Today s Lecture 2/25/10. Truth Tables Continued Introduction to Proofs (the implicational rules of inference)

Transcription

1 Today s Lecture 2/25/10 Truth Tables Continued Introduction to Proofs (the implicational rules of inference)

2 Announcements Homework: -- Ex 7.3 pg. 320 Part B (2-20 Even). --Read chapter 8.1 pgs Ex 8.1 pgs Part A (1-10 All) --Ex 8.1 pgs Part B (1-20 All) -- if the inference is incorrect, just state that it is, you don t need to say why. Quiz on Tues (Mar 2nd): Be ready to state (write out) the 8 implicational rules. See pg. 361.

3 HW Ex 7.3 pg. 320 Part B (1-19 Odd) (See Tuesday s slides for #1 and #3)

4 # 5 A B A! B B " A T T T T T T F F F T F T T T F F F T F F invalid by row 3

5 # 7 A B A " (A # B) ~(A B) T T T F T F T T F T F T F F F F invalid by row 1

6 # 9 F G ~F # ~G " ~(F # G) T T F F T F T F F T T F F F T T invalid by rows 2 and 3

7 # 11 valid A B ~(A! B) " A ~B T T F F T F T T F T F F F F F F

8 # 13 valid A B ~(A $B) " (A ~B) # (B ~A) T T F F T F T T F T T T F F F F

9 # 15 P Q S P! Q S! Q ~Q " ~P ~S T T T T T F F T T F T T F F T F T F F T F T F F F T T F F T T T T F F F T F T T F T F F T T F T F F F F T T T T valid

10 # 17 L M N ~(L # M) ~M $ ~N " ~N T T T F T F T T F F F T T F T F F F T F F F T T F T T F T F F T F F F T F F T T F F F F F T T T valid

11 # 19 A B C A! (B! C) " (A B)! C T T T T T T T F F F T F T T T T F F T T F T T T T F T F T T F F T T T F F F T T valid

12 Proofs --We are now going to turn to a new method for determining the validity of symbolic arguments: proofs (or derivations). --A proof is a way of showing that on the assumption that the premises are true, the conclusion must be true. More specifically, --A proof is a series of steps proceeding from the premises of a symbolic argument to its conclusion via principles of correct reasoning: inference rules.

13 Inference Rules --Inference rules are rules of permission: they give us permission to proceed from one step of the proof to another. --We can think of inference rules as little valid arguments that we can employ in the course of a derivation. As such, if the premises of a rule are true, the conclusion of the rule must also be true.

14 Inference Rules For any proof of some symbolic argument A, we can view each line in the proof that is derived from an inference rule-- including the last line which contains the conclusion of A-- as being the conclusion of a valid argument, the premises of which are preceding lines. Each line then is guaranteed to be true given the relevant lines from which it follows. If the proof is done correctly, the conclusion of A has been shown that it must be true given the previous lines in the proof, which contain the premises of A. Thus if the proof is done correctly, our goal of showing that the conclusion of A must be true assuming the truth of the premises of A has been accomplished.

15 Inference Rules Note that failure to construct a legitimate derivation does not show that an argument is invalid. It might be that although a derivation is possible, we re simply not able to think of it. Thus truth-tables are still required to test for validity or invalidity if no proof is forthcoming.

16 8 Particular Inference Rules These should look familiar!! Modus Ponens (MP)! Modus Tollens (MT)! Disjunctive Syllogism (DS)! Hypothetical Syllogism (HS) These are new! Simplification (Simp)! Conjunction (Conj)! Addition (Add)! Constructive Dilemma (CD)

17 Modus Ponens (MP) If you have a conditional on a line, and the antecedent of that conditional on another line, you may infer the consequent: p! q p q

18 Modus Tollens (MT) If you have a conditional on a line, and the negation of the consequent on another line, you may infer the negation of the antecedent. p! q ~q ~p

19 Hypothetical Syllogism (HS) This rule takes us from pairs of conditionals to a new conditional. p! q q! r p! r

20 Disjunctive Syllogism If you have a disjunction on a line, and the negation of one of the disjuncts on another line, you may infer the disjunct that is not negated. p v q p v q ~p ~q q p

21 Constructive Dilemma (CD) This rule takes us from a disjunction and a pair of distinct conditionals to another distinct disjunction. p v q p! r q! s r v s

22 Simplification (Simp) This rule could be called conjunction elimination. It allows us to break a conjunction down into its parts. It has two forms: p " q p p " q q The rule is valid because if a conjunction is true, it follows that both of its conjuncts are true.

23 Conjunction Conjunction could also be called conjunction introduction. It allows us to introduce a conjunction into a derivation. p q p " q

24 Addition (Add) This rule allows us to introduce new disjunctions into a derivation. It has two forms. p p v q p q v p This rule is valid because disjunctions are true if one of their disjuncts are true. So if a statement is true, any disjunction involving it as a disjunct must also be true.

25 Addition (Add) Continued Gizmo is a pretty cat (p). So, Gizmo is a pretty cat or it is raining outside (p v q). --The inferred disjunction is true in virtue of the premise it was inferred from. --It s impossible for the premise to be true while the conclusion is false.

26 Note Well Note that the lower-case letters used to state these rules are statement variables. So long as they are replaced uniformly, they can be replaced by symbolic atomic or compound statements of our formal language. So, for example, the following are instances of modus ponens: (P v Q)! ~R P! (Q! R) P v Q P ~R Q! R

27 Inference Rules And this is an instance of disjunctive syllogism: (P Q) v (R " S) ~ (R " S) P Q Note that this is not an instance of modus tollens: P! ~Q Q ~P ~~Q is the negation of ~Q, so we d need that to use MT on P! ~Q.

28 Constructing Proofs Each proof has three columns: --The leftmost column is used to number the lines of the proof. --The middle column contains the content of the proof this is where we write symbolic statements. --The rightmost column contains annotations. Each step of the proof must be justified by an application of one of the inference rules. In this column we note which rule is appealed to in that step.

29 Constructing Proofs --We start by listing the premises. These have no annotation (this indicates that they are the premises). In the third column of the line where the last premise appears, we write the conclusion. This serves as a reminder of what we are trying to prove. --We then start applying our inference rules. Our goal, again, is to prove (or show) that the conclusion must be true given our premises i.e. that it logically follows from our premises. (We want the last line of our proof to contain the conclusion).

30 Example Derivation Suppose we want to construct a proof for the following argument: 1. P! (Q! R) 2. P " Q.. R P! (Q! R), P " Q, " R Start by writing the premises and conclusion, then start applying inference rules.

31 Example Derivation Suppose we want to construct a proof for the following argument: P! (Q! R), P " Q, " R Start by writing the premises and conclusion, then start applying inference rules. 1. P! (Q! R) 2. P " Q.. R 3. P 2, Simp 4. Q! R 1, 3, MP 5. Q 2, Simp 6. R 4, 5, MP