Updated 09 November 2000

LOGIC

- study of valid arguments

VALID ARGUMENT

- if the premises are true, the conclusion must be true

Socrates is a person.

All persons are mortal.

\Socrates is mortal.

~~~~~~~~~~~~~~~~~

Hague is a politican.

All politicans are amusing.

\Hague is amusing.

All students are wise.

Icabod is a student.

\Icabod is wise.

~~~~~~~~~~~~~~~~~~~~~~

Icabod is rich.

Icabod is a student.

\All students are rich.

The weather is cold and the weather is wet.

\The weather is wet.

~~~~~~~~~~~~~~~~

! The weather is cold or

! the weather is wet.

! \The weather is wet.

All zemindars are rich.

Icabod is a zemindar.

\Icabod is rich.

~~~~~~~~~~~~~~~~~

All quarks have charm.

This is a quark.

\ This has charm.

Icabod has a marcel.

\Icabod has a marcel or Icabod has a dog.

~~~~~~~~~~~~~~~~~~~~~

Isabel has an ai.

\ Isabel has an ai or Isabel has a Porsche.

logic is the study of valid arguments whose validity arises from form not content.

You can do logic without the aid of a dictionary.

Icabod is a bachelor.

\Icabod is unmarried.

~~~~~~~~~~~~~~~~~

Henry is red all over.

\Henry is not green all over.

If Icabod fails Prelims, Icabod will be sent down.

Icabod fails prelims.

\Icabod will be sent down.

~~~~~~~~~~~~~~~~~

If Isabel gets a distinction, she will not have to do physics.

Isabel gets a distinction.

\Isabel will not have to do physics.

F : Icabod fails Prelims.

D: I will be sent down.

If F then D

~~~~~~~~~~~~~~~~~

G : Isabel gets a distinction.

H: Isabel will not have to do physics

If G then H

10.

P : The weather is cold.

Q : The weather is wet.

P and Q

\ P

! P or Q

! \P

11.

Premises ? Conclusion ?

use sentences

not sentences

statements [propositions]

“What is meant, said, conveyed by a sentence.

12.

Il pleut

It is raining

~~~~~~~~~~~~~~~~~~~

Caesar stabbed Brutus.

Brutus was stabbed by Caesar.

~~~~~~~~~~~~~~~

It is now raining.

He has an enlightened economic policy.

13.

WHAT IS TRUTH ???

Property of a statement

Any statement either has or lacks it.

True or false.

Truth-value

NOT EPISTEMOLOGY

Billy Hague wears short trousers to Shadow cabinet meetings.

14.

Logic : study of arguments

Validity

Defined conditionally

If the premises are true, then the conclusion must be true.

15.

False premises and false conclusion.

Oxford students are rich.

Cambridge students are poor.

\Oxford students are rich and Cambridge students are poor.

16.

Icabod is a smart Balliol s.

Isabel is a smart Balliol s.

\All Balliol students are smart.

NOT VALID !!!

possible circumstance: George W. Bush enrols to do PPE at Balliol.

17.

VALID

If premises are true, the conclusions must be true.

Any possible circumstances in which the premises are true, the conclusion is true.

18.

WHY ?

1. It excites philosophers.

Validity is truth-preserving.

Descartes

I think

...

\ Goc exists.

Proceed logically from true premises, never err.

2. Pass Prelims.

3. Be the Demon Reasoner of the JCR.

- spot fallacies

- force SCR to a conclusion

Fantasy ???

19.

4. Representing explicitly procedures followed implicitly.

linquistics

sentences vs non-sentences

system of rules

no improvement ?

nice (certainly harmless)

practical ?

machine translation

theorem provers

20.

LOGIC

- system of rules

- represent explicitly

- mathematical model

- computer program

5. Please your parents

Getting a distinction.

21.

6. “mistakes will occur”

Russell walks in Bagley Wood and everything changes.

logicans are pedants.

check everything rigourously and explicitly.

7. A tool in philosophy

8. Ask your tutor.

22.

More to life than validity

Lots of arguments are not even trying.

1. smoking story

2. trees and acid rain

Premises support but do not entail

Inductive not deductive

23.

Inductive

good or bad

tonic water story

mice and hash story

risky, messy business which we will ignore.

24.

Hodges Logic :Logic is about consistency of beliefs.

Belief--------mental state

--------content

- mental states have dates.

- different states for difference folk.

What is believed - the content - that a proposition or statement is true.

Consistency of set of statements

possible circumstance in which all are true

consistency ¹ truth

“Balliol” student

Inconsistent set - not all true.

25.

A consistent set :

Hague wears a toupe.

The VC is a rock star.

CC students are poor.

Inconsistent set:

Icabod is a tutor.

Icabod is not a tutor.

Inconsistent set:

Fred is a person

All persons are mortal

Fred is immortal

26.

Form

{Icabod is a Balliol student, All Balliol students are arrogant, Icabod is not arrogant}

Content

{Henry is red all over, Henry is green all over}

27.

Consistency is nice.

Mark Thatcher is honest.

Mark Thatcher is not honest.

Wednesday:

We will not raise taxes

Friday:

We will raise taxes

fickle - logic is no help

28.

Take care !

It is and it isn’t.

It is raining and it is not raining.

Drizzle

God is one and God is three.

Enforce ambiguity.

re-interpret

Grass is good to smoke.

29.

Validity

form not content

definition (if premises are true, conclusion must be true)

test

initially very limited

30.

NS went to Buda or NS stayed in Oxford.

NS did not stay in Oxford.

\ NS went to Buda.

B or O

not-O

no way

B or O |

not-O | all true

not-B |

31.

set {B or O, not-O, not-B} is inconsistent.

counter-example set

formed from premises plus negation of conclusion.

32.

CES

Set formed by taking the premises of the argument and the negation of the conclusion.

Argument is valid just in case its CES is inconsistent

33.

Argument

B or O, not-O \ B

valid ???

Counter example set

{B or O, not-O, not-B}

inconsistent ???

34.

CES is inconsistent

{B or O, not-O, not-B}

No way can all be true.

B or O true

not-O true

not-B NOT TRUE -

FALSE

So B is true

And the argument is valid.

35.

ARGUEMENT

B or O, not-O \ B

valid

CES

{B or O, not-O, not-B}

B or O true

not-O true

B true

So CES is inconsistent.

36.

An argument is valid if and only if its counter-example set is inconsistent.

An argument is valid exactly when its CES is inconsistent.

37.

P and not P

\ P

\ P or Q

\ ØP

38.