Existential Import

A proposition is said to have existential import if the truth of the proposition requires a belief in the existence of members of the subject class.  I and O propositions have existential import; they assert that the classes designated by their subject terms are not empty. But in Aristotelian logic, I and O propositions follow validly from A and E propositions by sub-alternation. As a result, Aristotelian logic requires A and E propositions to have existential import, because a proposition with existential import cannot be derived from a proposition without existential import.  Thus, for Aristotle, if we believe it is that "All unicorns have one horn" then we are committed to believing that unicorns exist.  For the modern logician or mathematician, this is an unacceptable result because modern mathematics and logic often deal with empty or null sets or with imaginary objects.  A modern mathematician might, for example, wish to make a true claim about all irrational prime numbers.  Since there are no irrational prime numbers, Aristotle would say that any claim about them is necessarily false.

George Boole developed an interpretation of categorical propositions solves the dilemma by denying that universal propositions have existential import. Modern logic accepts the Boolean interpretation of categorical propositions.  This interpretation has the following consequences:

• Because A and E propositions have no existential import, sub-alternation is not valid.
• Because A and E propositions have no existential import, super-alternation is not valid.
• Contraries are eliminated because A and E propositions can now both be true when the subject class is empty.
• Sub-contraries, on the other hand, are retained because I and O propositions always have existential import.
• Some immediate inferences are preserved: conversion for E and I propositions, contraposition for A and O propositions, and obversion for any proposition.
• Any argument that relies on the mistaken assumption of existential import commits the existential fallacy.

The Boolean Square of Opposition

Why Reject Existential Import?

Modern logic rejects existential import for a number of reasons.  The most significant for our purposes have to do with the nature of universal claims and our understanding of what it means to say of a proposition that it is false.  Starting with the latter, ask yourself what would have to be the case about the world for you to claim that an A type proposition is false.  Consider the claim "All swans are white".  In order for that claim to be false, we need to know that there is at least one non-white swan.  Imagine how you might argue with someone who insists that it is true that "A'' swans are white".  You would produce as evidence for the falsity of the claim the existence of a non-white swan.  "No," your might argue, "not all swans are white, for here is a swan that is brown."  But now suppose for a minute that there were no swans at all.  What sort of evidence could you produce, in the total absence of any swans, against the claim that all swans are white?  Obviously you couldn't produce a non-white one because there aren't any swans at all.  In the absence of any evidence for a falsifying instance to the universal claim, you should accept the claim.  But now extend that reasoning to universal claims about empty classes and non-existent objects.  Universal claims about empty sets are all true, because there are no falsifying instances.

NOTE:  Claims about empty sets are trivially true.  Sure, "all irrational prime numbers are odd" because there are no irrational prime numbers, but it is equally true that "all irrational prime numbers are even".

Universal Conditionals

Another reason for adopting the modern approach toward existential import, related to the previous one, has to do with the interpretation of a universal claim.  Instead of seeing a universal claim as one about the members of 2 sets, modern logic sees A and E type propositions as universally quantified conditionals.  "All S's are P's" is understood by the modern logician as saying "For any object x, if x has property S, then x has property P".  Suppose there are no S's in the universe.  Then, no matter what value we assign to x, "x has property S" will be false.  But a conditional with a false antecedent is true, so a universal conditional whose subject class is empty is true.

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