Unit 3: Basic Sentential Logic
This unit introduces a topic that many students find intimidating: formal logic. Although it sounds difficult and complicated, formal (or symbolic) logic is actually a fairly straightforward way of revealing the structure of reasoning. By translating arguments into symbols, you can more readily see what is right and wrong with them and learn how to formulate better arguments. Advanced courses in formal logic focus on using rules of inference to construct elaborate proofs. Using these techniques, you can solve many complicated problems simply by manipulating symbols on the page. In this course, however, you will only be looking at the most basic properties of a system of logic. In this unit, you will learn how to turn phrases in ordinary language into well-formed formulas, draw truth tables for formulas, and evaluate arguments using those truth tables.
Completing this unit should take you approximately 13 hours.
Upon successful completion of this unit, you will be able to:
- contrast formal logic and informal logic;
- identify declarative, interrogative, and imperative sentences;
- define and identify several kinds of logical statements: negations, conjunctions, disjunctions, conditionals, and biconditionals;
- identify the scope and main connective for a well-formed formula;
- create truth-tables for several kinds of statements, sentences and arguments, such as negations, conjunctions, disjunctions, conditionals, and biconditionals;
- translate ordinary statements into logical language; and
- explain the limitations of truth-tables as assessment tools.
3.1: The Basics of Logic
Read this tutorial, which describes some basic logic concepts: validity, topic neutrality, necessity, and the difference between formal and informal reasoning.
3.1.1: Logical Statements, Connectives, and Relations
Read this section for an introduction to formal logic. Formal logic gives us a framework for objective, logical evaluations of conclusions. It can help you make valid inferences for certain kinds of statements. This section will not go deeply into how to do this type of logic but rather explain why it is important and give some basic examples.
Statements are the fundamental units of arguments and proofs in logic. These tutorials explain how to identify statements and introduce some of the basic ways that statements may be related to one another.
Complete the exercises and check your answers.
3.1.2: Logic Puzzles
Try your hand at some fun and tricky logic puzzles. Check your answers after you have solved them.
Complete this exercise, which will allow you to solve a difficult logic puzzle. Watch the video on this puzzle for a discussion of the solution.
3.2: A Little Bit of Formal Logic
3.2.1: How to Write Sentences in Sentential Logic
Read these four sections to learn how to identify and apply propositional (or sentential) logic functions. Using these symbols, you should be able to turn statements into symbolic formulas to more clearly see the logical connections taking place and determine when the conclusions are valid. It can look confusing at first, but moving slowly through these units will allow you to make valid logical proofs.
As you go, complete the exercises, then check your answers against the answer keys.
Note that the symbols used in some places can differ slightly from those used elsewhere. This is because there is not one standard set of symbols used for sentential logic, but a few. This table shows you the differences and helps translate between them.
In the resources in this course, the symbols for disjunction and negation are the same in both systems, but the symbols for conjunction, conditional, and biconditional are different.
Name Meaning Symbol 1 Symbol 2 Conjunction and & • Disjunction or v v Negation not ~ ~ Conditional if/then → ⊃ Biconditional if and only if ↔ ≡ In this section, you will read about how formal systems of logic work and what they are useful for. You will first be introduced to the elements of a simple system of logic called SL, and then you will learn how to construct statements called well-formed formulas (WFFs) in SL.
Complete the exercises and check your answers.
3.2.2: Connectives and Truth Tables
Read these sections to learn how to interpret, make, and apply truth tables to sentential logic formulas, note conditional statements in sentential logic, and translate the word "unless" into sentential logic. Be sure to note the difference between an antecedent and a consequent and between a necessary and sufficient condition.
Complete the exercises, checking your answers against the key.
Read this tutorial, which will introduce you to truth tables. Truth tables are an objective way of determining the validity of an argument as a whole when the argument is expressed symbolically.
Complete the exercises for this tutorial and check your answers.
3.2.3: How to Draw Truth Tables for More Complicated Statements
Read this tutorial to expand your knowledge of truth tables. The last tutorial showed you how to construct truth tables for the basic connectives in sentential logic (SL). This tutorial extends the same technique to more complex well-formed formulas, which approximate the kinds of statements that might be part of an argument in ordinary language.
Complete the exercises for this tutorial and check your answers.
3.2.4: Properties of Individual Well-Formed Formulas and Relations Between Them
Read these sections to learn more about relationships among truth statements and using and constructing logical proofs.
These sections review materially equivalent propositions and three other relationships among statements: tautological, contradictory, and contingent relationships. They also review the eight valid forms of inference: modus ponens, modus tollens, hypothetical syllogism, simplification, conjunction, disjunctive syllogism, addition, and constructive dilemma. They show how to construct proofs, including strategies for working forward or backward, depending on which is easier according to your premises. Finally, they summarize everything you have learned about sentential and propositional logic.
Complete the exercises as you study, then check your answers against the key.
Read this tutorial, which presents the concepts of consistency, entailment, and equivalence introduced in subunit 4.1.2 but defines them now in terms of their truth tables in SL. These are all relations between WFFs. This tutorial introduces the concepts of tautology, contingency, and inconsistency as properties of individual WFFs that can also be defined by their truth tables.
Complete the exercises for this tutorial and check your answers.
3.2.5: Understanding Truth Tables
This page reviews the nature of truth tables, the definitions of basic logical connectives, the rules for constructing truth tables, and the methods for using truth tables to determine whether a well-formed formula is tautologous, inconsistent, self-consistent, or contingent.
The material also discusses methods for using truth tables to determine whether two well-formed formulas are logically equivalent, contradictory, or consistent. Finally, it covers methods for using truth tables to determine whether an argument is valid.
3.2.6: How to Translate Ordinary Statements into Symbolic Formulae
Read this tutorial on formalization, which means turning statements and arguments in ordinary language into their symbolic counterparts; we might just as well call it translation. Notice that ordinary language contains hint words, letting us know when we will likely need one of the logical connectives.
Complete the exercises for this tutorial and check your answers.
3.2.7: Formalization Practice
Read this section, which reviews and elaborates upon procedures for translating ordinary statements into the language of symbolic logic, which the text calls propositional logic.
Complete the exercises to test your understanding.
3.2.8: Two Methods for Determining the Validity of an Argument
Read these tutorials, which provide information on determining whether an argument – or sequent – is valid in SL. Because using truth tables to establish validity is time-consuming, the second tutorial presents a shortcut version of the method.
Complete the exercises for both tutorials and check your answers.
3.2.9: Why Sentential Logic Is Not Enough
Read this tutorial on limitations. Some statements cannot be captured in sentential logic, especially statements involving words like every and all (like "all men are mortal"). This tutorial explains why and introduces the idea of predicate logic.
Complete the exercises for this tutorial and check your answers.