1. Copyright © 2003-2014 Curt Hill Rules of Inference What is a valid argument?

2. 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? Copyright © 2003-2014 Curt Hill

3. 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 Copyright © 2003-2014 Curt Hill

4. 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 soon Copyright © 2003-2014 Curt Hill

5. 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 Copyright © 2003-2014 Curt Hill

6. Four Types of Statements Each denoted by a letter Universal affirmative –All S is P –A–A Universal negative –No S is P –E–E Particular affirmative –Some S is P –I–I Particular negative –Some S is not P –O–O

7. Copyright © 2003-2014 Curt Hill 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

8. Copyright © 2003-2014 Curt Hill 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

9. Copyright © 2003-2014 Curt Hill 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

10. 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 Copyright © 2003-2014 Curt Hill

11. Pictorially Copyright © 2003-2014 Curt Hill P P  Q _____  Q This is the same as: (P  (P  Q))  Q

12. 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 Copyright © 2003-2014 Curt Hill

13. Modus Ponens Truth Table Copyright © 2003-2014 Curt Hill PQ PQPQP  (P  Q)(P  (P  Q))  Q TTTTT TFFFT FTTFT FFTFT

14. 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 Copyright © 2003-2014 Curt Hill

15. 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 Copyright © 2003-2014 Curt Hill

16. 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 Copyright © 2003-2014 Curt Hill

17. 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: –rsl–rsl –st–st –t–t Copyright © 2003-2014 Curt Hill

18. Example (3 of 3) – Argument 1.s  t – hypothesis 2.  t – hypothesis 3.  s – modus tollens using 1,2 4.r  s  l – hypothesis 5.r  s – simplification of 4 6.  r – modus tollens using 3, 5 7.r – double negative using 6 Copyright © 2003-2014 Curt Hill

19. 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 Copyright © 2003-2014 Curt Hill

20. Resolution The resolution rule of inference has received considerable attention –((p  q)  (  p  r))  (q  r) It has been used as the single inference rule for automated systems –Prolog programing language –Theorem provers Copyright © 2003-2014 Curt Hill

21. 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 Copyright © 2003-2014 Curt Hill

22. Recall Copyright © 2006-2014 - Curt Hill pq p  q Implication q  p Converse ¬p  ¬q inverse ¬q  ¬p contrapositive TTTTTT TFFTTF FTTFFT FFTTTT

23. Copyright © 2003-2014 Curt Hill Converse Consider any number of implications involving a subset belonging to a superset All cats are mammals, which is c  m and true While the converse is not: all mammals are cats, m  c 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

24. Copyright © 2003-2014 Curt Hill The inverse Negate both sides of the implication –Thus the inverse of is p  q 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 c  m and true –If not a cat then not a mammal is false,  c   m

25. Copyright © 2003-2014 Curt Hill The contrapositive This is the inverted converse, we negate both items and reverse the antecedent and consequence –Thus if the implication is p  q 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

26. 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 Copyright © 2003-2014 Curt Hill

27. Example Given that: –  x(P(x)  Q(x)) –  x(  P(x)  Q(x)  R(x)) Show that –  x(  R(x)  P(x)) Copyright © 2003-2014 Curt Hill

28. Example continued 1.  x(  P(x)  Q(x)  R(x)) - Given 2.(  P(x)  Q(x)  R(x) – univ instant 3.  R(x)  (  P(x)  Q(x)) – contrapos 4.  R(x)  (  P(x)  Q(x)) – deMorgans 5.  R(x)  (P(x)  Q(x)) – double neg 6.  x(P(x)  Q(x)) – given 7.P(x)  Q(x) – universal instant 8.  R(x)  (P(x)  Q(x)  P(x)  Q(x)) – Add 5,7 9.  R(x)  (P(x)  P(x)) – Resolution 10.  R(x)  (P(x)) – Idempotency 11.  x(  R(x)  P(x)) – univ gen Copyright © 2003-2014 Curt Hill

29. Exercises 1.6 5, 9, 23, 27 Copyright © 2003-2014 Curt Hill