Introductory logic course designed for students from a broad range of disciplines, from mathematics and computer science to drama and creative writing.

# Logic & Proofs

$80

- Description
- What students will learn
- Learning objectives by module
- Course assessments, activities, and outline
- Other course details
- System requirements
- Included instructor tools

### Description

Logic is a remarkable discipline. It is deeply tied to mathematics and philosophy, as correctness of argumentation is particularly crucial for these abstract disciplines. Logic systematizes and analyzes steps in reasoning: *correct* steps guarantee the truth of their conclusion given the truth of their premise(s); *incorrect* steps allow the formulation of counterexamples, i.e., of situations in which the premises are true, but the conclusion is false.

Recognizing (and having conceptual tools for recognizing) the correctness or incorrectness of steps is crucial in order to critically evaluate arguments, not just in philosophy and mathematics, but also in ordinary life. This skill is honed by working in two virtual labs. In the *ProofLab* you learn to construct complex arguments in a strategically guided way, whereas in the *TruthLab* the emphasis is on finding counterexamples systematically.

**Who should take this course?**

This is an introductory course designed for students from a broad range of disciplines, from mathematics and computer science to drama and creative writing. The highly interactive presentation makes it possible for any student to master the material. Concise multimedia lectures introduce each chapter; they discuss, in detail, the central notions and techniques presented in the text, but also articulate and motivate the learning objectives for each chapter.

**Open & Free Version**

The Open & Free, Logic & Proofs course includes the first five chapters of Logic & Proofs, providing a basic introduction to sentential logic. A full version of Logic & Proofs, including both sentential and predicate logic, is also available without technical or instructor support to independent users, for a small fee. No credit is awarded for completing either the Open & Free, Logic & Proofs course or the full, unsupported Logic & Proofs course.

**Academic Version**

Academic use of Logic & Proofs provides a full course on modern symbolic logic, covering both sentential and predicate logic, with identity. Optional suites of exams are available for use in academic sections.

#### Additional Course Details

**Topics Covered:**

The notions of statement and argument; Logical analysis of informal arguments; Syntax and semantics of: sentential logic, predicate logic, and identity; Natural deduction-style proofs, with an emphasis on effective and strategic proof construction; and Truth-trees, with an emphasis on systematic construction of counterexamples from completed trees.

#### In-Depth Description

Logic & Proofs is an introduction to modern symbolic logic, covering sentential and predicate logic (with identity). The course is highly interactive and engaging. It brings a fresh perspective to classical material by focusing on developing two crucial logical skills: **strategic construction of proofs** and the **systematic search for counterexamples**.

Concise multimedia lectures introduce each chapter of the course and discuss in detail central notions and techniques presented in the text. The introductory lectures articulate and motivate the learning objectives for each chapter.

**LAB EXERCISES**

The two crucial logical skills are developed via numerous exercises in two lab environments:

In the **ProofLab**, the main workbench of Logic & Proofs, students practice proof construction in a natural deduction framework. Their learning is supported by an intelligent and dynamic automated tutor. This tutor helps students, in a dialogue, to think through arguments in a strategic and systematic fashion. This **Proof Tutor** is making use of the automated proof search mechanism AProS; **see the AProS site here**.

In the **TruthLab**, the semantic counterpart to the ProofLab, students practice techniques for a semantic analysis of formulae and arguments. They begin with chasing truth up a parse tree, then complete truth-tables, and ultimately learn to build truth-trees for predicate formulae involving identity. The emphasis is on reading off counterexamples to invalid arguments from completed trees.

**CONTENT STRUCTURE**

Each chapter features both review materials and homework assignments, including quizzes and lab problems. The end-of-chapter quizzes and practice questions provide fully automated feedback to the student; the ample practice lab problems offer tutoring, while the problems in the chapter’s lab assignment do not, providing students with the opportunity to demonstrate mastery of the skills developed in completing the practice problems.

**USING THE COURSE AT ACADEMIC INSTITUTIONS**

At Carnegie Mellon and elsewhere, Logic & Proofs is offered as a semester-long introductory logic course. In addition to working through the online material at a specified rate (approximately one chapter per week), the class meets in small groups once a week for an instructor-led discussion session. Active student participation is not only encouraged, but required.

At some institutions, Logic & Proofs has been offered as a fully self-paced course, with online and drop-in instructor and TA support, but without regular class meetings. At yet other institutions, Logic & Proofs is used as the main resource for a course with traditional weekly lectures, as well as meetings with a TA. In all three modes of use, Logic & Proofs has been found pedagogically effective. See the relevant research by Patchan, Schunn, Sieg and McLaughlin at the **AProS project site**.

The course has been taken since 2007 by **more than 12,000 students for credit** at various institutions including Carnegie Mellon University, Carnegie Mellon Qatar, IUPUI, Francisco Marroquín University (Guatemala), Haverford College, University of British Columbia, University of Nevada in Las Vegas, Kent State University, College of Lake County, University of Washington, University of Maryland (College Park), and the University of South Florida. The course is is now also being offered through Stanford’s EPGY Program to high-school students.

## What students will learn

By the end of this course, students will have learned how to critically evaluate arguments, not just in philosophy and mathematics, but also in ordinary life. They will learn to:

- explain what an argument is, and determine whether or not a given passage constitutes an argument.
- identify the premises and conclusion of an argument, and represent the argument in standard form.
- determine whether a given argument is a bad or a good argument, and why this is the case.
- create diagrams that graphically depict the structure of arguments.

## Learning objectives by module

**Part 1: Introduction**

**Chapter 1: Statements and Arguments**- Determine whether or not a sentence of English expresses a statement, and if so,

to identify the statement expressed. - Explain what a statement is, and discuss how statements are related to

sentences. - Explain what an argument is, and determine whether or not a givenpassage

constitutes an argument. - Identify the premises and conclusion of an argument, andrepresent the argument

in standard form. - List the criteria an argument must meet in order to be considereda good

argument, and explain why each criterion is necessary.

- Determine whether or not a sentence of English expresses a statement, and if so,

**Part 2: Introduction**

**Chapter 2: Statements, Premises, and Conclusions**- Create diagrams that graphically depict the premises-and-conclusion structure of

arguments. - Determine whether or not a sentence of English expresses a statement, and if so,

to identify the statement expressed. - Determine whether premises support the conclusion jointly or independently.
- Explain what a statement is, and discuss how statements are related to

sentences. - Explain what an argument is, and determine whether or not a givenpassage

constitutes an argument. - Identify the premises and conclusion of an argument.

- Create diagrams that graphically depict the premises-and-conclusion structure of
**Chapter 3: Arguments, Validity, and Structure**- Analyze logically structured support that can link premises and conclusion of an

argument. - Create diagrams that graphically depict the structure of arguments.
- Determine whether a given argument is a bad or a good argument, and why this is

the case. - Explain what a structured statement is and give examples of statement structures

that are logically important.

- Analyze logically structured support that can link premises and conclusion of an

**Part 3: Sentential Logic**

**Chapter 4: Syntax and Symbolization**- Construct and identify formulae of sentential logic.
- Construct and use parse trees.
- Discern the logical structure of English sentences.
- Explain the grammar of the logical language of sentential logic.
- Symbolize English sentences as formulae of sentential logic.

**Chapter 5: Semantics**- Construct a truth-table for a given formula or argument.
- Determine the truth-value of a formula relative to a given truth-value

assignment. - Explain what a truth-value assignment is.
- Explain what tautological, contingent, and contradictory formulae are.
- Find a counterexample to an invalid argument, using a truth-table or truth-tree.
- Give the truth-conditions for the logical connectives.
- Use truth-tables to analyze arguments and formulae.

**Chapter 6: Derivations**- Apply and identify applications of rules of inference within a derivation.
- Establish the validity of rules of inference.
- Explain the structure of derivations.

**Chapter 7: Indirect Rules**- Apply and identify applications of the inference rules for negation.
- Explain the structure of indirect rules of inference.
- Find contradictions to use in applications of indirect rules.

**Chapter 8: Strategies and Derived Rules**- Apply and identify applications of derived rules.
- Approach proof construction problems in a strategic fashion.
- Provide explanations of and explain some significant properties of the logical

connectives.

**Chapter 9: Elementary Metamathematics**- Explain how a biconditional can be considered logically equivalent to a formula

in disjunctive normal form. - Explain the connection between truth-tables and Boolean circuits.
- Find a disjunctive normal form equivalent to any formula.
- Show that two formulae are logically equivalent just in case their biconditional

is a tautology.

- Explain how a biconditional can be considered logically equivalent to a formula

**Part 4: Predicate Logic**

**Chapter 10: Syntax and Semantics I**- Analyze the internal logical structure of English sentences.
- Construct atomic formulae from predicates and singular terms according to

syntactic rules. - Identify and distinguish between predicates and singular terms.
- Semantically interpret and evaluate formulae constructed using predicates and

singular terms.

**Chapter 11: Syntax and Semantics II**- Analyze the internal logical structure of English sentences involving quantity

terms. - Construct arbitrary predicate formulae according to the syntactic rules.
- Semantically interpret and evaluate predicate formulae.

- Analyze the internal logical structure of English sentences involving quantity
**Chapter 12: Derivations**- Apply and identify applications of the inference rules for the quantifiers.

**Chapter 13: Strategies and Derived Rules**- Convert an arbitrary formula into an equivalent formula in prenex normal form.
- Develop, apply, and identify applications of derived rules.
- Extend the strategic considerations for the construction of proofs to predicate

logic.

**Chapter 14: Identity and Functions**- Apply and identify applications of the inference rules for identity.
- Give the Russellian analysis of sentences involving definite descriptions.
- Translate English sentences involving identity into the extended language of

predicate logic. - Use function symbols in translating both English sentences and mathematical

statements.

## Course assessments, activities, and outline

*Preface*

**Logic & Proofs**

PART 1: Introduction

Chapter 1: Statements and Arguments

**Quiz:** Statements and Arguments

PART 2: Introduction

Chapter 2: Statements, Premises, and Conclusions

**Quiz**: Statements, Premises, and Conclusions

**Lab**: Statements, Premises, and Conclusions

Chapter 3: Arguments, Validity, and Structure

**Quiz**: Arguments, Validity, and Structure

**Lab**: Arguments, Validity, and Structure

PART 3: Sentential Logic

Chapter 4: Syntax and Symbolization

**Quiz**: Syntax and Symbolization

Chapter 5: Semantics

**Quiz**: Semantics

**Lab**: Semantics

Chapter 6: Derivations

**Quiz**: Derivations

**Lab**: Derivations

Chapter 7: Indirect Rules

**Quiz**: Indirect Rules

**Lab**: Indirect Rules

Chapter 8: Strategies and Derived Rules

**Quiz**: Strategies and Derived Rules

**Lab**: Derived Rules

Chapter 9: Elementary Metamathematics

**Quiz**: Elementary Metamathematics

**Lab**: Elementary Metamathematics

PART 4: Predicate Logic

Chapter 10: Syntax and Semantics I

**Quiz**: Syntax and Semantics I

**Lab**: Syntax and Semantics I

Chapter 11: Syntax and Semantics II

**Quiz**: Syntax and Semantics II

**Lab**: Syntax and Semantics II

Chapter 12: Derivations

**Quiz**: Derivations

**Lab**: Predicate Derivations

Chapter 13: Strategies and Derived Rules

**Quiz**: Strategies and Derived Rules

**Lab**: Predicate Derived Rules

Chapter 14: Identity and Functions

**Quiz**: Identity and Functions

**Lab**: Identity and Functions

**Functions & Computations**

PART 5: Set Theory

Chapter 15: Sets and Operations

**Chapter Test**: Sets and Operations

Sets and Operations: **Problem Set** **1**

Chapter 16: Cartesian Products and Relations

**Chapter Test**: Cartesian Products and Relations

Cartesian Products and Relations: **Problem Set** **1**

Chapter 17: Functions

**Chapter Test**: Functions

Functions: **Problem Set** **1**

Chapter 18: Sizes of Sets

**Chapter Test**: Sizes

Sizes: **Problem Set** **1**

Chapter 19: ZF

**Chapter Test**: Principles for Sets

ZF: **Problem Set** **1**

Chapter 20: Calculable Functions

**Chapter Test**: Calculable Functions

**Formal Set Theory**

PART 6: Logical Framework for Set Theory

Chapter 21: Formal Proofs

**Lab Assignment: **Formal Proofs

Chapter 22: Quantifier and Identity

**Lab Assignment: **Quantifier and Identity

PART 7: Incompleteness: Set-Theoretic Representation

Chapter 23: Natural Numbers as Sets

**Lab Assignment: **Natural Numbers as Sets

Chapter 24: Functions and Sizes

**Lab Assignment: **Functions and Sizes

Chapter 25: Internalization of Syntax

**Lab Assignment: **Internalization of Syntax

Chapter 26: Gödel’s Theorems

**Lab Assignment: **Gödel’s Theorems

**Logic & Proofs Exams**

PART 8: Exams

Chapter 27: Package A

Midterm Exam #1

Midterm Exam #1 – Argument Diagramming

Midterm Exam #1 – Truth-Tables

Midterm Exam #1 – Truth-Trees

Midterm Exam #2

Midterm Exam #2 – Truth-Tables

Midterm Exam #2 – Truth-Trees

Midterm Exam #2 – Derivations

Midterm Exam #3

Midterm Exam #3 – Truth-Trees

Midterm Exam #3 – Derivations

Final Exam

Final Exam – Argument Diagramming

Final Exam – Truth-Trees

Final Exam – Derivations

Chapter 28: Package A

Midterm Exam #1

Midterm Exam #1 – Truth-Tables

Midterm Exam #1 – Truth-Trees

Midterm Exam #2

Midterm Exam #2 – Truth-Tables

Midterm Exam #2 – Truth-Trees

Midterm Exam #2 – Derivations

Midterm Exam #3

Midterm Exam #3 – Truth-Trees

Midterm Exam #3 – Derivations

Final Exam

Final Exam – Truth-Trees

Final Exam – Derivations

Chapter 29: Package B

Midterm Exam

Midterm Exam – Truth-Tables

Midterm Exam – Truth-Trees

Final Exam

Final Exam – Truth-Trees

Final Exam – Derivations I

Final Exam – Derivations II

Chapter 30: Package C

Final Exam

Final Exam – Argument Diagramming

Final Exam – Truth-Trees

Final Exam – Derivations

Chapter 31: Package C

Final Exam

Final Exam – Truth-Trees

Final Exam – Derivations

**Resources**

## Other course details

This is a semester-long course, taking one week to cover each of the 12 core chapters of the material. That includes the homework assignments at the end of each chapter.

The course can also be used as an accelerated, 4 to 6 week introduction to or review of logic, at the rate of two to three chapters per week (possibly omitting some chapters or sections), to be followed by additional topics for the remainder of the semester.

January, 2018

Coming soon.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

## System requirements

OLI system requirements, regardless of course:

- internet access
- an operating system that supports the latest browser update
- the latest browser update (Chrome recommended; Firefox, Safari supported; Edge and Internet Explorer are supported but not recommended)
- pop-ups enabled
- cookies enabled

Some courses include exercises with exceptions to these requirements, such as technology that cannot be used on mobile devices.

This course’s system requirements:

- none listed (subject to change)

## Included instructor tools

Instructors who teach with OLI courses benefit from a suite of free tools, technologies, and pedagogical approaches. Together they equip teachers with insights into real-time student learning states; they provide more effective instruction in less time; and they’ve been proven to boost student success.

If you’d like to update an OLI course for your students, or even develop a new course or program of study, contact OLI Support for information about the OLI Author platform.

Learning and participation data is displayed in the Learning Dashboard. Read more.

Learning Engineering is the virtuous cycle of iterative improvement of learning content, instructional technologies, and the greater scientific research community. Read more

OLI courses can be deployed with preferences in dozens of areas, including: