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.

definitions

Steps through the definition of a logically valid argument. Quickly runs through the terms sentence, name and predicate.

Exercises for this topic (regular)

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definitions

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

Reading: §2.1

Exercises for this topic (regular)

Exercises for this topic (fast)

Alternative textbook exercises (regular): 2.3, 2.4

definitions

Explains what sentence letters are and why we use them.

zoxiy

There are logic exercises associated with each lecture. This is how to complete them ...

zoxiy ex-create

Describes how to create possible situations in logic-ex.

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.

Reading: §2.5

Exercises for this topic (regular)

Exercises for this topic (fast)

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

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

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.

Reading: §2.2

Exercises for this topic (regular)

Exercises for this topic (fast)

Alternative textbook exercises (regular): 2.5, 2.6

definitions

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

Reading: §3.1, §3.2, §3.3

Exercises for this topic (regular)

Exercises for this topic (fast)

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

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

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.

Reading: §3.3, §3.5

Exercises for this topic (regular)

Exercises for this topic (fast)

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

Alternative textbook exercises (fast): 3.14, 3.15

truth-tables

Illustrates the use of truth tables to determine logical validity.

Reading: §4.3

Exercises for this topic (regular)

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Alternative textbook exercises (regular): 5.1-5.4

definitions zoxiy ex-tt

Illustrates and defines the terms ‘contradiction’, ‘logical truth’ and ‘logical possibility’. Describes how to recognise these from truth tables. Demonstrates using zoxiy to construct truth-tables and answer questions about contradictions, logical truths and logical possibilities.

Reading: §2.2

Exercises for this topic (regular)

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Alternative textbook exercises (regular): 2.5, 2.6

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.

Reading: §4.1, §4.2, §5.4

Exercises for this topic (regular)

Exercises for this topic (fast)

Alternative textbook exercises (regular): 4.1, 4.2

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

truth-tables

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

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.

proof zoxiy ex-proof

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

Reading: §5.1, §6.1

Exercises for this topic (regular)

Exercises for this topic (fast)

Alternative textbook exercises (regular): 6.1

Alternative textbook exercises (fast): 5.3--5.6

proof zoxiy ex-proof ex-trans

When you use zoxiy you can type words instead of using symbols. But which words stand for which symbols? Here’s a list.

proof

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

Reading: §6.1

Exercises for this topic (regular)

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definitions

Describes how to write proofs.

proof

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

Reading: §2.2

Exercises for this topic (regular)

Exercises for this topic (fast)

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.

Reading: §3.6, §4.2

Exercises for this topic (regular)

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

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.

Reading: §8.1, §8.2

Exercises for this topic (regular)

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Alternative textbook exercises (fast): 8.20--8.25

proof proof-example

Discusses (but does not complete) a relatively hard proof involving the rule →Intro. The proof is from premise A∨B to conclusion ¬B→A.

Exercises for this topic (regular)

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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.

Reading: §8.1, §8.2

Exercises for this topic (regular)

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proof

States the rules of proof for ∨, ∨Intro and ∨Elim.

Reading: §6.2

proof proof-example

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

Reading: §5.2, §6.2

Exercises for this topic (regular)

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

proof

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

Reading: §6.3

Date given: Tuesday 21st January 2014

slides , handout [pdf], recording [warwick only]

Exercises for this lecture:

semantics

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

Reading: §7.1

Exercises for this topic (regular)

Exercises for this topic (fast)

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

definition syntax

Describes a mistaken application of the rule ¬Elim. Asks why this application is mistaken.

Exercises for this topic (regular)

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Alternative textbook exercises (regular): 6.8

proof proof-example

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

Reading: §5.3, §6.3

Exercises for this topic (regular)

Exercises for this topic (fast)

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

proof syntax

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

Reading: §3.5

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

truth-tables

Whiz through truth tables for (A ∧ B) ∨ C and A ∧ (B ∨ C).

truth-tables

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

Exercises for this topic (regular)

Exercises for this topic (fast)

Alternative textbook exercises (regular): 3.14, 3.15

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

syntax

What is the source of ambiguity in natural languages? Depicts two structures for Groucho Marx' utterance 'I shot an elephant in my pyjamas'.

syntax

States the rules of syntax for awFOL. Illustrates these with tree diagrams. Links to truth tables.

Reading: §9.3

Exercises for this topic (regular)

definition syntax

Defines the notion of scope using trees; explains its application in constructing proofs and truth-tables.

Reading: §3.5

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Alternative textbook exercises (regular): 3.14, 3.15

truth-tables semantics

Explains the notion of a truth-functional connective. Discussion of why 'because' could not have a truth table.

Reading: §7.0 (the text before §7.1)

Exercises for this topic (regular)

Exercises for this topic (fast)

Alternative textbook exercises (regular): 7.9

Alternative textbook exercises (fast): 7.9

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.

Exercises for this topic (regular)

Exercises for this topic (fast)

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.

Reading: §9.1

quantification

Quantifiers are for talking about things without naming them.

Reading: §9.1, §9.2

Exercises for this topic (regular)

Exercises for this topic (fast)

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

Alternative textbook exercises (fast): 9.8–-9.10

Date given: Tuesday 2nd February 2016

slides , handout [pdf], recording [warwick only]

Exercises for this lecture:

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.

Reading: §9.1

quantification

Quantifiers are for talking about things without naming them.

Reading: §9.1, §9.2

Exercises for this topic (regular)

Exercises for this topic (fast)

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

Alternative textbook exercises (fast): 9.8–-9.10

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).

Reading: §12.2, §13.2

Exercises for this topic (regular)

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proof zoxiy ex-proof

Explains how to do proofs involving quantifier rules with zoxiy. (Where you need to put the name ‘a’ in a box, write ‘[a]’.)

quantification

Looks at translating 'All squares and blue' and 'Some squares are blue' into awFOL using quantifiers.

Reading: §9.2, §9.3, §9.5

Exercises for this topic (regular)

Exercises for this topic (fast)

Alternative textbook exercises (fast): 9.16.10--9.16.15, 9.17.7--9.17.15

semantics

Explains the meaning of ∀ with an example.

Reading: §9.4

Exercises for this topic (regular)

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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).

Reading: §12.1, §12.3, §13.1

Exercises for this topic (regular)

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

proof

Quick recap of the four main quantifier rules.

Reading: §13.1, §13.2

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

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.

Reading: §9.5, §9.6

Exercises for this topic (regular)

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

Alternative textbook exercises (fast): 9.12-–9.13

semantics

Explains the meaning of ∃ with a simple example.

Reading: §9.4

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semantics

Illustrates how quantifiers bind variables (but does not define this). Contrast ∀x(Square(x) → Blue(x)), all squares are blue, with ∀x Square(x) → ∀x Blue(x), if everything is square then everything is blue.

Reading: §9.3

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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.

Reading: §9.5, §9.6

Date given: Tuesday 9th February 2016

slides , handout [pdf], recording [warwick only]

Exercises for this lecture:

quantification

Practices translating an English argument involving quantifers into awFOL.

Reading: §9.2, §9.3, §9.5

Exercises for this topic (regular)

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quantification

Illustrates giving a counterexample to an argument involving quantifiers.

Exercises for this topic (regular)

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.

Reading: §11.1

Exercises for this topic (regular)

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Alternative textbook exercises (fast): 11.2, 11.4

quantification semantics translation

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

Reading: §11.1

Exercises for this topic (regular)

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Alternative textbook exercises (regular): 11.2

quantification relations counterexamples

Explains what it is for a relation to be transitive. Describes how to show, using counterexamples, that the relation NotAdjacent (which holds between two objects just when the first is not adjacent to the second) is not transitive. Also describes how to express the counterexample formally.

Reading: §15.1, §15.6

Exercises for this topic (regular)

Exercises for this topic (fast)

quantification relations

Describes how quantifiers can be used to express the claim that a particular relation is reflexive, symmetric or transitive. Also introduces dot-arrow diagrams to describe relations.

Reading: §15.1, §15.6

Exercises for this topic (regular)

Exercises for this topic (fast)

Alternative textbook exercises (regular): 15.33--15.40 (second edition)

Alternative textbook exercises (fast): 15.33, 15.37--15.39 (second edition)

counterexamples relations

Describes how quantifiers can be used to express the claim that a particular relation is reflexive, symmetric or transitive.

Reading: §15.1, §15.6

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quantification semantics translation

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

Reading: §11.2, §11.3

Exercises for this topic (regular)

Exercises for this topic (fast)

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

translation quantification

How can we translate sentences involving quantifiers and negation from English into our formal language awFOL? Discusses translating 'Something is not dead' and 'Nothing is dead'; also 'Everything is not broken' and 'Not everything is broken'. New: now includes discussion of first-order counterexamples.

Exercises for this topic (regular)

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Alternative textbook exercises (regular): 9.12

Alternative textbook exercises (fast): 9.18--9.19

Date given: Tuesday 23rd February 2016

slides , handout [pdf], recording [warwick only]

Exercises for this lecture:

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.

Reading: §10.1, §10.3, §10.4

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proof revision proof-example

Describes how to prove that not everything is not dead given that something is dead.

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Alternative textbook exercises (fast): 13.43--13.45

proof revision proof-example

Describes how to prove that something is not dead from the premise that not everything is dead.

Alternative textbook exercises (fast): 13.49--13.50

translation quantifiers

Describes how we can use knowledge of equivalences among propositional sentences to get quantifier equivalences with an example. The example is the inference from this: A → B ⫤⊨ ¬B → ¬A, to this: ∀x(Square(x) → Broken(x)) ⫤⊨ ∀x(¬Broken(x) → ¬Square(x)).

Reading: §10.3

Exercises for this topic (regular)

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Alternative textbook exercises (regular): 10.20, 10.22

quantification semantics

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

Reading: §14.1

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translation

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

Exercises for this topic (regular)

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

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

translation

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

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Exercises for this topic (fast)

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).

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quantification semantics translation

Continues the discussion of how to translate English sentences involving a combination of universal and existential quantifiers into awFOL with an example, 'There is a store for everything.' Also discusses structural ambiguities related to quantifier scope and how to use awFOL to pin down this sort of ambiguity.

Reading: §11.4, §11.5

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proof

Discussion of an incorrect proof which appears to apply ∀Intro to prove this invalid argument: (∀x Square(x)) → (∀x Blue(x)) therefore ∀x(Square(x) → Blue(x)).

Reading: §13.1, §13.2

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meta

Statement of the substitution theorem. Suppose that φ, ψ and χ are sentences of awFOL. Suppose that φ is logically equivalent to ψ. Let χ[φ/ψ] be the result of replacing, in χ, zero or more occurrences of φ with ψ. Then the subsitution theorem says that χ[φ/ψ] is logically equivalent to χ.

Reading: §4.5, §10.3

Exercises for this topic (regular)

Exercises for this topic (fast)

Alternative textbook exercises (regular): 7.25

Alternative textbook exercises (fast): 4.31, 7.25

meta

Explains what it is for a system of proof, such as Fitch, to be sound; and what it is for a system of proof to be complete.

Reading: §8.3, §13.4

Date given: Tuesday 1st March 2016

slides , handout [pdf], recording [warwick only]

Exercises for this lecture:

quantifiers translation

Discusses translating English sentences involving obscured quantifiers into awFOL with an example.

Reading: §11.2

Exercises for this topic (regular)

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meta truth-tables

Explains what it is for a set of connectives to be truth-functionally complete. Proves that {¬, ∧, ∨} is truth-functionally complete. ‘A set of truth-functors is said to be expressively adequate (or sometimes functionally complete) iff, for every truth function whatever, there is a formula containing only those truthcfunctors which express that truthcfunction, i.e. which has as its truthctable the truthctable specifying that function.’ (Bostock, Intermediate Logic p. 45).

Reading: §7.4

Exercises for this topic (regular)

Exercises for this topic (fast)

Alternative textbook exercises (regular): 7.25, 7.26, *7.28, 7.29

Alternative textbook exercises (fast): 7.25, 7.26, *7.28, 7.29

meta

Explains how to prove proofs about proofs. Examples include: (i) if A ⊢ B then ⊢ A → B, (ii) if A ⊢ B then A ⊢ ¬¬B

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meta

Identifies a property of the rules of proof in Fitch: no rule allows us to go from truths to a falsehood. Illustrates by contrasting ∧Elim with ∧Fubar.

Reading: §8.3

Alternative textbook exercises (regular): 7.32

Alternative textbook exercises (fast): 7.32

meta

Sketches a proof of the soundness theorem for the propositional part of our formal system of proof, Fitch.

Reading: §8.3

Exercises for this topic (regular)

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meta

Outlines how the completeness theorem for the propositional part of our formal proof system, Fitch, will work.

Reading: §8.3

meta

Proves the following claim, where Γ is a set of sentences of awFOL: if for every sentence letter P, either Γ⊢P or Γ⊢¬P, then for every sentence of awFOL φ, either Γ⊢φ or Γ⊢¬φ.

Reading: §8.3

Exercises for this topic (regular)

Exercises for this topic (fast)

meta

Sketches a proof of the completeness theorem.

Reading: §8.3, §17.1, §17.2

quantification relations

Describes how quantifiers can be used to express the claim that a particular relation is reflexive, symmetric or transitive. Also introduces dot-arrow diagrams to describe relations.

Reading: §15.1, §15.6

Alternative textbook exercises (regular): 15.33--15.40 (second edition)

Alternative textbook exercises (fast): 15.33, 15.37--15.39 (second edition)

quantification translation

Identifies a problem translating some sentences involving comparisons into our formal language, awFOL.

Reading: §14.1, §14.3

quantification translation

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

Reading: §14.3

Exercises for this topic (regular)

Exercises for this topic (fast)

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