Simple Quantified Propositions

 The branch of logic we're introducing here is called Quantificational Logic because it has the ability to express propositions which employ the notions of some and all. Some and all are called quantifiers. Compare the following sentences:

Sentence 1 is a singular proposition, the kind of proposition you learned to translate in the previous section. In contrast, sentences 2 and 3 do not contain names. In the place of "Belinda" we have "Everyone" and "Someone" respectively. "Everyone" and "someone" do not name particular individuals. "Everyone" refers to all individuals. "Someone" means "at least one."  Hence we can't use individual constants to represent quantifiers.   Instead, we introduce a new symbol to represent "all", which we call the universal quantifier and another symbol to represent "some," which is called the existential quantifier:

The two quantifers function syntactically like prefix operators such as the tilde. We place it in front of the propositional function it operates on or quantifies over. We translate "Everyone is tall" by putting the universal quantifier in front of the propositional function Tx, and so the translation is

(x)Tx

which we read as: "For every x, x is tall."  "Someone is tall" is translated as:

(∃x)Tx

which translates as "There is an x, such that x is tall."

Note that the existential quantifer commits us to the existence of at least one thing which has the property in question.  When we say "Some apples are green" we may intend to convey that several, that is more than one, apple is green. But logically we're committed to the minimal claim that at least one apple is green. In logic we want to capture the minimal claim made in a proposition. Hence we translate "some" as "at least one."

The first exercise for this section will hone your skill at distinguishing simple quantified propositions from singular propositions. Exercise 2 will have you doing translations of these two types of propositions in Quantificational Logic.