejects Baroco XLL and Bocardo LXL — is located within this classification.
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References
THOM 1981 §18.
OCKHAM II.24.27–48 argues that “the Philosopher in the Prior Analytics speaks of the conversion of necessity-propositions only when they are taken in the composite sense [i.e. de dicto] or equivalently.”
JOHNSON 1989.
STAHL 1976 p.209 uses similar diagrams.
PATTERSON 1990 is mistaken when he says (n.20) that Li-conversion “is not valid on the reading (de re) one needs if Darii NAN [our LXL] is to be valid”. Our de re semantics validates both Li-conversion and Darii LXL.
BRENNER p.8.
See McCALL 1963 §20. McCall’s approach will be discussed in §25.1.
See THOMASON p.112. Thomason’s ideas will be discussed in §25.1.
These are all features of system A in THOM 1981 Chapter Six.
CAMPSALL 14.04.
The basic idea of introducing modalised terms and special principles relating them to unmodalised terms is in STAHL 1976 p.179, and before that in GEACH pp.202–203.
A proof can be adapted from that in THOM 1981 §46.
See SMILEY; THOM 1981 §48.
JOHNSON 1989 p.274.
THOM 1991, 1.5.
THOMASON Definition 2.
JOHNSON 1989 (d,e) p.274.
THOMASON Definition 3.
JOHNSON 1989 (c) p.274.
JOHNSON 1989 p.274.
JOHNSON 1989 (c) p.274.
JOHNSON 1989 (c) p.274.
JOHNSON 1989 (c) p.274.
See FREDDOSO p.18.
JOHNSON 1989 p.274. THOMASON p.116. GEACH p.202 proposes a superstructure identical with this, except that Lo-forms are read as in superstructure A.
BURIDAN 2,6,8: “In any divided sentence de necessario the subject is ampliated to supposit for those things which can be.”
BURIDAN 4.2.3: “In the first figure, there is always a valid syllogism from a pair of premisses de necessario… to a conclusion with the same mode….”
BURIDAN 4.2.5: “In the second figure… there is always a valid syllogism from a pair of premisses de necessario… to a conclusion which is de necessario….”
BURIDAN 4.2.8: “In the third figure… a conclusion de necessario always follows from two premisses which are de necessario….”
BURIDAN 4.2.53: “In the first figure… from a major de necessario and an assertoric minor there is always a valid syllogism to a particular conclusion de necessario….”
BURIDAN 2.6.27: “(a) From no sentence de necessario does there follow another sentence de necessario by conversion of the terms, except (b) a universal negative follows from a universal negative.”
BURIDAN 2.6.14: “From no sentence de necessario does there follow an assertoric sentence, or conversely, unless an assertoric universal negative follows from a universal negative de necessario.”
BURIDAN 4.2.53: “In the first figure… from a major de necessario and an assertoric minor there is always a valid syllogism… but… not to a universal conclusion de necessario.”
BURIDAN 4.2.57.
BURIDAN 4.2.65.
OCKHAM IIIa.31.15–16 etc. Buridan allows that the normal implication of an ampliateci subject can be cancelled by inserting the phrase ‘what is’ before the subject; and Ockham allows that the normal implication of a subject that is neither ampliated nor restricted can be cancelled by an explicit ampliation of the subject.
OCKHAM IIIc. 11.3–6. These inferences are said to be valid whether the modal propositions are taken ‘in sensu compositionis’ or ‘in sensu divisionis’.
OCKHAM IIIa.31.8–43. These syllogisms are said to be valid if (but only if) the modal propositions are taken ‘in sensu divisionis’.
OCKHAM II.24.49–79.
§41. See Ockham IIIa.33.54–62.
NORTMANN 1994, p. 117. For a critique of Theophrastus, see PATTERSON 1995 §4.1.
See BOCHENSKI, Chapter 5 §3.
ŁUKASIEWICZ p.189.
ŁUKASIEWICZ p.137.
KNUUTTILA p.163.
BECKER, p.90.
For critiques of the Becker approach, see McCALL 1963 §8, PATTERSON 1989 §1.
ANGELELLI, pp.176–213.
ANGELELLI p.207.
ANGELELLI p.210.
NORTMANN 1994, 120–121. Nortmann mistakenly refers here to Ferio LQM, a mood which, though arguably Aristotelian, is not explicitly proved in the text.
ANGELELLI §12.
This analysis occurs in SCHMIDT, p.82.
Neither NORTMANN 1990 nor NORTMANN 1994 explicitly includes an analysis of Li- or Lo-forms. The analysis I have suggested implies the convertibility of Li-forms, as shown above.
NORTMANN 1990 p.73. For a critique of Nortmann’s approach, see PATTERSON 1995 pp.178ff.
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Thom, P. (1996). System AL . In: The Logic of Essentialism. The New Synthese Historical Library, vol 43. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1663-0_7
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