copyright © 2003-2014 curt hill rules of inference what is a valid argument?

Copyright © 2003-2014 Curt Hill

Rules of Inference

What is a valid argument?

Introduction

• I used to have a friend who did not like mushrooms

• Therefore he argued:– Everyone who has cancer ate

mushrooms– Therefore mushrooms cause cancer

• Is this a valid argument?– Why or why not?


Valid Arguments• An argument is a sequence of

statements• It ends in a conclusion• Each of the statements should be

– Given as true or obviously true in their own right

– Follow from preceding statements

• A valid argument is trustworthy• A fallacy is not

– Usually violates the rules


Argument

• An argument contains premises and a conclusion

• The last statement is the conclusion

• All the previous statements are premises

• Showing that the form is valid is important– This will be the basis of a proof, which

we shall consider soonCopyright © 2003-2014 Curt Hill

Rhetoric

• Much of what we have in an argument is originally derived from what is and is not valid in a lawyer’s argument to a jury

• Aristotle noticed that a slick lawyer could argue that right was wrong and any number of other fallacies

• He publishes which arguments are valid and which are not



Four Types of Statements• Each denoted by a letter• Universal affirmative

– All S is P– A

• Universal negative– No S is P– E

• Particular affirmative– Some S is P– I

• Particular negative– Some S is not P– O


Four types (continued)

• In each of these statements:– S which is the subject– P is the predicate

• All or no have the obvious meanings

• Some means one or more


Syllogism

• Aristotle's main form was a syllogism

• Each syllogism consisted of two premises (a major and minor) and one conclusion

• The premises and conclusion are of one of previous four statement types


Syllogism

• Example– All cats eat mice– Felix is a cat– Therefore Felix eats mice

• Statement types– First is universal affirmative– Second is a particular affirmative– Third is a particular affirmative

Rules of Inference• A valid argument form has been

proven to be trustworthy– Often a syllogism

• One of the rules of inference is modus ponens– Also known as the law of detachment

• The form is that we have an implication and assert the antecedent

• This guarantees the consequent• A table of inference rules is in Rosen


Pictorially


P PQ _____QThis is the same as:(P(PQ)) Q

Modus Ponens

• This is just one of many syllogisms that is a tautology

• A tautology is an expression that is always true– The truth table may only have Trues

for the column of the expression

• Consider Modus Ponens again


Modus Ponens Truth Table


P Q PQ P(PQ) (P(PQ))Q

T T T T T

T F F F T

F T T F T

F F T F T

Truth Tables• You may prove things with truth

tables• However there are problems with such

an approach• Many variables make the truth table

large• The calculation of each cell is

somewhat error prone– Real mathematicians disdain such an

approach

• Instead we use arguments


Rules of Inference

• An inference rule allows us to assert the conclusion

• Another way to consider the rules of inference are as rewrite rules

• That is if we have the two propositions of an inference rule then we can rewrite these two as a new proposition

• We now build arguments using this process


Example (1 of 3) – Definitions

• Consider the following hypotheses– If it does not rain then the sailboat

race will occur and a lifesaving demonstration will occur

– If the sailboat race occurs then a trophy will be awarded

– The trophy is not awarded

• We want to show it rained


Example (2 of 3) – Variables

• Let rain be r• Let sailboat race be s• Let lifesaving demonstration be l• Let trophy be t• Our given hypotheses are:

rsl– st t


Example (3 of 3) – Argument

1. st – hypothesis2. t – hypothesis 3. s – modus tollens using 1,24. rsl – hypothesis 5. rs – simplification of 4 6. r – modus tollens using 3, 57. r – double negative using 6


Example commentary

• Each proposition was either:– A hypothesis – A new proposition that was derived

from previous propositions and inference rules

• We move in a step by step manner from the hypotheses to a conclusion

• Multiple conclusions are possible– For example we may conclude also

that the life saving demonstration was not held


Resolution

• The resolution rule of inference has received considerable attention – ((pq) (pr))(qr)

• It has been used as the single inference rule for automated systems– Prolog programing language– Theorem provers


Fallacy

• Just as there are many valid rules of inference there are many invalid rules of inference

• These invalid rules are typically not tautologies– They have one or more false values in

the relevant column of the truth table

• Lets consider the variations of the implication



Converse• Consider any number of implications

involving a subset belonging to a superset

• All cats are mammals, which is cm and true

• While the converse is not: all mammals are cats, mc

• This is not to say that p q and q p may not have the same truth value– They may or may not depending on the p

and q that is chosen– When p and q are equivalent they do have

same truth value


The inverse

• Negate both sides of the implication– Thus the inverse of is pq is p q

• The inverse also does not have the same truth value as the implication

• Consider again subset – All cats are mammals, which is cm

and true– If not a cat then not a mammal is

false, c m


The contrapositive

• This is the inverted converse, we negate both items and reverse the antecedent and consequence– Thus if the implication is pq then

the contrapositive is q p

• This one is a winner in that it has the same logical value as the implication

• Consider again subset – If not a mammal then not a cat is

true, m c

Quantified Statements• Just as there are inference rules for

propositions, there are also inference rules involving quantification and propositions

• These four are shown as Table 2 in Rosen– Universal instantiation and generalization– Existential instantiation and

generalization

• Other manipulations of quantification also exist


Example

• Given that: x(P(x) Q(x)) x(P(x) Q(x) R(x))

• Show that x(R(x) P(x))


Example continued1. x(P(x) Q(x) R(x)) - Given2. (P(x) Q(x) R(x) – univ instant3. R(x) (P(x) Q(x)) – contrapos4. R(x) (P(x) Q(x)) – deMorgans5. R(x) (P(x) Q(x)) – double neg 6. x(P(x) Q(x)) – given7. P(x) Q(x) – universal instant8. R(x) (P(x) Q(x)P(x)Q(x)) – Add 5,79. R(x) (P(x)P(x)) – Resolution10. R(x) (P(x)) – Idempotency11. x(R(x) P(x)) – univ gen


Exercises

• 1.6• 5, 9, 23, 27


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