PHIL 2140 Introduction to Logic

Course Overview

This course is an introduction to formal logic: using formal (symbolic) analysis and other formal methods to determine what logically follows from what. The course covers the two best known systems of formal logic: propositional logic (also called truth-functional or sentential logic) and first-order logic (also called predicate or quantificational logic). Students will learn the language of both systems, and be able to construct truth-tables, formal proofs, and counterexamples to logical arguments.

Course Content

This course follows the following basic schedule of topics:

Arguments, Logic, and Formal Logic

An argument is a piece of reasoning: one provides reasons for something being the case. However, are those reasons provided actually good reasons? Logic is the study of exactly that, and it has been around for thousands of years, since this is of course a very basic and important question: reasoning is great ... if you get it right! In the 19th century, though, systems of formal logic started to appear: systems that started to analyze reasoning in a purely symbolic, mathematical, manner, starting with George Boole and Boolean Logic. These systems provide enormous precision and power to the study of logic, but it should also be recognized that they are of limited scope. In particular, while formal logic is a great tool to figure out whether something necessarily follows from something else, it does not deal well with plausibility and uncertainty. In short, formal logic works well where things are clean, nice, precise, and crisp, but not where things are fuzzy, uncertain, and messy. Indeed, applications of formal logic are almost exclusively in mathematics and computer science, and rarely in 'real', day to day life.

Please Read: An important note to the students: Logic vs Critical Thinking!!

Truth-Functional Logic

Truth-functional (also called propositional or sentential) logic is the logic at the level of propositions or sentences: it analyzes how larger sentences may be composed of smaller sentences using words like 'and, 'or', 'not, and 'if ... then ...', and studies the logic of arguments involving such sentences through the use of truth-functional connectives, which make up the language of truth-functional logic. Important skills and methods the students will need to learn are symbolization (taking English sentences and symbolizing them in the language of truth-functional logic), truth-tables (analyzing truth-functional properties and relationships of truth-functional logic expressions), formal proofs (step-by-step derivations of statements from other statements, using purely symbolic and thus super strict inference rules), and a few other methods and representations.

Quantificational Logic

Quantificational (also called predicate or first-order) logic is an extension of truth-functional logic. It digs a little deeper at the nature of sentences, allowing analysis at the level of subjects (objects) and predicates, and also allows basic quantification ('all' and 'some') of objects. As such, it is more powerful than truth-functional logic, and is indeed such a powerful, yet precise language that it is often used in mathematics and computer science simply for the sake of more clearly representing complicated claims or theorems. Students will therefore spend quite a bit of time on symbolization, thus learning to 'speak' the language of first-order logic. Formal methods for analysis include Venn diagrams, formal proofs, and a few others.

Applications of Logic

Scattered throughout the course will be applications of logic. Formal logic has important applications to computer science in the form of computer circuity ('logic gates'), programming languages (from boolean expressions to complete logic-based programming languages), automated theorem proving, and program certification. Logic is also used in mathematics to precisely express theorems and axioms, and to derive those theorems from axioms, thus not only certifying difficult mathematical proofs, but also showing how exactly different claims and ideas relate to each other. Other applications of logic are to solve logic puzzles (always good for preparing for the LSAT!), and occsionally we will find an application of formal logic in other sciences or real life arguments.

Typical Grade Distribution

Connections to Other Courses

There are no prerequisites for this course, but an affinity with mathematics and logical thinking will definitely help. Also, formal logic is covered in various MATH and CSCI courses, especially in CSCI 2200 Foundations in Computer Science. Follow-up courses to this course are PHIL-4140 / MATH-4140 Intermediate Logic and PHIL-4420 / MATH-4030 / CSCI 4420 Computability and Logic. CSCI students are allowed to use CSCI 2200 Foundations of Computer Science as a prerequisite for these more advanced logic courses instead, but if they want to really become proficient in formal logic, it might be a good idea to take Introduction to Logic as well, which incidentally can be counted as a Humanities requirement. Finally, formal logic is used in CSCI 4150 Introduction to Artificial Intelligence.

Syllabus

Detailed Schedule

Typical Weekly Schedule