Semantics of First-Order Logic

The 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.

What you'll learn

• This 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.