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FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 15. MAIN MODAL SYLLOGISMS.
We called a mood of syllogism, a combination of formally fully specified
premises and conclusion in a given figure (e.g. 1/AAA).
We will call mode, any combination of symbols which does not by itself fully
specify a syllogistic form, but which abstracts a specific aspect of such, in a
given figure (e.g. 1/uuu). It was
shown, in Aristotelean logic, that the primary valid modes of polarity and
quantity are as in the following table. Table
15.1 Valid
modes of Polarity and Quantity.
We can at the outset, prior to systematic validation, predict that the
valid modes for natural and temporal modality will be the following, by analogy
to the results obtained for extensional modality. Table
15.2 Valid
Modes of Natural and Temporal Modalities.
Note the slight difference between quantity modes and modality modes. The
modes aaa and mmm
are valid in all figures, whereas sss
is not (3/ssp is exceptional, and
anyway does not yield an s
conclusion). This is due to modality standing outside the relationship between
the terms, whereas quantity concerns the subject more directly.
Natural and temporal modality being essentially analogous, we can
concentrate on developing the theory of syllogism for the former, and then
generalize the results to the latter. Apart from the above we will need to
investigate the valid modes of mixed, natural and temporal syllogism.
In the broadest sense, of course, all syllogism is modal. But for the
sake of convenience we will often find it useful to call nonmodal, syllogism
both of whose premises are actual or momentary (aaa
or mmm); so that syllogism with one
or both premises necessary or possible, can be called modal. Aristotelean logic
can then be said to have concerned nonmodal syllogism, while this thesis
concerns modal syllogism.
If we combine together the valid modes of polarity and quantity for a
given valid mode of modality, in each of the figures, we should obtain the valid
moods of syllogism. Let us now do so, using the valid natural modality modes, to
develop a full list of natural syllogism, including both the nonmodal
(Aristotle's achievement) and the modal (the new contribution). This is the
principal goal of our whole formal research. The notation system used for this,
consists in applying modality subscripts (n,
p, a) to the six standard
symbols, A, E,
I, O,
R, G.
We see in the list below that only 56 primary moods emerge as logically
valid, not counting derivative syllogism. There are 18 valid moods in each of
the first three figures, and 2 in the fourth. Since 19 of the above moods are
actual, only 37 are original forms. Table
15.3 Primary
Valid Moods of Natural Syllogism.
We
will now present these 37 valuable new forms in full, for the record. a.
First
Figure. Form: M-P, S-M, S-P.
b.
Second
Figure. Form: P-M,
S-M, S-P.
c.
Third
Figure. Form: M-P, M-S, S-P.
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