**Unit 4: 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, we
can more readily see what is right and what is wrong with them, and we
can learn how to formulate better arguments. *

*Advanced courses in formal logic focus on using rules of inference to
construct elaborate proofs. Using these techniques, we can solve many
complicated problems simply by manipulating symbols on the page. In this
course, however, we will only be looking at the most basic properties of
a system of logic. In this unit we 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. *

**Unit 4 Time Advisory**

This unit should take you approximately 17 hours.
☐
Subunit 4.1: 3.5 hours

☐ Subunit 4.2: 13.5 hours

☐ Subunit 4.2.1: 1 hour

☐ Subunit 4.2.2: 2 hours

☐ Subunit 4.2.3: 1 hour

☐ Subunit 4.2.4: 1 hour

☐ Subunit 4.2.5: 2 hours

☐ Subunit 4.2.6: 1 hour

☐ Subunit 4.2.7: 3 hours

☐ Subunit 4.2.8: 2 hours

☐ Subunit 4.2.9: 0.5 hours

☐ Subunit 4.2.10:

**Unit4 Learning Outcomes**

Upon successful completion of this unit, you will be able to:
- contrast the study of the principle for correct reasoning and the
study of the psychology of reasoning;
- describe the difference between formal logic and informal logic;
- explain why the principles of reasoning are topic neutral and
necessary;
- contrast and illustrate three kinds of statements: declarative,
interrogative, and imperative;
- define and illustrate the notions of consistency, inconsistency,
self-defeat, entailment, and logical equivalence;
- describe and illustrate the components of a system of logic;
- identify some reasons for creating and studying a system of logic;
- describe the rules for constructing well-formed formulae;
- identify and construct well-formed formulae;
- define and identify several kinds of logical statements: negations,
conjunctions, disjunctions, conditionals, and biconditionals;
- define the notion of scope for a well-formed formula;
- identify the scope of a well-formed formula;
- define the notion of a main connective;
- identify the main connective of a well-formed formula;
- read, construct, and analyze truth-tables;
- identify and apply the truth-tables for several kinds of logical
statements: negations, conjunctions, disjunctions, conditionals, and
biconditionals;
- apply truth-tables to determine whether specific statements are true
of particular situations;
- define the notions of tautology, inconsistent well-formed formula,
and contingent well-formed formula;
- identify well-formed formulae that are tautologous, inconsistent, or
contingent;
- apply truth-tables to determine whether a well-formed formula is
tautologous, inconsistent, or contingent;
- apply truth-tables to determine whether one well-formed formula
entails another;
- apply truth-tables to determine whether two well-formed formulae are
logically equivalent, contradictory, or consistent;
- apply truth-tables to determine whether arguments are valid or
invalid;
- translate ordinary statements into logical language; and
- describe and illustrate the limitations of truth-tables as
assessment tools.

**4.1 Logic Basics**
**4.1.1 What Logic Is (And What It Is Not)**
- **Reading: University of Hong Kong’s Critical Thinking Web: Joe Lau
and Jonathan Chan’s Basic Logic: “Tutorial L01”**
Link: University of Hong Kong’s Critical Thinking Web: Joe Lau and
Jonathan Chan’s

*Basic Logic*: “Tutorial L01” (HTML)

```
Instructions: Read this tutorial, which describes some basic
concepts of logic: validity, topic neutrality, necessity, and the
difference between formal and informal reasoning.
If you are using the HTML version, be sure to run the animated
portion of the tutorial.
Complete the exercises for this tutorial. Then, check your answers
against the [“Tutorial L01 Answer
Key”](http://www.saylor.org/site/wp-content/uploads/2014/02/PHIL102-Subunit4.1.1-TutorialL01AnswerKey-FINAL.pdf)
(PDF).
Reading this tutorial and completing the exercises should take
approximately 30 minutes.
Terms of Use: This resource is licensed under a [Creative Commons
Attribution-Non-Commercial-Share-Alike License
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/). It is
attributed to Joe Lau and Jonathan Chan, and the original version
can be found [here](http://philosophy.hku.hk/think/).
```

**4.1.2 Logical Statements, Connectives, and Relations**
- **Reading: University of Hong Kong’s Critical Thinking Web:
Professor Joe Lau’s and Professor Jonathan Chan’s “Tutorial L02:
Statements” and “Tutorial L03: Basic Concepts”**
Link: University of Hong Kong’s Critical Thinking Web: Joe Lau and
Jonathan Chan’s *Basic Logic*: “Tutorials
L02,
L03, and
L04” (HTML)

Instructions: Read these three tutorials. 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 for each tutorial. Then, check your answers
against the following: “Tutorial L02 Answer
Key”
(PDF) and “Tutorial L03 Answer
Key”
(PDF).

```
Reading these tutorials and completing the exercises should take
approximately 1.5 hours.
Terms of Use: This resource is licensed under a [Creative Commons
Attribution-Non-Commercial-Share-Alike License
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/). It is
attributed to Joe Lau and Jonathan Chan, and the original version
can be found [here](http://philosophy.hku.hk/think/).
```

**4.1.3 Logic is Fun!**
- **Reading: University of Hong Kong’s Critical Thinking Web:
Professors Joe Lau and Jonathan Chan’s Basic Logic: “Tutorial L05
and Tutorial L06”**
Link: University of Hong Kong’s Critical Thinking Web: Professors
Joe Lau and Jonathan Chan’s

*Basic Logic*: “Tutorial L05 and Tutorial L06” (HTML)

Instructions: Try your hand at some fun and tricky logic puzzles, as well as one exceptionally difficult one. For tutorial L05, complete puzzles 1–3, and then, check your answers against the “Tutorial L05 Answer Key.” Optionally, you can explore the Knights and Knaves questions, though note that there are no answers provided. For the answer to the challenging puzzle in the L06 tutorial, follow the embedded link to the Google Book.

Completing this activity should take approximately 1 hour.

Terms of Use: This resource is licensed under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0. It is attributed to Joe Lau and Jonathan Chan, and the original version can be found here.

**4.1.4 Review of Logics Basics**
- **Assessment: University of Hong Kong’s Critical Thinking Web: Joe
Lau and Jonathan Chan’s Critical Thinking Exercises: “Logical
Equivalence”**
Link: University of Hong Kong’s Critical Thinking Web: Joe Lau and
Jonathan Chan’s

*Critical Thinking Exercises*: “Logical Equivalence” (PDF)

```
Instructions: Complete this exercise, which will give you an
opportunity to better understand logical equivalence. Check your
answers against the [“Review of Logic Basics Answer
Key”](http://www.saylor.org/site/wp-content/uploads/2014/02/PHIL102-Subunit4.1.4-ReviewofLogicBasicsAnswerKey-FINAL.pdf)
(PDF).
Completing this assessment should take approximately 30 minutes.
Terms of Use: This resource is licensed under a [Creative Commons
Attribution-Non-Commercial-Share-Alike License
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/). It is
attributed to Joe Lau and Jonathan Chan, and the original version
can be found [here](http://philosophy.hku.hk/think/).
```

**4.2 A Little Bit of Formal Logic**
**4.2.1 About Formal Logic Systems and How to Write Sentences in SL**
- **Reading: University of Hong Kong’s Critical Thinking Web: Joe Lau
and Jonathan Chan’s Sentential Logic: “Tutorial SL01 and Tutorial
SL02”**
Link: University of Hong Kong’s Critical Thinking Web: Joe Lau and
Jonathan Chan’s

*Sentential Logic*: “Tutorial SL01 and Tutorial SL02” (HTML)

Instructions: Read these two tutorials about how formal systems of logic work and what they are useful for. The first tutorial introduces the elements of a simple system of logic called SL, and the second tutorial demonstrates how to construct statements, called well-formed formulas (WFFs), in SL.

Complete the exercises for tutorial SL02. Then, check your answers against the “Tutorial SL02 Answer Key” (PDF).

Reading these tutorials and completing the exercises should take approximately 1 hour.

Terms of Use: This resource is licensed under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0. It is attributed to Joe Lau and Jonathan Chan, and the original version can be found here.

**4.2.2 Connectives and Truth Tables**
- **Reading: University of Hong Kong’s Critical Thinking Web: Joe Lau
and Jonathan Chan’s Sentential Logic: “Tutorial SL03”**
Link: University of Hong Kong’s Critical Thinking Web: Joe Lau and
Jonathan Chan’s

*Sentential Logic*: “Tutorial SL03” (HTML)

Instructions: Read this tutorial, which will introduce you to truth-tables.

*Truth-tables*are an objective way of determining the validity of anargument as a whole when the argumentis expressed symbolically.

Complete the exercises for this tutorial.

Reading this tutorial and completing the exercises should take approximately 2 hours.

Terms of Use: This resource is licensed under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0. It is attributed to Joe Lau and Jonathan Chan, and the original version can be found here.

**4.2.3 How to Draw Truth-Tables for More Complicated Statements**
- **Reading: University of Hong Kong’s Critical Thinking Web: Joe Lau
and Jonathan Chan’s Sentential Logic: “Tutorial SL04”**
Link: University of Hong Kong’s Critical Thinking Web: Joe Lau and
Jonathan Chan’s

*Sentential Logic*: “Tutorial SL04” (HTML)

Instructions: Please 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 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.

Reading this tutorial and completing the exercise should take approximately 1 hour.

Terms of Use: This resource is licensed under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0. It is attributed to Joe Lau and Jonathan Chan, and the original version can be found here.

**4.2.4 Properties of Individual Well-Formed Formulas and Relations
between Them**
- **Reading: University of Hong Kong’s Critical Thinking Web: Joe Lau
and Jonathan Chan’s Sentential Logic: “Tutorial SL05”**
Link: University of Hong Kong’s Critical Thinking Web: Joe Lau and
Jonathan Chan’s

*Sentential Logic*: “Tutorial SL05” (HTML)

Instructions: Read this tutorial, which presents the same 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.

Reading this tutorial and completing the exercises should take approximately 1 hour.

Terms of Use: This resource is licensed under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0. It is attributed to Joe Lau and Jonathan Chan, and the original version can be found here.

**4.2.5 Understanding Truth-Tables**
- **Reading: Butte College: Wu Wei-Ming’s iLogic: “Chapter 3,
Section 2: Truth Tables”**
Link: Butte College: Wu Wei-Ming’s

*iLogic*: “Chapter 3, Section 2: Truth Tables” (HTML)

Instructions: Read section 2 of chapter 3. The material reviews the nature of truth-tables, the definitions of basic logical connectives, the rules for constructing truth-tables, 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 discusses methods for using truth-tables to determine whether an argument is valid. Use the interactive arrows to follow the examples.

Reading this section should take approximately 2 hours.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. It is attributed to Wu Wei-Ming, and the original version can be found here.

**4.2.6 How to Translate Ordinary Statements into Symbolic Formulae**
- **Reading: University of Hong Kong’s Critical Thinking Web: Joe Lau
and Jonathan Chan’s Sentential Logic: “Tutorial SL06”**
Link: University of Hong Kong’s Critical Thinking Web: Joe Lau and
Jonathan Chan’s

*Sentential Logic*: “Tutorial SL06” (HTML)

Instructions: 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 are likely to need one of the logical connectives.

Complete the exercises for this tutorial.

Reading this tutorial and completing the exercises should take approximately 1 hour.

Terms of Use: This resource is licensed under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0. It is attributed to Joe Lau and Jonathan Chan, and the original version can be found here.

**4.2.7 Formalization Practice**
- **Reading: Butte College: Wu Wei-Ming’s iLogic: “Chapter 3:
Introduction” and “Chapter 3, Section 1: Symbolization”**
Link: Butte College: Wu Wei-Ming’s

*iLogic*: “Chapter 3: Introduction” (HTML) and “Chapter 3, Section 1: Symbolization” (HTML)

Instructions: Read the introduction and section 1 of chapter 3. The material reviews and elaborates upon procedures for translating ordinary statements into the language of symbolic logic, which the text calls

*propositional logic*. Complete the exercises using the interactive features at the bottom of the page to test your understanding.

Reading these sections and completing the exercises should take approximately 3 hours.

Terms of Use: These resources are licensed under Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported Licenses. There resources are attributed to Wu Wei-Ming, and the original versions can be found here and here.

**4.2.8 Two Methods for Determining the Validity of an Argument**
- **Reading: University of Hong Kong’s Critical Thinking Web: Joe Lau
and Jonathan Chan’s Sentential Logic: “Tutorial SL07 and Tutorial
SL08”**
Link: University of Hong Kong’s Critical Thinking Web: Joe Lau and
Jonathan Chan’s

*Sentential Logic*: “Tutorial SL07 and Tutorial SL08” (HTML)

Instructions: Read these two tutorials, which provide information on how to determine if an argument – or

*sequent*– is valid or not 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. It may also help to copy them down with pencil and paper. Check your answers against the “Tutorial SL07 Answer Key” (PDF) and “Tutorial SL08 Answer Key” (PDF).

Reading these tutorials and completing the exercises should take approximately 2 hours.

```
Terms of Use: This resource is licensed under a [Creative Commons
Attribution-Non-Commercial-Share-Alike License
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/). It is
attributed to Joe Lau and Jonathan Chan, and the original version
can be found [here](http://philosophy.hku.hk/think/).
```

**4.2.9 Why Sentential Logic Is Not Enough**
- **Reading: University of Hong Kong’s Critical Thinking Web: Joe Lau
and Jonathan Chan’s Sentential Logic: “Tutorial SL10”**
Link: University of Hong Kong’s Critical Thinking Web: Joe Lau and
Jonathan Chan’s

*Sentential Logic*: “Tutorial SL10” (HTML)

Instructions: Read this tutorial on limitations. There are some statements that cannot be captured in sentential logic, especially statements involving words like

*every*and

*all*(e.g., “all men are mortal”). This tutorial explains why and introduces the idea of predicate logic.

Complete the exercises for this tutorial. Then, check your answers against the “Tutorial SL10 Answer Key” (PDF).

Reading this tutorial and completing the exercises should take approximately 30 minutes.

```
Terms of Use: This resource is licensed under a [Creative Commons
Attribution-Non-Commercial-Share-Alike License
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/). It is
attributed to Joe Lau and Jonathan Chan, and the original version
can be found [here](http://philosophy.hku.hk/think/).
```

**4.2.10 Review of Formal Logic**
- **Reading: Purdue University: Charles A. Bouman's “Review of Formal
Logic”**
Link: Purdue University: Charles A. Bouman's “Review of Formal
Logic”
(PDF)

```
Instructions: Click on the “Logic.pdf” link for a brief review of
formal logic.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```