Deductive Arguments: Universal and Existential Quantifiers

Written by Levi Clancy

First published January 14, 2010 Last modified October 10, 2011

Quantificationally Valid Forms

Not all valid argument forms are truth-functional. Some use universal quantifiers (ie, every) and some use existential quantifiers (ie, some). However, it is imperative that a universal quantification is not ambiguous, as in “everything is not blue” — is everything non-blue, or is it that some things non-blue?

Argument	Construct	Overview
Universal Instantation	Every F is G. If α is F, Then α is G.	A counterexample refutes a universal instantiation, for example by providing an α that is F but not G.
Existential Quantification	Some F is G. another example, Most F are G.	An existential quantification is not subject to counterexamples; it can only be disproven by examining every F and showing that none are G. This is because existential quantifications yield no entailments concerning any given item.
Universal Syllogism	Every F is G. α is F. ∴ α is G.	Universal Syllogism is valid, as assured by the validity of Universal Instantiation (If &alpha is F, it is G) and Modus Ponens (&alpha is F, ∴ &alpha is G).

Quantificationally Invalid Forms

Argument	Construct	Overview
Universal Negation	Everything is not G another example, Every F is not G.	Is everything a non-G, or are some things G and some things non-G? This form is ambiguous.
“Some” and “Not All”	Some F are G. ∴ Not all F are G.	By reductio ad absurdum, this form is clearly invalid. For example, at an all-girls school where all its students are female, it is still true that some students are female.

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