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There are four qualifiers in term logic, organized according to two distinctions: particular/universal (some versus all) and affirmative/negative (permitting none and not all).

There are 256 combinations of these quantifiers in the form of a syllogism. One is 'AAA', or (all x are y) and (all y are z) implies (all x are z).

Out of these 256, only 24 combinations are valid. Is there an effective strategy to generate these 24 valid forms, possibly from simpler rules relating to the quantifiers? (For instance: all implies some...?)

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  • a quibble: there is no quantification in term logic, so no quantifiers. quantification was a radical innovation invented in the 19th c.
    – user20153
    Commented Dec 3, 2016 at 20:42
  • Thanks @mobileink -- is there a better way to call them? (I notice there's also indefinite/singular modes of these as well as universal, existential, etc)
    – Joseph Weissman ♦
    Commented Dec 3, 2016 at 22:32
  • not that I know of, alas. I suspect the right idea is that these were qualifiers, grammatical rather than logical operators, serving to modulate the sense of the sentence as a whole. but that's a guess - I've looked around a good bit and haven't found anything much. I know Fred Sommers is a contemporary philosopher who did a lot of work in term logic, so you might start there, but I haven't read his stuff.
    – user20153
    Commented Dec 5, 2016 at 19:53
  • I agree with mobileink. As a matter of fact, when I read the sentence, I read it as "There are four qualifiers..." not "quantifiers." If you make a map of the possible "combinations" and identify the 24 valid ones, you might be able to use combinatorial logic to obtain them.
    – Guill
    Commented Dec 6, 2016 at 0:08

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Yes there is. The first thing to note is that there are four "figures" of syllogism which are defined by where the major, minor and middle terms fall in the syllogism. Then what you do is memorize a variety of ways in which various valid syllogisms can be "reduced" to one of these four valid figures. These rules are codified in a little Latin song made of nonsense names "Barbara, Celarent, etc". For details see Paul Vincent Spade *Thoughts, Words, and Things", p. 20-25. http://pvspade.com/Logic/docs/Thoughts,%20Words%20and%20Things1_2.pdf

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