# Existential Import

How is it that universal propositions (from the Boolean standpoint) don't commit us to the existence of the subject term while particular propositions actually do? Also, why particularly take or focus on the subject term? What about the predicate term?

• There are no quantifiers in Boolean algebra, only in predicate calculus, and it does not allow quantification of predicates. And it does not commit us to anything, this is just a semantic convention to match its rules of inference. – Conifold Nov 27 '19 at 22:37
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• i think there''s an interesting question here, but it's a bit unclear as it stands -- can you explain some of the terms you use, for the neophyte? – user38026 Nov 28 '19 at 3:24

It follows from how the formal semantics of the quantifiers and the connectives used with these quantifiers is defined.

An existential statement

There is a P which is Q

is formalized as

∃x(P(x) ∧ Q(x))

A universal statement

All P are Q

is formalized as

∀x(P(x) → Q(x))

The semantics of the quantifiers themselves and the logical constants ∧ and → used with them is defined as follows:

(1) ∃xφ is true iff there is at least one individual a such that φ is true of a
(2) ∀xφ is true iff for all individuals a, φ is true of a
(3) φ∧ψ is true iff and φ is true and ψ is true
(4) φ→ψ is true iff φ is false or ψ is true

This leads us to the following interpretations:

∃x(P(x) ∧ Q(x)) is true
⇔ there is an individual a such that P(a) ∧ Q(a) is true                       (def. (1))
⇔ there is an individual a such that P(a) is true and Q(a) is true         (def. (3))

So an existentially quantified statement entails, by the way the interpretation of ∃ and ∧ is defined, that there be an actual individual a of which P(a) is true.

However,

∀x(P(x) → Q(x)) is true
⇔ for all individuals a, P(a) → Q(a) is true                       (def. (2))
⇔ for all individuals a, P(a) is false or Q(a) is true           (def. (4))

For any individual to which property P does not apply, the implication P(a) → Q(a) becomes true -- we're only saying that if a has property P, then it should also have property Q, but if it doesn't have property P in the first place, then we don't care; it doesn't make the claim P(a) → Q(a) false, and since in classical logic there is only true or false, the immediate consequence is that P(a) → Q(a) is true of that non-P inidividual. That's just how → is defined.
Now if there is no individual at all to which P applies, then for all individuals the implication P(a) → Q(a) gets true. And since we have that P(a) → Q(a) is true for all individuals a, then by the definition of ∀, ∀x(P(x) → Q(x)) is true as well.
The only way for a universally quantified statement to become false is if there were an individual for which P(a) → Q(a) is not true, which would be the case only if P(a) is true but Q(a) is false. If that were to happen, we would have a counterexample to the claim "If x is P then x is Q". But if there is no individual which has property P at all, then there is nothing to falsify that claim, and in classical logic, anything that's not false is true. So if there are no counterexamples which are P but not Q, we say that the statement "All x which are P are also Q" is (vacuously) true.

The reason why we "focus on the subject term" is that as soon as the implication P(a) → Q(a) gets true because P(a) is false, we no longer need to care whether Q(a) is true. a could or could not have property Q -- it doesn't matter; if the left-hand side of an implication is false, then the implicational statement is true no matter whether the right-hand side is true or false, so we can stop our evaluation and proceed to the next individual as soon as we encounterd that a does not have property P anyway. Only in case the left-hand side P(a) is true, the truth value of Q(a) becomes interesting, because then Q(a) needs to be true as well in order for the implication to become true.
So as soon as we found out that the subject term does not apply to the individual, the predicate term becomes irrelevant, because we're only interested in those individuals to which P does apply, about the others we don't need to worry.
However, when the subject term does apply to an individual then we do also care about the truth value of the predicate term, because that's the cases we're interested in, in which the truth of the universal depends on whether Q is also true of a. We focus on the subject term first and filter out those individuals which we are not talking about anyway, for those we do, we check whether the predicate term applies as well.