Fallacy of Quantificational Logic

Taxonomy: Logical Fallacy > Formal Fallacy > Fallacy of Quantificational Logic

Exposition:

Quantificational logic is an extension of propositional logic, and a fallacy of quantificational logic is a logical mistake specific to the features of quantificational logic that go beyond those found in propositional logic. The notion of "quantification" involved in quantificational logic is not usually numerical, but typically limited to "all", "none", and "some".

Quantificational logic begins by analyzing the internal structure of propositions that propositional logic treats as "atomic", that is, unanalyzable. Consider the proposition:

Socrates is wise.

This is a simple, unanalyzable proposition in propositional logic because it does not contain any propositional parts joined by truth-functional connectives, such as "and" or "not". However, in quantificational logic, it has an internal structure consisting of a name and a predicate. "Socrates", of course, is a name. In addition to names, quantificational logic has individual variables which stand in for names in a way similar to pronouns in English. So, in the example, "x is wise" is the predicate, with the variable "x" acting as a placeholder for a name; replace the variable with the name "Socrates" and we get the example proposition. For this reason, quantificational logic is sometimes called "predicate logic".

The final new grammatical element in quantificational logic, which gives it that name, is the category of quantifier. There are many quantifiers, just as there are many truth-functional connectives, but the two most frequently encountered in quantificational logic, and which you need to know about to understand quantificational fallacies, are the following:

Universal Quantifier Existential Quantifier
All x Some x
Examples
All philosophers are wise. Some philosophers are silly.

These quantifiers can be combined in complex ways that go beyond what can be done with the categorical propositions used in the familiar categorical syllogisms. For example: "Some dog hates all cats." This may look at first glance like an I-type categorical proposition, but it also contains within it the quantifier "all" of an A-type proposition, making it a mash-up of A- and I-type propositions. This isn't possible in the traditional Aristotelian logic, which must be extended in some way to accommodate multiple quantifiers, and quantificational logic is just such an extension. In quantificational logic, there is no upper limit on how many quantifiers may occur in a statement, for example: "every boy loves some dog that hates all cats."

This entry is for the most general fallacy of quantificational logic; for specific types of quantificational fallacy, see the Subfallacies, below.

Source:

Robert Audi (General Editor), The Cambridge Dictionary of Philosophy (Second Edition), 1995, pp. 272-3.

Resource:

Howard Pospesel, Introduction to Logic: Predicate Logic (1976). A good introduction to quantificational logic with an emphasis on how it relates to English.

Subfallacies: