Lecture 19: Minimal vs. Intuitionistic vs. Classical Logic 1 Goals for Today ✤ First we introduce the repeat rule and give some examples. ✤ This rule plus the other rules we learned earlier are called minimal logic. We’ll also learn today about intuitionistic and classical logic. ✤ Practically, this just amounts to learning two new rules (the EFSQ- Rule and the ¬¬-Rule) and to introducing some notation for the three logics (minimal, intuitionistic, classical). ✤ We’ll also talk about some traditional reservations about adding these two new rules to our logic. 2 Repeat Rule ✤ This doesn’t come up so often, ✤ In a picture, the rule is simple: but you will see it occasionally. ✤ ℓ₁. ϕ ✤ In prose, the rule says: ✤ ℓ. ϕ (repeat ℓ). ✤ If you have ϕ on line ℓ1, then you can write ϕ again on any ✤ Initially it’s unclear why one subsequent line ℓ>ℓ₁. would ever need/want to use this rule. ✤ In applying this, stay out of closed boxes. That is, don’t use ✤ This rule is mostly useful as an this rule to repeat things in aid or helper to applying the closed boxes outside of them. introduction rule for ➝. 3 Example of Repeat Rule ✤ Show: q⊢(p➝q) ✤ What’s the idea here: well, we wanted to get p➝q, and ✤ Proof: so we try to go from p to q inside a box. ✤ 1. q (assumption) ✤ How are we going to get q? ✤ 2. p (assumption) Well, it’s kind of a strange question, because q is one of ✤ 3. q (repeat rule) our assumptions. ✤ ✤ 4. p➝q (I➝ 2-3) So the idea of repeat rule is we can repeat things we already have at later lines. 4 EFSQ Rule ✤ The acronym EFSQ stands for ✤ In a picture, this rule looks like: the latin phrase “ex falso sequitur quodlibet” which translates as ✤ ℓ₁. ⊥ “from a contradiction, anything follows”. ✤ ℓ. ψ (EFSQ ℓ₁). ✤ In prose, this rule says: ✤ In applying this rule, keep in mind that instances of ⊥ that ✤ If you have falsum ⊥ on any appear in closed off boxes line ℓ1, then you can write above line ℓ are off limits: don’t any formula ψ on any appeal later on to anything in subsequent line ℓ>ℓ₁. an already closed-off box. 5 First Example of EFSQ rule ✤ Show: ¬p ⊢(p➝q) ✤ Again, the way to understand this example is to start at the ✤ Proof: bottom. ✤ 1. ¬p (assumption) ✤ We want to get p➝q, and so the way that we do this is by going ✤ 2. p (assumption) from p to q inside a box. ✤ 3. ⊥ (E¬ 1,2) ✤ ✤ 4. q (EFSQ rule) But now we have a new tool for getting to the end of the box q: ✤ 5. p➝q (I➝ 2-4) namely, just secure ⊥ first and then apply EFSQ rule. 6 Second Example of EFSQ rule ✤ Show: p∨q, ¬p ⊢ q. ✤ Here’s the idea. One of our ✤ 1. p∨q (assumption) assumptions is a disjunction, so ✤ 2. ¬p (assumption) we try to apply E∨. ✤ 3. p (assumption) ✤ So we try to get lines with the ✤ 4. ⊥ (E¬ 2,3) first disjunct implying the ✤ 5. q (EFSQ 4) conclusion (p➝q) and the ✤ 6. p➝q (I➝ 3-5) second disjunct implying the conclusion (q➝q). ✤ 7. q (assumption) ✤ 8. q (repeat 7) ✤ For p➝q, we go from p to q ✤ 9. q➝q (I➝ 7-8) inside a box by going from p to ✤ 10. q (E∨ 1,6,9) ⊥ and then to q. 7 Double Negation Rule, aka ¬¬Rule ✤ The last rule we learn is the ✤ Here’s a simple example. double-negation rule, aka ¬¬ Recall in last lecture, we gave rule. In prose, it reads: a really long proof to show that ¬¬¬p ⊢¬p. ✤ If you have ¬¬ϕ at line ℓ1, then you can write ϕ at any ✤ Well, there’s a super simple subsequent line ℓ> ℓ1. proof using the ¬¬ rule: ✤ Here’s the picture version: ✤ 1. ¬¬¬p (assumption) ✤ ℓ1. ¬¬ϕ ✤ 2. ¬p (¬¬ 1). ✤ ℓ. ϕ (¬¬ ℓ1). 8 Famous Example of ¬¬-Rule ✤ Show: ⊢p∨¬p. ✤ What’s the idea of the proof? Well, you want to get ✤ 1. ¬(p∨¬p) (assumption) something, so you try to get its double negation. ✤ 2. p (assumption) ✤ 3. p∨¬p (I∨ 2) ✤ So you try to get ¬¬(p∨¬p). This ✤ 4. ⊥ (E¬ 1,3) starts with a negation, so you try to go from ¬(p∨¬p) to ⊥ ✤ 5. ¬p (¬I 2-4) inside a box. ✤ 6. p∨¬p (I∨ 5) ✤ 7. ⊥ (E¬ 1, 6) ✤ How are you going to get ⊥? ✤ 8. ¬¬(p∨¬p) (I¬ 1-7) Well, you try to also get ¬p, so ✤ 9. p∨¬p (¬¬ 8) that you can get p∨¬p. 9 Famous Example of ¬¬-Rule ✤ Show: ⊢p∨¬p. ✤ There’s admittedly something strange about this proof ✤ 1. ¬(p∨¬p) (assumption) procedure. Namely, we were trying all along to get p∨¬p. ✤ 2. p (assumption) ✤ 3. p∨¬p (I∨ 2) ✤ But when we look at the ✤ 4. ⊥ (E¬ 1,3) finished proof, we see that p∨¬p appears already at line 3. ✤ 5. ¬p (¬I 2-4) ✤ 6. p∨¬p (I∨ 5) ✤ Why couldn’t we just stop at ✤ 7. ⊥ (E¬ 1, 6) line 3? Well, it’s inside a box, ✤ 8. ¬¬(p∨¬p) (I¬ 1-7) and if you start a box, you have ✤ 9. p∨¬p (¬¬ 8) to finish it. 10 Please take note ✤ On the final exam, you will be required to show ⊢p∨¬p. Answering this question will literally involving giving all nine lines of the proof on the previous slides with the justifications of all the steps. 11 Second Example of ¬¬-Rule ✤ Show: ¬q ➝ ¬p ⊢ p➝q. ✤ What’s the idea of this proof? Well, you need to go from p to q ✤ 1. ¬q ➝ ¬p (assumption) inside a box. ✤ 2. p (assumption) ✤ Idea is to get q by getting ¬¬q. ✤ 3. ¬q (assumption) How are you going to get ¬¬q? ✤ 4. ¬p (E➝1, 4) Well, it starts with a negation, so you try to go from ¬q to ⊥ ✤ 5. ⊥ (E¬ 2,4) inside a box. ✤ 6. ¬¬q (I¬ 3-5) ✤ 7. q (¬¬ 6) ✤ We know what to do with ¬q: for, our assumption tell us that ✤ ➝ ➝ 8. p q (I 1-7) ¬q ➝ ¬p. 12 Enumerating the Rules for Propositional Logic ✤ So now we know all the rules for propositional logic! ✤ We can enumerate all the rules as follows: ✤ I∧, E∧, I∨, E∨, I➝, E➝, I¬, E¬, Repeat Rule ✤ EFSQ ✤ ¬¬ Rule 13 Minimal, Intuitionistic, Classical ✤ One calls minimal logic the deductive system which has only the introduction and elimination rules for the connectives ∧,∨,➝,¬ plus the repeat rule. If a proof ϕ1, ϕ2, . ., ϕn ⊢ ψ uses only these rules, we write ϕ1, ϕ2, . ., ϕn ⊢M ψ ✤ One calls intuitionistic logic the deductive system that extends minimal logic with the EFSQ rule. If a proof ϕ1, ϕ2, . ., ϕn ⊢ ψ uses only the rules of minimal logic plus the EFSQ rule, then one writes ϕ1, ϕ2, . ., ϕn ⊢I ψ ✤ One calls classical logic the deductive system that uses all the rules, including both the EFSQ rule and ¬¬ Rule. So ϕ1, ϕ2, . ., ϕn ⊢ ψ just means the same thing as ϕ1, ϕ2, . ., ϕn ⊢C ψ. 14 Limits of Intuitionistic Logic ✤ One thing that you can’t do in ✤ We already know why intuitionistic logic is prove that ⊢C p∨¬p p∨¬p. This is called the law of the Indeed, the famous proof excluded middle or LEM. from a couple of slides back showed just that. ✤ We indicate that LEM can be proven in classical logic but ✤ We don’t yet have the tools can’t be proven in minimal to show that logic by writing ⊬I p∨¬p ✤ ⊢C p∨¬p ✤ ⊬I p∨¬p 15 Motivation for Intuitionistic Logic ✤ ✤ The intuitionists -- implicitly Likewise, ⊢I ϕ∨ψ just means Brouwer, explicitly Heyting-- that ⊢I ϕ or ⊢I ψ, or more basic rejected the idea that the aim of still, that a proof of a logic was the truth. Their idea disjunction is either a proof of was rather that logic should the one disjunct or a proof of aim to explicate proof. the other disjunct. ✤ On this view, ⊢I ϕ∧ψ just ✤ Hence, on this view, it’s natural means that ⊢I ϕ and ⊢I ψ, or that one should reject ⊢I p∨¬p. more basic still, that a proof of a For, there are many situations in conjunction is a proof of the one which we simply don’t have a conjunct together with a proof proof of p or its negation. E.g. of the other disjunct. p = it will rain tomorrow. 16 Philosophical Motivation for Minimal Logic ✤ Here’s one philosophical motivation for rejecting EFSQ, and hence for sticking with minimal logic. With EFSQ we can prove (p∧¬p)➝q, regardless of what p and q are. ✤ You might reasonably have the view that if you can prove r➝q, then there should be some relationship between r and q. 17 Ω 18.