Word of the Week Wednesday: Existential Instantiation

by Max Andrews

The Word of the Week is: Existential Instantiation

Definition: A rule of inference that introduces existential quantifiers.  The symbol for an existential quantifier is (∃x).

More about the term: The existential quantifier indicates that there is at least one thing in a categorical reference.  Instantiation is an operation that removes a quantifier and replaces every variable bound by the quantifier with that same instantial letter.  There are eight rules of inference to derive a conclusion of an argument via deduction:

  1. Modus Ponens: p ⊃ q … p… .:q
  2. Modus Tollens: p ⊃ q … ~q … .: ~p
  3. Pure Hypothetical Syllogism: p ⊃ q … q ⊃ r … .: p ⊃ r
  4. Disjunctive Syllogism: p v q … ~q … .:p
  5. Constructive Dilemma: (p ⊃ q) & (r ⊃ s) … p v r … .: q v s
  6. Simplification: p & q… .: p
  7. Conjunction: p … q … .: p & q
  8. Addition: p … .: p v q

As long as an existential quantifier is attached to a line of argument the above actions cannot be done.

Example of use: All professors are college graduates.  Some professors are logicians.  Therefore, some logicians are college graduates.

  1. (x) (Px ⊃ Cx)
  2. (∃x) (Px & Lx)
  3. .: (∃x) (Lx & Cx)

This is how we can get to this conclusion by removing the universal and existential quantifiers (let’s let x become m for Max).

  1. (x) (Px ⊃ Cx)
  2. (∃x) (Px & Lx)
  3. Pm & Lm          2, EI
  4. Pm ⊃ Cm         1, UI
  5. Pm                   3, Simp
  6. Cm                  4, 5, MP
  7. Lm & Pm          3, Commutativity (not discussed here but you can see how it works)
  8. Lm                  7, Simp
  9. Lm & Cm        8, Conj
  10. (∃x) (Lx & Cx)  9, EG

For more on logic and existential instantiation see any edition of Patrick Hurley’s Introduction to Logic.


2 Comments to “Word of the Week Wednesday: Existential Instantiation”

  1. “entitites”? Sounds like alien beings! (Check your disclaimer.)

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