Deductive Validity, Lecture notes of Reasoning

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Critical Reasoning:

A Romp Through the Foothills of

Logic

Lecture Four: Deductive Validity

Marianne Talbot Department for Continuing Education University of Oxford Michaelmas Term 2012

Last week we learned: ! that critical reasoning is normative not descriptive ! that there are two types of ‘following from’ ! that deductive arguments are: ! truth preserving (when good) ! such that their being good is an either/or matter ! such that we can determine a priori whether they are good or not ! that inductive arguments are: ! not truth preserving ! such that their being good is a matter of degree ! such that we can determine whether they are good or not only a posteriori

So we now know what arguments are… …and we know how to analyse them and set them out logic-book-style… …and we know the key characteristics of… …both deductive and inductive arguments

This week we shall learn how to evaluate deductive arguments

A deductive argument is valid… …and its premises entail its conclusion… …when and only when there is no logically possible situation… … in which its premises are true… …and its conclusion false

And a deductive argument is INVALID… …whenever its premises fail to entail its conclusion… …whenever there is a logically possible situation… … in which its premises are true… …and its conclusion false

Argument forms that are always valid

include:

! Modus Ponens: If P , then Q****. P****. Therefore, Q****. ! Modus Tollens: If P , then Q****. Not-Q****. Therefore, Not-P****. ! Generalisation: P, therefore P or Q (or Q, therefore P or Q) ! Specialisation: P. Q. Therefore P and Q ! etc., etc., etc…

Argument forms that are never valid include:

! Denying the Antecedent: If P then Q. Not P. Therefore, not Q. ! Affirming the Consequent: If P, then Q. Q. Therefore, P. ! etc., etc., etc…

They can also be evaluated by means of truth

tables (or tableaux):

P Q If P then Q P l- Q T T T T T T F F T _ F F T T F _ T F F T F _ F

But both these methods involve… …learning how to formalise arguments… …to eliminate the English and replace it with symbols…

…and we are doing informal not

formal logic

In informal logic the best way to evaluate a deductive argument… … is to set it out logic book style… …construct the counterexample set… … and ask whether the sentences of the counterexample set… … are consistent

On the LHS is an argument and on

the RHS is its counterexample set

Deepak is a banker All bankers are rich Therefore Deepak is rich Deepak is a banker All bankers are rich It is not the case that Deepak is rich

On the LHS is another argument and

on the RHS is its counterexample set

All bankers are rich Deepak is rich Therefore Deepak is a banker All bankers are rich Deepak is rich It is not the case that Deepak is a banker

For a logician it is a sufficient condition

of a deductive argument’s being a

good argument that it is valid

In every day life we want more of a

deductive argument before we will

count it as a good argument

Are these arguments valid?

Grass is green Therefore 2+2= 2+2= Therefore grass is green

Are these arguments ‘good’?