Updated 09 November 2000

LOGIC

 

 - study of valid arguments

 

VALID ARGUMENT

 

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

1.

 


Socrates is a person.

 

All persons are mortal.

 

\Socrates is mortal.

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

Hague is a politican.

 

All politicans are amusing.

 

\Hague is amusing. 

2.

 

 

All students are wise.

 

Icabod is a student.

 

\Icabod is wise.

 

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

Icabod is rich.

 

Icabod is a student.

 

\All students are rich.

3.

 

 

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.

4.

 


All zemindars are rich.

 

Icabod is a zemindar.

 

\Icabod is rich.

 

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

All quarks have charm.

 

This is a quark.

 

\ This has charm.    

5.

 

 

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.

6.

 

 

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

 

You can do logic without the aid of a dictionary.                                         

7.

 

 

Icabod is a bachelor.

 

\Icabod is unmarried.

 

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

Henry is red all over.

 

\Henry is not green all over.

8.

 

 

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.

9.

 


F : Icabod fails Prelims.

D:  I will be sent down.

 

If F then D

F

\D

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

G : Isabel gets a distinction.

H: Isabel will not have to do physics

 

G

If G then H

\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

            \B

 

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

 

                                    \Q

38.