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*To*: jboske@yahoogroups.com*Subject*: Re: [jboske] Quantifiers, Existential Import, and all that stuff*From*: pycyn@aol.com*Date*: Tue, 5 Mar 2002 11:26:20 EST

The traditional logic (Aristotle as interpreted by a couple millennia of logicians, not all of marked competence) had the following pattern A All S is P E No S is P I Some S is P O Some S is not P The traditional relations among these were Contradictories: A is equivalent to ~O and conversely E is equivalent to ~I and conversely Contraries: A and E cannot both be true but can both be false A implies ~E Subcontraries O and I cannot both be false but can both be true ~O implies I Subalternation: A implies I E implies O The tradition that arrived at the middle of the 19th century (coming through a period devoid of competent logicians) was this, with the understanding that all four forms had existential import. This, it is easy to see, will work only if it is always assumed that S is non-empty (otherwise, only subalternation and contraries remain), a restriction that many found inherently implausible and that gave clearly wrong results in some cases. So, logicians from about 1858 on started holding that the universal propositions (A and E) did NOT have existential import, though the particular did. This destroyed most of the basic features of the old system (contraries, subcontraries and subalternation all dropped out, leaving only contradictories), but got rid of a number of undesirable inferences and the need for a hidden assumption for many cases (8 out of 24 traditionally valid ones). And gave a tidier system, that could be tied in with other developments in algebra and logic. But, of course, it also gave some undesirable inferences in ordinary -- non-mathematical -- situations: the fact that there are no Ss meant that all Ss are whatever you want them to be. In recent years logicians and historians of logic have reexamined Aristotle and the logically more competent commentators on him and decided that the system he proposed and that was most used in the good times (after Aristotle, a couple of Thursdays in the 12th and 13th centuries) is one in which the affirmative propositions (A and I) have existential import, but the negatives (E and O) do not (Carroll almost got to this but could not conceive of O not being importing). This system restores the whole of the traditional structure noted at the beginning, without any additional assumptions needed. It also allows a way to define the propositions of any other system of these propositions that differ in terms of existential import: a negative predicate term inserted into any form, creates the form with the same quantity (universal or particular) and import but opposite quality. Thus a non-importing universal affirmative is just No S is ~P (A-) and an importing particular negative is Some S is ~P (O+). Thus you get two complete systems of AEIO, the + with existential import and the - without. As a practical matter, no one seems ever to have used I- (or even figured out just what it would mean). But aside from that we are free to mix and match quantifiers from the two lists ad lib. The + system is just the 19th century traditional system, without the assumption that there are Ss. The "modern" system uses A- and E-, I+ and O+. The Aristotelian system is A+, E-, I+, O-. Carroll's system, which got him into trouble and to a lot of hand-waving, was A+. E-, I+, O+. Lojban is now on the brink of being able to use the complete set of these quantifiers: the + group is {Q (lo) broda cu brode}, the - group is {Q da poi broda cu brode}. Assuming that {ro} and {su'o} behave properly for A+, A- and I+ and that {no} works for E+ and E- and that O+ is just {su'o S cu naku P}, we need only a new form for O-. {na'e ro} fills the bill, for even if S is empty, the value will be different from {ro}. It is probably easier to use this new form for O throughout, eliminating the {naku}. Negation shifting rules are now simple -- moving a negation across a quantifier (or a quantifier across a negation) changes everything about: quantity (particular -- universal), quality (affirmative -- negative) and import. The last is, of course, the hardest, since it requires rewriting a sizable chunk from (or to) {Q da poi broda} to (or from) {Q (lo) broda}, but the additional expressiveness is probably worth the effort. |

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