Welcome to Mathematical Logic. In this course you will be introduced to the concepts and techniques used in mathematical logic. We will start right from the beginning, assuming no prior exposure to this or similar material, and progress through discussions of the proof and model theories of propositional and first-order logic.
The ability to reason is fundamental to human beings. Whatever the discipline or discourse it is important to be able to distinguish correct reasoning from incorrect reasoning. The consequences of incorrect reasoning can be minor, like getting lost on the way to a birthday party, or more significant, for example launching nuclear missiles at a flock of ducks, or permanently losing contact with a space craft.
The fundamental question that we will address in this course is "when does one statement necessarily follow from another" --- or in the terminology of the course, "when is one statement a logical consequence of another". This is an issue of some importance, since an answer to the question would allow us to examine an argument presented in an article, for example, and to decide whether it really demonstrates the truth of the conclusion of the argument. Our own reasoning might also improve, since we would also be able to analyze our own arguments to see whether they really do demonstrate their conclusions.
We will proceed by giving a theory of truth, and of logical consequence, based on a formal language called FOL (the language of First-Order Logic). We adopt a formal language for making statements, since natural languages (like English, for example) are far to vague and ambiguous for us to analyze sufficiently. Armed with the formal language, we will be able to model the notions of truth, proof and consequence, among others.
While mathematical logic is technical in nature, the key concepts in the course will be developed by considering natural English statements, and we will focus the relationships between such statements and their FOL counterparts. The goal of the course is to show how natural English statements and arguments can be formalized and analyzed for correctness and truthfullness for example.
This section develops a simple theory of truth and proof that we will later extend to full first-order logic. The focus here is on simple English statements involving words like and, or, not and constructions like if...then... and ...unless.... We will also examine arguments whose validity depends on the meanings of these words and introduce rules for determining which arguments are valid and which invalid.
Here we extend propositional logic to the full theory of first-order logic. This will allow us to formalize English words like every, some, most and the and constructions like at least seventeen, more than twelve. As we will see, first-order logic is not capable of expressing every construction in English though. We will examine some such constructions and why first-order logic cannot capture them. We will introduce formal rules for determining the validity of reasoning using first-order logic sentencs.
In this section of the course we will look at some applications of first-order logic. We will show how some basic mathematical concepts can be expressed in first-order logic. We will also continue our examination of the theory that we have developed, and outline some of its strengths and limitations.
In this section of the course we will look beyond first-order logic to investigate logics which allow us to account for a wider selection of arguments. Modal logic is concerned with sentences in natural languages which involve modal operators such as \emph{always}, \emph{should} and \emph{obligatory}. Diagrammatic logic recognizes that information is frequently represented in non-textual forms. Everyday reasoning can involve the use of spreadsheets, tables, circuit diagrams or Venn diagrams for example, and accounts of logic based only on sentences will be unable to capture this type of reasoning.
Teaching Assistants will lead discussion sections for the class. The times and locations of discussion sections are posted on the Coursework web site. Attendance at sections is optional but recommended.
My office hours are 11am--noon on Mondays and 2--3pm on Thursdays. I will also try to be available immediately after classes. My office is in Cordura Hall, number 213. You can also call 723-9030 for an appointment outside these times, or drop by and take the chance that I am available.
Students Enrolled in Philosophy 150X will join the class on 10/26, at which point the class will be at the end of Part II of the course, studying complex uses for the quantifiers. My assumption is that you will either already know, or be able to quickly to get up to speed with, the material from the earlier parts of the course.
Students will complete the final two assignments, due on (11/20 and 12/4) and the final exam. The assignments will both be counted, and comprise 35% of your final grade, while the final will comprise the remaining 65%.
Previous instructors of 150/150X have observed that 150X students do worse on average than their 150 counterparts. The suggestion is that this is because it can be quite difficult to essentially pick up a new course in the middle of the quarter, from a scheduling/workload point of view. I hope that you will bear this in mind when you join us in November.
Homework assignments wil be posted here. All assignments are due by 5pm on the stated day. This means that all electronic work must be received by the Grade Grinder, and all written work should be in the Philosophy 150 mailbox in Building 90, by this time.
We strongly recommend that you do as many exercises from the Language, Proof and Logic textbook as you can. Much of what we will learn is a matter of practice, and the availability of the Grade Grinder as an automated TA means that you can quickly get a sense of how you are doing.
Suggested optional exercises Exercises 1.7, 2.17--19, 3.18, 4.24
Suggested optional exercises Exercises 5.1--5, 5.22, 6.5, 6.6, 7.12, 8.18--23, 8.25.
Suggested optional exercises Exercises 9.10, 9.11, 10.9, 11.7.