# PH133 Logic: Slides and Handouts

You can find slides and handouts below, together with an outline of each lecture.

Please note that these may be revised even after the lecture occurred.

## Lecture 1

Date given: Wednesday 22nd April 2015

Exercises for this lecture:

### Learning Objectives (013)

Graphical depiction of learning objectives for this course.

### Quick Intro to awFOL (02)

definitions

Gives some examples of the formal language awFOL and explains their relation to ordinary English. Also explains terms like 'name' and 'predicate'.

Alternative textbook exercises (regular): 1.1--1.5, *1.6, 1.8--1.10

### Logically Valid Arguments (03)

definitions

Explains the notion of logically valid argument. This is the central notion for this course.

Alternative textbook exercises (regular): 2.3, 2.4

### Counterexamples (04)

definitions zoxiy ex-create

Explains the notion of a counterexample. Discusses a counterexample to a simple argument. Demonstrates using zoxiy to construct a counterexample involving shape and size properties.

Alternative textbook exercises (regular): 2.8, 2.10, 2.12, 2.21

Alternative textbook exercises (fast): 2.8, 2.10, 2.12, 2.21

### Identity (06)

definitions zoxiy ex-create

Introduces the notion of identity used in studying first-order logic. Includes example argument and the two principles. Also demonstrates how to assign names in zoxiy, and using zoxiy to make identity statements true and false.

Alternative textbook exercises (regular): 2.5, 2.6

### Sentence Letters (012)

definitions

Explains what sentence letters are and why we use them.

### Truth Tables (071)

definitions

Introduces truth tables for conjunction, disjunction and negation. Truth tables give the meanings of connectives in our formal language awFOL.

Alternative textbook exercises (regular): 3.1, 3.2, 3.5, 3.7

Alternative textbook exercises (fast): 3.1, 3.3, 3.5, 3.7

### Complex Truth Tables (60)

truth-tables

Illustrates how to construct complex truth tables with an example, not(P and Q).

## Lecture 2

Date given: Thursday 23rd April 2015

Exercises for this lecture:

### Formalizing Arguments (135)

translation

Illustrates how for formalise simple arguments with the 'pigs' illustration.

### Logical Validity and Truth Tables (14)

truth-tables

Illustrates the use of truth tables to determine logical validity.

Alternative textbook exercises (regular): 5.1-5.4

### Why Logic? (011)

definitions

Provides two reasons for studing logic, one abstract the other practical.

### Complex Truth Tables (60)

truth-tables

Illustrates how to construct complex truth tables with an example, not(P and Q).

### Complex Truth Tables (09)

truth-tables

Illustrates how to construct complex truth-tables with an example, the truth-table for (P and Q) or R. Also describes how to order the reference columns.

Alternative textbook exercises (regular): 3.12, 3.13, 4.4-4.7, 4.12-14

Alternative textbook exercises (fast): 3.14, 3.15

### The Storm Clouds on the Horizon Were Getting Nearly Directly Overhead (162)

truth-tables

Illustrates and defines the terms 'tautology' and 'contradiction' with some quotes from George Bush.

### Contradictions, Logical Truths and Logical Validity (160)

truth-tables

Continues exploring the use of truth-tables to establish whether sentences are contradictions or logical truths, and to demonstrate logical validity. Notes that logical validity is not what you think it is.

Alternative textbook exercises (regular): 4.1, 4.2

Alternative textbook exercises (fast): 4.1, 4.2, 4.12--4.16

### Formal Proof (211)

proof

Marks the transition from truth-tables, in which the notion of truth plays a central role, to fitch proofs which are purely formal entities.

## Lecture 3

Date given: Tuesday 28th April 2015

Exercises for this lecture:

### Formal Proof (211)

proof

Marks the transition from truth-tables, in which the notion of truth plays a central role, to fitch proofs which are purely formal entities.

### Translating a Simple Argument (213)

translation

Illustrates how to translate an argument from English into our formal language awFOL.

### Formal Proof: ∧Elim and ∧Intro (21)

proof zoxiy ex-proof

Introduces and illustrates the use of rules of proof for conjunction. Explains how to create proofs using logic-ex.

Alternative textbook exercises (regular): 6.1

Alternative textbook exercises (fast): 5.3--5.6

### ∧Intro and ∨Intro: Compare and Contrast (212)

proof

Contrasts two rules of proof and explores how their differences are related to truth-tables.

### How to Write Proofs (130)

definitions

Describes how to write proofs.

### Rules of Proof for Identity (110)

proof

Introduces and illustrates the rules of proof for identity, =Intro and =Elim.

### Logic Makes Me Die Inside (161)

definition

Is the definition of logical validity flawed?

### ¬, ⊥ (270)

proof

Quick introduction to two connectives, ¬ and ⊥, together with illustration of rules of proof for ⊥.

### A ∧ B ∨ C (151)

proof syntax

Distinguishes lexical from structural ambiguity. Asks how we can know there is no structural ambiguity in our formal language awFOL.

Alternative textbook exercises (regular): 3.20, 3.21, 3.22

### A ∧ B ∨ C: the Truth-tables (153)

truth-tables

Illustrates how to construct complex truth tables for (A ∧ B) ∨ C and A ∧ (B ∨ C).

### A ∧ B ∨ C: They Are Different (154)

truth-tables

Uses two arguments to show that (A ∧ B) ∨ C and A ∧ (B ∨ C) really are different.

Alternative textbook exercises (regular): 3.14, 3.15

Alternative textbook exercises (fast): 3.14, 3.15, 7.2, 7.5, 7.6

### Formal Proof (111)

proof

Explains that Fitch is a formal (non-sematic) system of proof by highlighting an incorrect use of =Elim.

## Lecture 4

Date given: Wednesday 29th April 2015

Exercises for this lecture:

### Everything Is Broken (471)

quantification

Quantifiers are for talking about things without naming them.

Alternative textbook exercises (regular): 9.1 odd numbers only, 9.2 even numbers only

Alternative textbook exercises (fast): 9.8–-9.10

### Get Some Quantifiers: Don't Economise (473)

quantification

We're going to be using quantifiers a lot. We understand them thanks to Frege and Taski. It's important to get a pair you like.

### What does ∃ mean? (491)

semantics

Explains the meaning of ∃ with a simple example.

### Formal Proof (111)

proof

Explains that Fitch is a formal (non-sematic) system of proof by highlighting an incorrect use of =Elim.

### =Elim Again (112)

proof

Returns to the discussion of what it is for a system of proof to be formal and explains how to do awkward proofs involving identity.

### →Intro, →Elim (390)

proof proof-example

Introduction to the rules of proof for →, →Intro and →Elim. This is the first time subproofs are used. Also illustrates a proof without any premises. The proof is from premises P→Q and Q→R to conclusion P→R.

Alternative textbook exercises (fast): 8.20--8.25

### Subproofs with zoxiy (395)

proof proof-example zoxiy ex-proof

Introduction to the rules of proof for →, →Intro and →Elim. This is the first time subproofs are used. Also illustrates a proof without any premises. The proof is from premises P→Q and Q→R to conclusion P→R.

### ∨Elim: An Example (221)

proof proof-example

Illustrates the use of ∨Elim, a rule of proof. Uses the inference from A∨B to B∨A as an example.

Alternative textbook exercises (regular): 5.1--5.6, 6.2--6.6

### ∨Elim and Soundness (226)

proof

Introduces the rule of proof ∨Elim and links it to the truth table for ∨.

## Lecture 5

Date given: Tuesday 5th May 2015

Exercises for this lecture:

### Not Or (603)

translation

Negating disjunctions is tricky. This contrasts ¬(A ∨ B) with ¬A ∨ ¬B.

Alternative textbook exercises (regular): 3.19, 5.18

### What does ‘→’ mean? (700)

semantics

Shows how to derive a truth table for → from the rules of proof for →, →Intro and →Elim.

Alternative textbook exercises (regular): 7.1--7.6, 8.1 (yes/no answers are ok)

Alternative textbook exercises (fast): 7.2, 7.5, 7.6

### I Met a Philosopher (501)

quantification

Explains how to express propositions like 'I met a philosopher' by using the existential quantifier together with conjunction.

Alternative textbook exercises (regular): 9.1 odd numbers only, 9.2 even numbers only, 9.4, 9.5, 9.8, 9.9, 9.10, 9.12

Alternative textbook exercises (fast): 9.8, 9.9, 9.10, 9.12, 9.13

### All Squares Are Blue (503)

quantification revision

Explains how to translate 'All squares and blue' and 'Some squares are blue' into awFOL using quantifiers.

Alternative textbook exercises (regular): 9.4--9.5, 9.8--9.10

### What does ∀ mean? (492)

semantics

Explains the meaning of ∀ with an example.

### Vegetarians Are Evil (502)

quantification

Practices translating an English argument involving quantifers into awFOL.

### Counterexamples with Quantifiers (502b)

quantification

Illustrates giving a counterexample to an argument involving quantifiers.

### We knew that it was invalid ... (163)

definition

Some arguments are logically valid even though the conclusion is unrelated to the premises.

### Not If (604)

translation

Negating sentences whose main connective is the arrow is tricky. This contrasts ¬(A → B) with A → ¬B.

### ↔ : truth tables and rules (567)

translation

Mentions the truth-tables and rules of proof for the biconditional, ↔.

### Does ‘if’ mean what ‘→’ means? (640)

semantics quirk

Argues that ‘if’ must mean what ‘→’ means, and that it cannot.

### Translation with Quantifiers (507)

translation

Discusses how to translate 'Some persuasive arguments are not valid' and 'All quadrumanous discordians weep and wail except Gillian Deleude' into first-order logic.

Alternative textbook exercises (regular): 9.4–-9.5, 9.8–-9.9

Alternative textbook exercises (fast): 9.12-–9.13

## Lecture 6

Date given: Wednesday 6th May 2015

Exercises for this lecture:

### DeMorgan: ¬(A ∧ B) ⫤⊨ ¬A ∨ ¬B (235)

truth-tables

Introduces DeMorgan's Laws relating ¬, ∧ and ∨. The laws can be stated as A ∧ B ⫤⊨ ¬(¬A ∨ ¬B) and A ∨ B ⫤⊨ ¬(¬A ∧ ¬B). Also mentions that A → B ⫤⊨ ¬A ∨ B.

Alternative textbook exercises (regular): 3.19, 4.15--18, 7.1--7.2, *7.3--7.6

Alternative textbook exercises (fast): 3.19, 4.31

### Negation and the arrow: A → ¬B ⊭ ¬(A → B) (237)

truth-tables

Distinguishes if A, not B (A → ¬B) from the negation of if A,B (¬(A → B)).

### ↔ : truth tables and rules (567)

translation

Mentions the truth-tables and rules of proof for the biconditional, ↔.

### All Cats Are Grey (505)

translation

Covers translation of simple sentences like 'All cats are grey' using single quantifiers.

### universal:arrow, existential:conjunction (506)

quantification revision

Reminder that the universal quanifier works with the arrow, whereas the existential quantifier works with conjunction.

### Don't use ∃ with → (245)

translation

Stresses that ∀ works with → by examining sentences with ∀ and ∧, and ∃ with →.

Alternative textbook exercises (regular): 9.10, 9.15--9.17, *9.18--9

Alternative textbook exercises (fast): 9.10

### ¬Intro (281)

proof proof-example

Introduction to the rule of proof ¬Intro with an illustration. Discusses the proof from premise A∧B to conclusion ¬(¬A∨¬B).

Alternative textbook exercises (regular): 6.7--6.10, *6.11--6.12

Alternative textbook exercises (fast): 6.7--6.12, 6.18--6.20, 6.24--6.27, *6.40--6.42

### ¬Intro Proof Example (283)

proof proof-example

Discusses an intermediate proof involving ¬Intro, from premises P→Q and ¬Q to conclusion ¬P.

Alternative textbook exercises (regular): 6.24--6.26

### Subproofs Are Tricky (224)

proof

Offers what appears to be a proof of an argument that is not logically valid in order to illustrate a restriction on the use of subproofs.

### ∀Elim (800)

quantification proof

Explains the rule of proof ∀Elim with an example.

## Lecture 7

Date given: Tuesday 12th May 2015

Exercises for this lecture:

### Watch Out, Here Come Multiple Quantifiers (744)

revision quantification semantics translation

Preliminary to discussion of multiple quantifiers.

### Something Is Above Something (740)

quantification semantics

Introduces a simple sentence involving multiple quantifiers. Illustrates how to apply the procedure for determining the truth of sentences involving quantifiers to sentences containing multiple quantifiers.

Alternative textbook exercises (fast): 11.2, 11.4

### Multiple Quantifiers: Something Makes Someone Want to Die Inside (741)

quantification semantics translation

Introduces translation from English to awFOL using multiple existential quantifiers with an example.

### There Is Exactly One (625)

translation

Mentions several ways of expressing the idea that there is exactly one creator in our formal language, awFOL.

Alternative textbook exercises (regular): 14.10--14.12, *14.13

Alternative textbook exercises (fast): 11.10, 11.13, 14.2

### ∃Intro (801)

quantification proof

Explains the rule of proof ∃Intro.

### ∃Elim (550)

proof

Introduces the rule of proof ∃Elim with an example. The example proof is from premise ∃x( Blue(x) ∧ Square(x) ) to conclusion ∃x Blue(x).

### Translation with Quantifiers (507)

translation

Discusses how to translate 'Some persuasive arguments are not valid' and 'All quadrumanous discordians weep and wail except Gillian Deleude' into first-order logic.

Alternative textbook exercises (regular): 9.4–-9.5, 9.8–-9.9

Alternative textbook exercises (fast): 9.12-–9.13

### Scope and Quantifiers (710)

semantics

Explores differences between ∃x(Square(x) ∧ Blue(x)), some squares are blue, and ∃x Square(x) ∧ ∃x Blue(x), something is square and something is blue.

### ∀Intro (570)

proof proof-example

Introduces the rule of proof ∀Intro with an example. The example proof is from premise ∀x(Square(x) → Blue(x)) to conclusion ∀x Square(x) → ∀x Blue(x).

Alternative textbook exercises (regular): 12.4--12.5, *12.6--12.7, 12.9--12.10

### Summary of Quantifier Rules (575)

proof

Quick recap of the four main quantifier rules.

Alternative textbook exercises (fast): 10.20, *10.24--10.7, 10.28--10.29, 13.2--13.3, 13.8--13.9, 13.11, 13.13, 13.15

### Two Things Are Broken (470)

quantification semantics

Explains how to translate sentences involving number into awFOL using quantifiers and identity.

## Lecture 8

Date given: Wednesday 13th May 2015

Exercises for this lecture:

### Subproofs Are Tricky: The Answer (227)

proof

Explains that in doing a proof, it is not allowed to cite lines from a closed subproof, thereby answering a question asked earlier.

### ∀Intro (570)

proof proof-example

Introduces the rule of proof ∀Intro with an example. The example proof is from premise ∀x(Square(x) → Blue(x)) to conclusion ∀x Square(x) → ∀x Blue(x).

Alternative textbook exercises (regular): 12.4--12.5, *12.6--12.7, 12.9--12.10

### There Is a Store for Everything (560)

quantification semantics translation

Explains how to translate English sentences involving a combination of universal and existential quantifiers into awFOL with an example.

Alternative textbook exercises (regular): 11.3, 11.4, 11.8, 11.9, 11.11, 11.13, *11.10

Alternative textbook exercises (fast): 11.8, 11.9, *11.11

### Variables (600)

translation syntax

Explains the role of variables in the formal language awFOL. Stresses that translations into English should not mention variables (e.g. 'there is an x such that ...' does *not* count as English for the purposes of this course).

### Loving and Being Loved (755)

quantifiers translation

Discusses translating English sentences about loving and being loved in order to illuminate the difference between ∃y∀x Loves(x,y) and ∃y∀x Loves(y,x).

### Somebody Is Not Dead (607)

translation quantification

Continues discussion of how to translate sentences involving quantifiers and negation from English into our formal language awFOL? Discusses translating 'Some person is not dead' and 'No person is dead'; also 'Every person is not dead' and 'Not every person is dead'.

### Quantifier Equivalences: ¬∀x Created(x) ⫤⊨ ∃x ¬Created(x) (764)

translation quantifiers semantics

Introduces the logical equivalence that allows us to turn ∀ into ∃, and conversely. Explains how to prove informally that the equivalence holds using the truth-conditions for quantified statements.

### The End Is Near (790)

quantification translation

Extremely brief introduction to definite descriptions. Hints at how sentences such as 'The winner is hungry' might be translated into awFOL.

Alternative textbook exercises (regular): 14.26, 14.28

Alternative textbook exercises (fast): 14.2, 14.4, 14.5, 14.10, 14.11, 14.26, 14.28