Introduction Valid Inference Steps Proof by Cases Introduction Valid Inference Steps Proof by Cases Announcements For 09.20 Methods of Proof for Boolean Logic 1 HW2 & 3 are due now! • Both involved only electronic submissions Inference Steps & Proof by Cases 2 The new version (11.5) of the software has been buggy • You can download version 2.7 from Blackboard William Starr • It is much more stable 3 HW1 grades will show up on Blackboard soon 09.20.11 William Starr j Phil 2310: Intro Logic j Cornell University 1/36 William Starr j Phil 2310: Intro Logic j Cornell University 2/36 Introduction Valid Inference Steps Proof by Cases Introduction Valid Inference Steps Proof by Cases Outline The Big Picture Where is Today? • You are taking a logic class • Logic is mainly about logical consequence 1 Introduction • It's about conclusions following (or not following) from premises 2 Valid Inference Steps • So far, we've explored two methods for understanding logical consequence: 3 Proof by Cases 1 Proof (Ch.2: informal, formal) 2 Tautological Consequence (Ch.4: truth tables) • However, we have not discussed how the methods of proof can be used for the Booleans • That will be our project for today William Starr j Phil 2310: Intro Logic j Cornell University 3/36 William Starr j Phil 2310: Intro Logic j Cornell University 6/36 Introduction Valid Inference Steps Proof by Cases Introduction Valid Inference Steps Proof by Cases The Big Picture Proofs But Wait... The Way Forward • We've studied the basics of proofs but we haven't done • We had to learn truth tables, why proofs too? any proofs involving the Boolean connectives • Truth tables are useful for the Booleans, but have • Today we'll be discussing the informal methods of significant limitations: proof for the Booleans 1 Impractical: Truth tables get extremely large. An interesting argument could have well over 14 atomic • Next class we'll extend our formal proof system (F) sentences, the table would be over 16,000 rows! with formal rules for the Boolean connectives 2 Limited Applications: as we learned Thursday, • These formal rules will closely mirror the informal there are logical consequences that aren't tautological proof methods discussed today consequences. Why? Truth tables are blind to the • Understanding the informal ones will help the formal logic of expressions other than the Booleans. ones make sense • The methods of proof fill this gap admirably • The informal ones will also be useful in everyday reasoning William Starr j Phil 2310: Intro Logic j Cornell University 7/36 William Starr j Phil 2310: Intro Logic j Cornell University 8/36 Introduction Valid Inference Steps Proof by Cases Introduction Valid Inference Steps Proof by Cases Proof Proof What is it? Steps, What? Proof A proof is a step-by-step demonstration which shows that a The Nature of Steps conclusion C must be true in any circumstance where some Each step of a proof appeals to certain facts about the premises, say P1 and P2 are true meaning of the vocabulary involved. These facts are what we implicitly appeal to when we say a step is obvious. 1 The step-by-step demonstration of C can proceed through intermediate conclusions • What kind of facts? 2 It may not be obvious how to show C from P1 and P2, • Facts which guarantee that the step will never lead us but it may be obvious how to show C from some other from something true to something false claim Q that is an obvious consequence of P1 and P2 • Let's consider an old example 3 Each step provides conclusive evidence for the next William Starr j Phil 2310: Intro Logic j Cornell University 10/36 William Starr j Phil 2310: Intro Logic j Cornell University 11/36 Introduction Valid Inference Steps Proof by Cases Introduction Valid Inference Steps Proof by Cases Proof Proof An Old Inference Step New Steps Indiscernibility of Identicals If n is m, then whatever is true of n is also true of m • So we need to think about which inference steps (where `n' and `m' are names) negation, conjunction and disjunction support • That is, we need to think about what they mean • We've already started doing this: • This is a fact about the meaning of is 1 ^ takes the `worst' truth value • This fact guarantees that if it is true that n is m, then 2 _ takes the `best' truth value it is true that whatever holds of n also holds of m 3 : flips the truth value • In other words it could never lead us from true claims • Now we just need to think about what these facts to false ones imply in terms of inference steps and proof methods • This is the essence of a valid inference step William Starr j Phil 2310: Intro Logic j Cornell University 12/36 William Starr j Phil 2310: Intro Logic j Cornell University 13/36 Introduction Valid Inference Steps Proof by Cases Introduction Valid Inference Steps Proof by Cases Conjunction Conjunction Elimination Elimination • Suppose you have proved a conjunction P ^ Q • From looking at the truth table for ^ or thinking • Conjunction elimination is pretty obvious: about the meaning of and, both P and Q are clearly 1 Jay walks and Kay talks consequences of P ^ Q • So Jay walks 2 Large(a) ^ Cube(a) • This inference pattern is called conjunction elimination • So Cube(a) Truth Table for ^ Conjunction Elimination • This inference is so obvious that we rarely take the time to mention that we are making it P Q P ^ Q 1 From P ^ Q you can • true true true infer P In your informal proofs you don't have to mention it true false false either, but you should be aware that you are making it false true false 2 From P ^ Q you can and why it's a valid inference step infer Q false false false William Starr j Phil 2310: Intro Logic j Cornell University 16/36 William Starr j Phil 2310: Intro Logic j Cornell University 17/36 Introduction Valid Inference Steps Proof by Cases Introduction Valid Inference Steps Proof by Cases Conjunction Disjunction Introduction Introduction • There's a similar inference step for inferring a • Suppose you have proven P conjunction from its conjuncts: • Then you can infer P _ Q, no matter what Q is • Why? Conjunction Introduction Truth Table for _ • If P is true, then P _ Q If you have proven both P and Q, you can P Q P _ Q is true, regardless of infer P ^ Q Q's truth value! true true true true false true • This is a valid • Again, so obvious it's never mentioned false true true inference step, since it • You don't have mention it in your proofs, but you false false false is guaranteed to lead should understand it us from true claims to true claims William Starr j Phil 2310: Intro Logic j Cornell University 18/36 William Starr j Phil 2310: Intro Logic j Cornell University 19/36 Introduction Valid Inference Steps Proof by Cases Introduction Valid Inference Steps Proof by Cases Disjunction Informal Proof Introduction A Rule of Thumb So _ supports this inference step: Rule of Thumb for Informal Proofs In an informal proof, it is always legitimate to move from P Disjunction Introduction to Q if both you & your audience already know that Q is a logical consequence of P 1 From A you can infer A _ B 2 From A you can infer B _ A • We've all learned the following equivalences: DeMorgan's Laws Double Negation • It may seem useless, to infer from: ::P , P 1 :(P ^ Q) ,:P _:Q (1) Likes(jay; circles) 2 :(P _ Q) ,:P ^ :Q that (2) Likes(jay; circles) _ Likes(kay; squares) • So in informal proofs for this class you could say: • But, we will find uses for these kinds of inferences \From :(Cube(a) ^ Tet(b)) it follows by DeMorgan's Laws that :Cube(a) _:Tet(b)..." William Starr j Phil 2310: Intro Logic j Cornell University 20/36 William Starr j Phil 2310: Intro Logic j Cornell University 22/36 Introduction Valid Inference Steps Proof by Cases Introduction Valid Inference Steps Proof by Cases Informal Proof An Example Proof A Rule of Thumb Argument 1 Argument 1 • Let's give an informal proof • Of course, if you are asked to prove one of DeMorgan's of this inference Laws, or you are talking logic with stranger you :(A _ B) shouldn't appeal to DeMorgan's Laws :A • So we have three new inference steps, some equivalences and a rule of thumb Proof of Argument 1 • So what? We are given :(A _ B), which is equivalent to :A ^ :B by • Well, now we can prove some stuff DeMorgan's Law (2). So :A follows immediately. • We used Conjunction Elimination in this last step, but there was no need to explicitly say so William Starr j Phil 2310: Intro Logic j Cornell University 23/36 William Starr j Phil 2310: Intro Logic j Cornell University 25/36 Introduction Valid Inference Steps Proof by Cases Introduction Valid Inference Steps Proof by Cases Another Example Proof Moving On Inference 2 Proof Methods Argument 2 • Let's give an informal proof of this inference too • We have discussed: a = b ^ :Cube(a) 1 Conjunction Intro and Elim :(Cube(a) _ Cube(b)) 2 Disjunction Intro • But what about: Proof of Argument 2 1 Disjunction Elim 2 Negation! We are given that a = b and :Cube(a), so by the indiscernibility of identicals :Cube(b).