A Shortcoming of Quantifiers

The main goal of translating natural language sentences (e.g. those in English) into logical languages (e.g. those in First Order Logic) is to make explicit the ways in which such sentences are either true or false. There are common keywords of natural languages that indicate how they are to be handled logically speaking. For example, the way that a sentence of English that conjoins two independent clauses via the keywords ‘and’ or a ‘but’ is true when and only when each independent clause is itself true, and false otherwise. For example, the sentence ‘My lawn is brown and it is too hot outside my door’ is true if and only if ‘My lawn is brown’ is true and ‘It is two hot outside my door’ is true. Since what the independent clauses say in particular is unimportant in regard to the way that such a sentence that they are a part of is true, the independent clauses are summarized by a capital letter, usually starting with ‘P’ and then ‘Q’ (and then with subscripts if the entire alphabet gets used), while the ‘and’ or ‘but’ is substituted for the logical connective of conjunction, which is typically symbolized as ‘^’ or ‘&’. Logical connectives just track the way that the sentences they connect contribute to the overall truth value of the sentence. So a first order logical sentence ‘P & Q’ is true if and only if P is true and Q is true, and false otherwise.

First Order Logic also makes explicit the truth values of those sentences that involve certain quantified expressions; in particular, sentences involving keywords such as ‘all’, ‘some’, or ‘no’ (as in, e.g. ‘No raindrops fall on my brown, dry lawn’).

If rain falls on my lawn, and I want to express this fact in First Order Logic, then this would be translated using the existential quantifier, which says roughly that there is at least one raindrop that falls on my lawn. More explicitly, it says: ‘There is at least one x such that x is a raindrop that falls on my lawn’, where ‘x’ indicates a variable that is substituted for a name (all names refer to one and only one thing, where ‘thing’ is perhaps meant in the loosest possible sense–e.g. a number). If I next want to express what is actually happening, that there is indeed no rain falling on my lawn, then an equivalent way to go is just to negate the above claims by saying that it is not the case that rain falls on my lawn, or it is not the case that at least one raindrop falls on my lawn. The explicit version would say: ‘It is not the case that there is at least one x such that x is a raindrop that falls on my lawn.’ How First Oder Logic works is that this sentence is true if and only if, if every object/thing (in the domain of discourse) is given a name, no substitution of ‘there is at least one x such that x’ for any of these names would result in the sentence’s being false. Every such replacement by a name in the domain of discourse (if all objects of the domain are named) has to result in a true sentence.

Consider what happens for the sentence ‘There is something’ or perhaps more specifically, ‘There is something there’. First of all, it is a little confusing for how to deal with such a sentence when there is indeed something present. Could it be saying that there is an x or that there is at least one x, unqualified? If so, then, if we are to parallel the way we did the replacement earlier, we would replace ‘there is at least one x’ with a name; yet this would prevent its being a sentence, since names aren’t sentences. This result points toward the way to satisfy this expression being that the name that replaces there being at least one x must be a thing. That is, the name names a thing, which is true by definition. Thus, the sentence would go, most explicitly, ‘There is at least one x such that x is a thing’. If there is a lawn, then the lawn is given a name, perhaps ‘Lawny McLawnface’, then when substituted for ‘there is at least one x such that x’, the sentence ‘Lawny McLawnface is a thing’ turns out to be true.

Next consider how we are to deal with the sentence ‘There is something’ when there is in fact nothing. If there is nothing (if it is possible for there to be nothing in some domain of discourse), then there is no thing upon which to bestow a name, and thus no substitution possible for the part that reads ‘There is at least one thing x such that’. Such a sentence is unevaluable for being truncated–‘is a thing’ just ain’t a sentence. Yet shouldn’t we be able to speak truthfully about a domain that happens to have nothing?

One result of this discussion is that the sentence ‘There is something’ is a logical truth of First Order Logic, since it is impossible to be false. As we saw, in the case where it should be false it is unevaluable. Yet, if I am right to think that it should be false in certain possible domains, then although a logical truth of First Order Logic, the expression is nonetheless not a logical necessity. It is sometimes false, although never so in First Order Logic as it standardly deals with quantification. The shortcoming is then that not every logical necessity of First Order logic is actually a logical necessity.

The way that this shortcoming is dealt with, as far as I’ve seen, is to stipulate that First Order Logic always applies to non-empty domains of discourse. This has the obvious result that empty domains of discourse are outside of the scope of First Order Logic. There are some expressions that are unable to be rendered explicit in first order logical terms.

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