# Semantics of First-Order Logic

First-order logic is a restricted, formalized language which is particularly suited to the precise expression of ideas. The language has uses in many disciplines including computer science, mathematics, linguistics and artificial intelligence.

We will describe how to write sentences in the language, how to determine when a sentence is true in a particular situation, how to recognize important relationships between sentences, and describe some limitations of the language.

##### Semantics of First-Order Logic

# About this course

Skip About this courseThe focus of this class is on the *language of first-order logic* , a formally defined language that allows us to make precise and unambiguous statements about any subject of interest.

Using the language of first-order logic we will investigate many foundational topics in logic. We will address such questions as what counts as a grammatical expression, and the circumstances under which it makes a claim about the world (whether it can be considered true or false, E.g. “the sky is brown”, as compared to “oh, my goodness!”).

For expressions that do make claims — we call these *sentences* — we can further examine whether they are true or false in particular situations. “Aristotle is alive” is a sentence that was once true, but became false around 2000 years ago, and has remained false ever since.

These questions fall into the study of *semantics* , or meaning.

Once we understand how sentences can be considered true or false, we can investigate important related questions. Some sentences are always true, that is true in every situation — we call such sentences *logical truths*. Sentences bear relationships with one another. For example, two sentences might be true in exactly the same situations - they are *logically equivalent.* We will demonstrate methods for determining when these properties and relationships hold as natural extensions to the semantic theory for first-order logic.

Finally, we will explore the limits of first-order logic. There are some sentences of English that are not expressible in the language, and it is important to know that this is the case, and to understand why it is so. This observation has led logicians to develop yet more powerful languages with more complex semantics. Almost all of these languages are based on the language of first-order logic and knowledge of first-order logic is fundamental to understanding them. So first-order logic is a basic building block for the study of these language and is a great place to begin the journey into the field of logic.

### At a glance

- Institution: StanfordOnline
- Subject: Philosophy & Ethics
- Level: Introductory
- Prerequisites:
None.

- Language: English
- Video Transcript: English

# What you'll learn

Skip What you'll learnThis class is an introduction to one of the basic tools used in the study of logic, a tool that is applied in a range of disciplines from computer science and math to linguistics and philosophy.

The course is divided into two halves. In the first we study a fragment of first-order logic called *propositional logic*. This language allows us to get our feet wet with the basic ideas of the course. These ideas include the specification of formal grammar rules for determining when an expression is *well-formed*. Well-formed expressions may make claims about the world, that is they may be considered *true* or *false*. You will learn how to determine whether a sentence is true in a particular situation. With the basic ideas in hand, you will then learn how to recognize relationships between sentences, the most important of which is *consequence*. One sentence is a consequence of another, or follows from another, if it is true whenever the other is.

At the end of the section on propositional logic, we will demonstrate that its expressiveness is limited, and that any attempt to increase the expressiveness of the language requires fundamentally new expressive devices.

In the second half of the course we expand the language of propositional language to the full language of first-order logic, providing the new semantic theory. Everything that you learned about propositional logic holds in the larger language, but new expressive abilities are added to the language. We again investigate concepts of grammaticality, truth and consequence for the larger language. We will see that as a consequence of increasing the expressiveness of the language, the required extension to the semantic theory is more complicated than the theory of propositional logic.

Nonetheless, there are still sentences of English that are not expressible in first-order logic. We will conclude by describing these limitations, setting the stage for further learning in the field of logic.

# About the instructors

# Frequently Asked Questions

Skip Frequently Asked QuestionsQ: Who should take this course?

A: This course is for you if you are interested in learning one of the

fundamental tools of logic: the language of first-order logic. You will learn how to use the language to make and analyze statements that are precise and unambiguous. And you will understand the basic relationships between sentences such as logical consequence. The course has no prerequisites.

Q: How is this course related to the “Language, Proof and Logic” course offered by the same instructors on this platform?

A: “Language, Proof and Logic” is a full fifteen week long course covering all of the material in a standard introductory logic course, including both proof and model theory. The current course, “The Semantics of First-Order Logic”, is a shorter course which does not include the topic of creating proofs, but instead focusses on the construction and meaning of sentences of first order logic (model theory). If you have successfully completed “Language, Proof and Logic” it is likely that you already know the content of this course.

If you are not sure whether you are interested in studying logic, then this course is probably a good place to start. Later you might want to take Language, Proof and Logic, if you are interested to learn more.

Q: How is the course structured?

A: In addition to watching highly entertaining video lectures, you will complete assignments approximately weekly, using software applications and an automated TA who will give you feedback on your progress whenever you want it.

Q: How will I be assessed?

A: Assessment is based on weekly assignments. All assignments are graded automatically.

Q: Do I need to buy a textbook?

A: No. The textbook and software for this course are provided free-of-charge ($30.00 value) to all registered learners. If you are on the verified track you will additionally receive access to the Grade Grinder automated assessment service. This service is used to grade your work for this course.

Q: Can I get Stanford University credit for this class?

A: No.

# Ways to take this course

Verified Track | Audit Track | |

## Price | $50 USD | Free |

## Access to course materials | Unlimited | Limited |

## World class institutions and universities | ||

## edX support | ||

## Shareable certificate upon completion | ||

## Graded assignments and exams |