09-First_Order_Logic

#notes
Prerequisite

note

First-Order Logic

The second dialect of logic, first-order logic (FOL), is more expressive than propositional logic and uses objects as its basic components. With first-order logic we can describe relationships between objects and apply functions to them. Each object is represented by a constant symbol, each relationship by a predicate symbol, and each function by a function symbol.

Note

  • Terms in first-order logic are logical expressions that refer to an object. The simplest form of terms are constant symbols. However we don’t want to define distinct constant symbols for every possible object.[1]
  • Atomic sentences in first-order logic are descriptions of relationships between objects, and are true if the relationship holds.[2]
  • Complex sentences of first order logic are analogous to those in propositional logic and are atomic sentences connected by logical connectives.
  • Quantifiers: The universal quantifier ∀, has the meaning “for all,” and the existential quantifier ∃,, has the meaning “there exists.”
  • Equality symbol: signify that two symbols refer to the same object.[3]

Tip

With propositional logic, we model our world as a set of symbols that are true or false. Under this assumption, we can represent a possible world as a vector, with a 1 or 0 for every symbol.

  • This binary view of the world is what is known as a factored representation. With first-order logic, our world consists of objects that relate to one another.
  • This second object-oriented view of the world is known as a structured representation, is in many ways more expressive and is more closely aligned with the language we naturally use to speak about the world.

The following table summarizes the first order logic syntax:

  1. For example, if we want to refer to John’s left leg and Richard’s left leg, we can do so by using function symbols like Leftleg(John) and Leftleg(Richard). Function symbols are just another way to name objects and are not actual functions. ↩︎

  2. An example of an atomic sentence is Brother(John,Richard) which is formed by a predicate symbol followed by a list of terms inside the parentheses. ↩︎

  3. For example, the incredible sentence （不懂的可以自行了解 Einstein）(Wife(Einstein)=FirstCousin(Einstein)∧Wife(Einstein)=SecondCousin(Einstein)) is true. ↩︎