Syllogism - Existential Import

Existential Import

If a statement includes a term so that the statement is false if the term has no instances (is not instantiated) then the statement is said to entail existential import with respect to that term. In particular, a universal statement of the form All A is B has existential import with respect to A if All A is B is false if there are no As.

The following problems arise:

(a) In natural language and normal use, which statements of the forms All A is B, No A is B, Some A is B and Some A is not B have existential import and with respect to which terms?

(b) In the four forms of categorical statements used in syllogism, which statements of the form AaB, AeB, AiB and AoB have existential import and with respect to which terms?

(c) What existential imports must the forms AaB, AeB, AiB and AoB have for the square of opposition be valid?

(d) What existential imports must the forms AaB, AeB, AiB and AoB have to preserve the validity of the traditionally valid forms of syllogisms?

(e) Are the existential imports required to satisfy (d) above such that the normal uses in natural languages of the forms All A is B, No A is B, Some A is B and Some A is not B are intuitively and fairly reflected by the categorical statements of forms Ahab, Abe, Ail and Alb?

For example, if it is accepted that AiB is false if there are no As and AaB entails AiB, then AiB has existential import with respect to A, and so does AaB. Further, if it is accepted that AiB entails BiA, then AiB and AaB have existential import with respect to B as well. Similarly, if AoB is false if there are no As, and AeB entails AoB, and AeB entails BeA (which in turn entails BoA) then both AeB and AoB have existential import with respect to both A and B. It follows immediately that all universal categorical statements have existential import with respect to both terms. If AaB and AeB is a fair representation of the use of statements in normal natural language of All A is B and No A is B respectively, then the following example consequences arise:

"All flying horses are mythological" is false if there are not flying horses.

If "No men are fire-eating rabbits" is true, then "There are fire-eating rabbits" is false.

and so on.

If it is ruled that no universal statement has existential import then the square of opposition fails in several respects (e.g. AaB does not entail AiB) and a number of syllogisms are no longer valid (e.g. BaC,AaB->AiC).

These problems and paradoxes arise in both natural language statements and statements in syllogism form because of ambiguity, in particular ambiguity with respect to All. If "Fred claims all his books were Pulitzer Prize winners", is Fred claiming that he wrote any books? If not, then is what he claims true? Suppose Jane says none of her friends are poor; is that true if she has no friends? The first-order predicate calculus avoids the problems of such ambiguity by using formulae that carry no existential import with respect to universal statements; existential claims have to be explicitly stated. Thus natural language statements of the forms All A is B, No A is B, Some A is B and Some A is not B can be exactly represented in first order predicate calculus in which any existential import with respect to terms A and/or B is made explicitly or not made at all. Consequently the four forms AaB, AeB, AiB and AoB can be represented in first order predicate in every combination of existential import, so that it can establish which construal, if any, preserves the square of opposition and the validly of the traditionally valid syllogism. Strawson claims that such a construal is possible, but the results are such that, in his view, the answer to question (e) above is no.