Is Logic Gibberish?

Is Logic Gibberish?

We are familiar with the standard notation of predicate logic in which we have what is called variable binding. Thus we have a symbol for (say) existence followed by an “x” and then a formula in which a predicate and bound variable occur (“Ex(Fx)”). How should we read this? It is common to hear it read as “There exists an x such that Fx”. What does this mean? The letter “x” is supposed to be replaceable by a singular term such as a proper name, a pronoun, or a demonstrative. So, a substitution instance might be “There exists a London such that London is crowded” or “There exists a him such that he is tall”. But these sentences are nonsense; it is not grammatical to place a singular noun in this position. What is wanted is clearly a predicate like “city” or “man”: then we can say “There exists a city such that that city is crowded” or “There exists a man such that that man is tall”. The correct logical form would then be something like “EF(that F is G)”, read as “There exists an F such that that F is G”, where “F” and “G” are predicate expressions. But this is not what logical notation says under the usual reading: this makes sense but that does not. Alternatively, we might try saying that the usual formula is to be read “There exists an object x such that x is F”, so that we have “There exists an object x such that x is a city and x is crowded”. That looks closer to the usual notation, but it contains a funny construction, viz. “an object x”. Put aside whether “object” is an adequate count noun and ask yourself what could be meant by following this word with an “x”. Presumably a substitution instance would be “an object London” or “an object it”: but these are also nonsense. That occurrence of “x” after “E” cannot be replaced by such an expression on pain of producing gibberish. It really doesn’t belong there at all; what belongs there is a count noun or predicate variable replaceable by a count noun, as in “There exists a city such that”. So, predicate logic should not be written in the standard way but rather along the lines of “EF(that F is G)”; otherwise the notation is gibberish. Even that formula has its drawbacks in the shape of the expression “that F”, since it is obviously not deictic and it has no anaphoric antecedent. One wants to say instead “There exists an F, x, such that Gx”, as in “There exists a city, x, such that x is crowded”, so as to have an antecedent to work with. But this also is gibberish, as can be seen by substituting a proper name or pronoun for “x” throughout. There is nothing wrong with saying “London is crowded”, but what is meant by “There exists a city, London”? Is this just a funny way of saying “London exists”? In fact, no one ever translates the standard formula this way, instead saying simply “There exists an x such that x is a city and x is crowded”—which takes us back to our first point. There is simply no intelligible way to interpret this initial occurrence of “x”. Hence predicate logic is gibberish and needs to be overhauled into something that actually makes sense. The whole quantifier-plus-variable symbolism is a mistake.[1]

[1] An Aristotelean might pounce and suggest that Aristotle’s subject-predicate logic is the way to go. Thus, we will have “Some men are philosophers” and “All men are mortal” where the first two words in these sentences are construed as logical subjects. Or else we will need to devise a whole new grammar and notation for expressing logical relations involving “some” and “all”.