F22H1 Logic and Proof Week 6

1. F22H1 Logic and Proof Week 6 Reasoning

2. How can we show that this is a tautology (section 11.2): The hard way: “logical calculation” The “easy” way: “reasoning” That’s what chapter 11 is all about: Logical Calculation: use well defined rules one by one and you will slowly get towards the thing that you are trying to prove. Reasoning: (logical calculation, but with a few hints, tips and additional rules) is more like common sense.

3. Of course, we can do a truth table to show that this is always T – however you can’t do that with, e.g., 100 logical variables. So we need to learn reasoning. We start with this. Why?? Because Lemma 7.3.4. Says this: So, if our calculation eventually gets to “P => R”, using steps that are either or then we will have proved what we want to prove

4. This is the next step: Understand? Next: Tricky! But straightforward

5. Next this: Understand it? Also known as False-elimination. Next, we can go a long way with rules from chap 7: Actually this only uses AND-weakening

6. The remainder is fairly quick and easy: So from Lemma 7.3.4. We have now proved the tautology we wanted to prove.

7. Koyaanasqatsi

8. A brand new dawn … Let’s just assume, for the time being, that the whole first bit is true {Assumption} (1) Now let’s make another assumption: • {Assumption} • P • {from (1) and (2)} • Q • {from (1) and (3)} • R • {we assumed P (2) and derived R (4), this means:} • 5) P => R {We assumed (1), and got (5), this means:} 6)

9. Inference This is the general rule for inference: You can be creative about when to make an inference and what to base it on, but of course it has to be valid.

10. Some valid inference rules • {Tautology} • {Implication and double negation} • (2) • {Follows from (1) and (2) – called } • (3) • {Assume} • … • {follows from (1) -- • (m) • {follows from (1) and (m) if there are no assumptions between (1) • and (m) that the derivation of (m) relies on -- • (m+1)

11. Reasoning by Contradiction This is an extremely useful inference rule, very often used, often called reductio ad absurdum The idea is: assume the negation of what you want to prove – If reasoning then leads to a contradiction (i.e. leads to False), then you can infer (i.e. you have proved) the negation of your assumption. • {Assume} • … {via valid inferences but no other assumptions we get …} • (m) False • {it follows that the original assumption must be False, so • we can infer the following} • (m+1)

12. E.g. Let’s prove that 10 is an even number. • To do this by Contradiction, we first assume that • “10 is even” is False: • {Assumption} • 10 is an odd number • {this follows from (1) • (2) 10 = 2k + 1 for some k in N • {this follows from (2), by simple algebra} • (3) 9 = 2k for some k in N • {simple algebra, follows from (3)} • (4) k = 4.5 and k in N • {This follows from (4), since it is a contradiction} • (5) False • {We have now proved this, via proof by contradiction} • (6) 10 is an even number

13. Prove that this: Implies: It stands to reason, when you look at it. Let’s prove it by contradiction • {Assume} • { Implication} • {De Morgan, double negation, and associativity} • {follows from (3) by AND-elimination, twice} • (4) • {we can see that (4) is a contradiction, so we have proved this: • (5)