# semantics of first-order logic

(1.4 hours to learn)

## Summary

The semantics of a first-order language is defined in terms of mathematical structures which give the meanings of all the constants, functions, and predicates in the language. In particular, one can recursively define a function which evaluates, given a structure and a first-order sentence, whether the structure satisfies the sentence. If all structures satisfying a set A of sentences also satisfy another set B of sentences, then A logically implies B.

## Context

This concept has the prerequisites:

- first-order logic
- propositional logic (The semantics for FOL includes the semantics for propositional logic.)
- recursion theorem (The Recursion Theorem is needed to define the semantics.)

## Goals

- define what it means for a mathematical structure to satisfy (or be a model of) a set of first-order sentences

- define what it means for one set of first-order sentences to logically entail another

## Core resources (read/watch one of the following)

## -Free-

→ Notes on Logic (2013)

Lecture notes for a course on first order logic.

## -Paid-

→ A Mathematical Introduction to Logic

An undergraduate textbook in mathematical logic, with proofs.

Location:
Section 2.2, "Truth and models," up through subsection "Logical implication," pages 80-89

→ A Course in Mathematical Logic

A graduate textbook in mathematical logic.

Location:
Section 2.1, "First-order semantics," pages 49-54

→ The Language of First-Order Logic

An undergraduate logic textbook aimed at philosophers, with an educational software package.

Location:
Chapter 11, "Advanced topics in FOL," up through subsection 11.3, "Truth and satisfaction, revisited," pages 258-268

## See also

-No Additional Notes-