logic1

INTRODUCTION TO LOGIC

1 Sets, Relations, and Arguments

Volker Halbach

Pure logic is the ruin of the spirit.Antoine de Saint-Exupéry

Practicalities

�e Logic Manual

web page for the book: http://logicmanual.philosophy.ox.ac.uk/Exercises Bookletslides of the lecturesworked examplespast examination papers with solutions

Mark Sainsbury: Logical Forms: An Introduction to PhilosophicalLogic, Blackwell, second edition, 2001

Practicalities

�e Logic Manual

web page for the book: http://logicmanual.philosophy.ox.ac.uk/Exercises Booklet

slides of the lecturesworked examplespast examination papers with solutions

Practicalities

�e Logic Manual

web page for the book: http://logicmanual.philosophy.ox.ac.uk/Exercises Bookletslides of the lectures

worked examplespast examination papers with solutions

Practicalities

�e Logic Manual

web page for the book: http://logicmanual.philosophy.ox.ac.uk/Exercises Bookletslides of the lecturesworked examples

past examination papers with solutionsMark Sainsbury: Logical Forms: An Introduction to PhilosophicalLogic, Blackwell, second edition, 2001

Practicalities

�e Logic Manual

Practicalities

�e Logic Manual

Practicalities

Why logic?

You’ll need it to understand philosophical texts and to discussphilosophy.

Philosophy is about arguments and reasoning. Logicians re�ecton how we reason and what constitutes correct reasoning.

Logic provides one with tools to show that certain claims do notfollow from certain assumptions.

Tools from logic are used not only in philosophy but also inlinguistics, mathematics, computer science, and other subjects.

Practicalities

Why logic?

Practicalities

Why logic?

Practicalities

Why logic?

Practicalities

Why logic?

1.5 Arguments, Validity, and Contradiction

Arguments

De�nitionSentences that are true or false are called declarative sentences.In what follows I will focus exclusively on declarative sentences.

De�nitionAn argument consists of a set of declarative sentences (thepremisses) and a declarative sentence (the conclusion) marked asthe concluded sentence.

Arguments

De�nitionSentences that are true or false are called declarative sentences.In what follows I will focus exclusively on declarative sentences.

De�nitionAn argument consists of a set of declarative sentences (thepremisses) and a declarative sentence (the conclusion) marked asthe concluded sentence.

ExampleI’m not dreaming if I can see the computer in front of me. I cansee the computer in front of me. �erefore I’m not dreaming.

‘I’m not dreaming if I can see the computer in front of me’ is the�rst premiss.

‘I can see the computer in front of me’ is the second premiss.

‘I’m not dreaming’ is the conclusion, which is marked by‘therefore’.

Occasionally the conclusion precedes the premisses or is foundbetween premisses. �e conclusion needn’t be marked as such.

Alternative ways to express the argument:

ExampleI’m not dreaming. For if I can see the computer in front of me I’mnot dreaming, and I can see the computer in front of me.

ExampleI’m not dreaming, if I can see the computer in front of me. �us,I’m not dreaming. �is is because I can see the computer in frontof me.30

�e point of ‘good’ arguments is that they take one from truepremisses to true conclusions. Many arguments with thisproperty exhibit certain patterns.

ExampleFiona can open the dvi-�le if yap is installed. yap is installed.�erefore Fiona can open the dvi-�le.

Logicians are interested in the patterns of ‘good’ arguments thatcannot take one from true premisses to a false conclusion.

CharacterisationAn argument is logically (or formally) valid if and only if there isno interpretation under which the premisses are all true and theconclusion is false.

ExampleZeno is a tortoise. All tortoises are toothless. �erefore Zeno istoothless.

ExampleSocrates is a man. All men are mortal. �erefore Socrates ismortal.

Features of logically valid arguments:�e truth of the conclusion follows from the truth of thepremisses independently what the subject-speci�cexpressions mean. Whatever tortoises are, whoever Zeno is,whatever exists: if the premisses of the argument are true theconclusion will be true.

�e truth of the conclusion follows from the truth of thepremisses purely in virtue of the ‘form’ of the argument andindependently of any subject-speci�c assumptions.It’s not possible that the premisses of a logically validargument are true and that its conclusion is false.In a logically valid argument the conclusion can be false (inthat case at least one of its premisses is false).

See also Sainsbury for a discussion.

Features of logically valid arguments:�e truth of the conclusion follows from the truth of thepremisses independently what the subject-speci�cexpressions mean. Whatever tortoises are, whoever Zeno is,whatever exists: if the premisses of the argument are true theconclusion will be true.�e truth of the conclusion follows from the truth of thepremisses purely in virtue of the ‘form’ of the argument andindependently of any subject-speci�c assumptions.

It’s not possible that the premisses of a logically validargument are true and that its conclusion is false.In a logically valid argument the conclusion can be false (inthat case at least one of its premisses is false).

Features of logically valid arguments:�e truth of the conclusion follows from the truth of thepremisses independently what the subject-speci�cexpressions mean. Whatever tortoises are, whoever Zeno is,whatever exists: if the premisses of the argument are true theconclusion will be true.�e truth of the conclusion follows from the truth of thepremisses purely in virtue of the ‘form’ of the argument andindependently of any subject-speci�c assumptions.It’s not possible that the premisses of a logically validargument are true and that its conclusion is false.

In a logically valid argument the conclusion can be false (inthat case at least one of its premisses is false).

Features of logically valid arguments:�e truth of the conclusion follows from the truth of thepremisses independently what the subject-speci�cexpressions mean. Whatever tortoises are, whoever Zeno is,whatever exists: if the premisses of the argument are true theconclusion will be true.�e truth of the conclusion follows from the truth of thepremisses purely in virtue of the ‘form’ of the argument andindependently of any subject-speci�c assumptions.It’s not possible that the premisses of a logically validargument are true and that its conclusion is false.In a logically valid argument the conclusion can be false (inthat case at least one of its premisses is false).

�e following argument isn’t logically valid:

ExampleEvery eu citizen can enter the us without a visa. Max is a citizenof Sweden. �erefore Max can enter the us without a visa.

However, one can transform it into a logically valid argument byadding a premiss:

ExampleEvery eu citizen can enter the us without a visa. Max is a citizenof Sweden. Every citizen of Sweden is a eu citizen. �erefore Maxcan enter the us without a visa.

�e following argument isn’t logically valid:

ExampleEvery eu citizen can enter the us without a visa. Max is a citizenof Sweden. �erefore Max can enter the us without a visa.However, one can transform it into a logically valid argument byadding a premiss:

ExampleEvery eu citizen can enter the us without a visa. Max is a citizenof Sweden. Every citizen of Sweden is a eu citizen. �erefore Maxcan enter the us without a visa.

Characterisation (consistency)A set of sentences is consistent if and only if there is a least oneinterpretation under which all sentences of the set are true.

Characterisation (logical truth)

A sentence is logically true if and only if it is true under anyinterpretation.

‘All metaphysicians are metaphysicians.’

Characterisation (consistency)A set of sentences is consistent if and only if there is a least oneinterpretation under which all sentences of the set are true.

Characterisation (logical truth)

A sentence is logically true if and only if it is true under anyinterpretation.

‘All metaphysicians are metaphysicians.’

Characterisation (contradiction)A sentence is a contradiction if and only if it is false under anyinterpretation.

‘Some metaphysicians who are also ethicists aren’tmetaphysicians.’

I’ll make these notions precise for the formal languages orpropositional and predicate logic.20

Characterisation (contradiction)A sentence is a contradiction if and only if it is false under anyinterpretation.

‘Some metaphysicians who are also ethicists aren’tmetaphysicians.’

I’ll make these notions precise for the formal languages orpropositional and predicate logic.20

1.1 Sets

�e following is not really logic in the strict sense but we’ll need itlater and it is useful in other areas as well.

CharacterisationA set is a collection of objects.

�e objects in the set are the elements of the set.

�ere is a set that has exactly all books as elements.�ere is a set that has Volker Halbach as its only element.

1.1 Sets

�ere is a set that has exactly all books as elements.

�ere is a set that has Volker Halbach as its only element.

1.1 Sets

Sets are identical if and only if they have the same elements.

Example�e set of all animals with kidneys and the set of all animals witha heart are identical, because exactly those animals that havekidneys also have a heart and vice versa.

1.1 Sets

�e claim ‘a is an element of S’ can be written as ‘a ∈ S’. One alsosays ‘S contains a’ or ‘a is in S’.

�ere is exactly one set with no elements. �e symbol for this setis ‘∅’.�e set {Oxford, ∅, Volker Halbach} has as its elements exactlythree things: Oxford, the empty set ∅, and me.

Here is another way to denote sets:

{ d ∶ d is an animal with a heart}

is the set of all animals with a heart. It has as its elements exactlyall animals with a heart.

1.1 Sets

�ere is exactly one set with no elements. �e symbol for this setis ‘∅’.

�e set {Oxford, ∅, Volker Halbach} has as its elements exactlythree things: Oxford, the empty set ∅, and me.

1.1 Sets

Example{Oxford, ∅, Volker Halbach} = {Volker Halbach, Oxford, ∅ }

Example{the capital of England, Munich} = {London, Munich, thecapital of England}

ExampleMars ∈ {d ∶ d is a planet }

Example∅ ∈ {∅}

1.1 Sets

1.2 Binary relations

Relations

�e set {London, Munich} is the same set as {Munich, London}.

�e ordered pair ⟨London, Munich⟩ is di�erent from the orderedpair ⟨Munich, London⟩.Ordered pairs are identical if and only if the agree in their �rstand second components, or more formally: