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First-Order Logic ​

Date: 02-27-2024
Link: https://crystal.uta.edu/~gopikrishnav/classes/common/4308_5360/slides/fol.pdf

Objects
Called constants Examples:

  • John
  • Mary

Relations and Functions ​

Should "sibling" be a relation or a function?

  • Relation: siblings(John, Mary)
  • Function: sibling(John) returns Mary

"Sibling" should be a relation, because someone can have many (or no) siblings. A function can only return one value.

Basic Elements of First Order Logic ​

In propositional logic we only had symbols and connectives.

In first-order logic we have NO SYMBOLS. Instead, we have:

  • Constants.
  • Predicates.
  • Functions.
  • Connectives (and, or, not, if, iff).
  • The equal sign = (a "special" predicate).
  • Variables.
  • Quantifiers, ∀ (for every), ∃ (there exists).

Demo ​

Timestamp:
02-27-2024
03:17 PM

Taller(Bob, x)
x is a free variable. It has no inherent truth value.
If you are using a variable, it has to be quantified.

  • universal quantifier
  • existential quantifier

If you universally quantify a variable, it does not matter what I plug in for x, it is true.
You have to be very careful when using quantifiers. If you say for all x, you mean for all x.

Universal Quantifier (∀x)[Taller(Bob, x)]

Existential Quantifier
(∃x)[Taller(Bob, x)]