Things to Remember when Symbolizing with Quantifiers

Some. Like the word "or", the word "some" is ambiguous in English, although most people don't notice its ambiguity. Just as "or" can signify inclusive or exclusive disjunction, let us distinguish the inclusive and exclusive senses of the word "some". (These are my terms; you won't find them in the literature.) Inclusively, "some" means "at least one, possibly all". Exclusively, it means "not all, possibly none". We use it inclusively in sentences like, "Some of you will earn an A on the final exam." We want to leave the door open to the possibility that all will earn A's, and close the door on the possibility that none will. We use it exclusively when we pass a bowl of nuts to a guest and say, "Please take some." We mean take a few, or none, but don't take them all.

Only some. In the sentence, "All even numbers are divisible by 2, but only some are divisible by 4", the phrase "only some" means "more than one and less than all" or "neither none nor all". Hence it is not equivalent to either the inclusive or the exclusive sense of "some" (Tip 20), although it is equivalent to their conjunction. We should translate the second half of our example sentence, 'only some even numbers are divisible by four', thus: '(x)(Ex Fx) (y)(Ey ~Fy)' —essentially, 'some are and some aren't'.

No A's are B's. We translate this as '(x)(Ax ~Bx)". By transposition, this is equivalent to "(x)(Bx ~Ax)"; hence in translation, use either one. If no A's are B's, then no B's are A's.

Not all / all not. English speakers frequently mix these up; hence you might do so too. For example, if you want to warn people that glittering things are sometimes gold, but sometimes not gold, then you should say "Not all that glitters is gold." Despite this, the proverb says "All that glitters is not gold," which is absurd; it says that no glittering things are gold, not even gold.

"And" sometimes means "or". Be prepared to translate "and" as "or". For example: "Men and women are welcome to apply." It's tempting to translate the "and" here as a conjunction: "(x)[(Mx Wx) Ax]". But this actually says that everything that is both a man and a woman is welcome to apply. But we mean that everything that is either a man or a woman (or both) is welcome to apply: "(x)[(Mx ▼ Wx) Ax]".

  1. Another perspective on this: We want to limit the universe of discourse to the subset of things that are either men or women (or both), not to the subset of things that are both men and women.

  2. We can avoid using "▼" if we translate the original sentence thus: "(x)(Mx Ax) (y)(Wy Ay)" (men are welcome to apply and women are welcome to apply). This, however, is equivalent to our translation containing "▼".

None but. We can paraphrase "None but ripe bananas are edible" in many equivalent ways. No bananas except ripe ones are edible. Only ripe bananas are edible. A banana is edible only if it is ripe. All edible bananas are ripe. "(x)((Bx Ex) Rx)". Don't let these many forms confuse you. 

Indefinite articles. The articles "a" and "an" sometimes take existential, sometimes universal, quantifiers. "A bat is a mammal" really means "All bats are mammals": "(x)(Bx Mx)". "A bat is on my neck" means "There exists a bat on my neck": "(x)(Bx Nx)". Because there is no hard and fast rule, paraphrase the English before symbolizing. 

Definite articles. "The" sometimes takes existential, sometimes universal, quantifiers. "The horse is a noble animal" really means "All horses are noble animals": "(x)(Hx Nx)". "The horse in the winner's circle is on drugs" really means "There exists a horse (namely, the one in the winner's circle) who is on drugs": "(x)(Hx Wx Dx)". Because there is no hard and fast rule, paraphrase the English before  symbolizing.

Any. "Any" sometimes takes existential, sometimes universal, quantifiers. "Any bat is a mammal" really means "All bats are mammals": "(x)(Bx Mx)". "If any bats are in the room, then I'm outtahere" means "If there exists a bat in the room...": "(x)(Bx Rx) Oi". Because there is no hard and fast rule, paraphrase the English before symboleizing.