Editing Syllogism (section)

==Existential import==
{{Multiple issues|{{Original research section|date=July 2020}}
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If a statement includes a term such that the statement is false if the term has no instances, then the statement is said to have ''existential import'' with respect to that term. It is ambiguous whether or not a universal statement of the form ''All A is B'' is to be considered as true, false, or even meaningless if there are no As. If it is considered as false in such cases, then the statement ''All A is B'' has existential import with respect to A.

It is claimed Aristotle's logic system does not cover cases where there are no instances. Aristotle's goal was to develop a logic for science. He relegates fictions, such as mermaids and unicorns, to the realms of poetry and literature. In his mind, they exist outside the ambit of science, which is why he leaves no room for such non-existent entities in his logic. This is a thoughtful choice, not an inadvertent omission. Technically, Aristotelian science is a search for definitions, where a definition is "a phrase signifying a thing's essence." Because non-existent entities cannot be anything, they do not, in Aristotle's mind, possess an essence. This is why he leaves no place for fictional entities like goat-stags (or unicorns).<ref>{{Cite web |url=http://www.iep.utm.edu/aris-log/ |title=Groarke, Louis F., "Aristotle: Logic", section 7. (Existential Assumptions), Internet Encyclopedia of Philosophy |access-date=2017-03-07 |archive-url=https://web.archive.org/web/20170204235013/http://www.iep.utm.edu/aris-log/ |archive-date=2017-02-04 |url-status=dead }}</ref>

However, many logic systems developed since ''do'' consider the case where there may be no instances. Medieval logicians were aware of the problem of existential import and maintained that negative propositions do not carry existential import, and that positive propositions with subjects that do not [[supposition theory|supposit]] are false.

The following problems arise:

{{Ordered list
| 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?
|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?
|What existential imports must the forms AaB, AeB, AiB and AoB have for the square of opposition to be valid?
|What existential imports must the forms AaB, AeB, AiB and AoB have to preserve the validity of the traditionally valid forms of syllogisms?
|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 AaB, AeB, AiB and AoB?
| type = lower-alpha
}}

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 mythical" is false if there are no flying horses.
:If "No men are fire-eating rabbits" is true, then "There are fire-eating rabbits" is true; 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 such ambiguity by using formulae that carry no existential import with respect to universal statements. Existential claims must 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 represented in first order predicate calculus in which any existential import with respect to terms A and/or B is either explicit 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 it can establish which construal, if any, preserves the square of opposition and the validity of the traditionally valid syllogism. Strawson claims such a construal is possible, but the results are such that, in his view, the answer to question (e) above is ''no''.