Predicate Functions and Quantifiers

This lesson builds an understanding of how predicate logic sentences are written (syntax) and how their meaning is resolved (semantics).

Elements of syntax

Predicate logic imports almost all syntactic ideas from propositional logic, e.g., it retains the idea of symbolically representing terms instead of relying on whole words and sentences. This symbolism helps logicians focus on forms instead of the content of the argument. Predicate logic also retains all the logical connectives that compound sentences. On top of that, it adds new elements to its syntax, such as predicate functions and quantifiers.

Symbols and logical connectives

One major point of departure is that, while in propositional logic, the fundamental unit for a symbol is a simple proposition“It is raining” gets replaced by the “r” symbol., and in predicate logic, it’s the predicate function. We will expand on this shortly.

To add complexity to the sentence structures of predicate logic, there’s an allowance for using all the logical operators of propositional logic. Predicate logic includes the following connectives:

• $¬$ (negation, which represents the logical NOT)

• $∨$ (disjunction, which represents the logical OR)

• $∧$ (conjunction, which represents the logical AND)

• $→$ (implication, which represents the logical IF-THEN)

• $↔$ (biconditional, which represents the logical IF AND ONLY IF)

Predicate functions

In linguistic terms, a predicate often refers to the part of a sentence that provides information about the subject, describing what the subject is doing or what qualities it possesses. In the context of predicate logic, a predicate function is like a mathematical function that takes subjects as its input and returns a truth value (either true or false) based on whether the subject meets the criteria asserted in the predicate.

Here’s a simple way to identify a predicate function in a sentence: we look for words or phrases that express qualities, actions, or connections between subjects. These are often verbs or adjectives placed around one or more nouns or pronouns. For example, in “Samantha is happy,” the word “happy” is the predicate function because it describes Samantha’s emotional state. Here,$happy$becomes a function that takes one subject as its input. That input, in this case, is a constant,$samantha$.

In syntax, “Samantha is happy” is translated in predicate logic as $happy(samantha)$. Firstly, this predicate function, $happy(samantha)$ ...