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. |