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Lecture 21: Soundness and Completeness Lecture 21: Soundness and Completeness 1 Lecture 21: Soundness and Completeness 2 Goal Today ✤ Our goal today is to state and define the soundness and completeness theorems. ✤ Then we will extract some rules of thumb for how to effectively use these results together to solve problems. 3 Initial Contrast: Provability vs. Validity ✤ So over the past couple of weeks, ✤ The validity relation holds when a we’ve learned about two different truth table containing them all notions: satisfies this property: on any row where all the premises ϕ1, ϕ2,…, ϕn ✤ provability: ϕ1, ϕ2, . ., ϕn ⊢ ψ are true, the conclusion ψ is also true. ✤ validity: ϕ1, ϕ2,…, ϕn ⊨ ψ ✤ There’s an analogous notion for predicate logic, but today we’ll ✤ Again, the idea is: provability stick to propositional logic. relation holds when there is a proof of the conclusion ψ from premises ✤ It’s natural at this juncture to ask ϕ1, ϕ2, . ., ϕn. how these two relate to one another. Before we state the general result, let’s look at some examples. 4 First Example ✤ Show: p, p➝q ⊢q ✤ Show: p, p➝q ⊨q ✤ Proof: ✤ Proof: ✤ 1. p (assumption) 1 2 3 4 5 ➝ ✤ 2. p➝q (assumption) p q p p q q 1 T T T T T ✤ 3. q (E➝ 1,2) 2 T F T F F 3 F T F T T 4 F F F T F 5 Second Example ✤ Show: p, q ⊢ p∧q ✤ Show: p, q ⊨ p∧q ✤ Proof: ✤ Proof: ✤ 1. p (assumption) 1 2 3 ✤ 2. q (assumption) p q p∧q 1 T T T ✤ 3. p∧q (I∧1,2) 2 T F F 3 F T F 4 F F F 6 Third Example ✤ Show: p,¬p ⊢ q ✤ Show: p,¬p ⊨ q ✤ Proof: ✤ Proof: ✤ 1. p (assumption) 1 2 3 3 4 ✤ 2. ¬p (assumption) p q p ¬p q 1 T T T F T ✤ 3. ⊥ (E¬ 1,2) 2 T F T F F ✤ 4. q (EFSQ 3) 3 F T F T T 4 F F F T F 7 The Major Result ✤ These examples should hopefully make the following general result very plausible: ✤ ϕ1, ϕ2, . ., ϕn ⊢ψ if and only if ϕ1, ϕ2,…, ϕn ⊨ ψ ✤ So given premises ϕ1, ϕ2, . ., ϕn and conclusion ψ, there is a proof from assumptions ϕ1, ϕ2, . ., ϕn to conclusion ψ if and only if the argument with premises ϕ1, ϕ2, . ., ϕn and this conclusion ψ is valid. ✤ Obviously the two notions ⊢ and ⊨ are very different: the first concerns proofs, and the second concerns what happens on every row of a truth table. But the major result says that these two things line up. 8 Two Parts of the Major Result ✤ The two parts of the major result are sometimes given special names: ✤ Soundness Theorem: if ϕ1, ϕ2, . ., ϕn ⊢ψ then ϕ1, ϕ2,…, ϕn ⊨ ψ ✤ Completeness Theorem: if ϕ1, ϕ2,…, ϕn ⊨ ψ then ϕ1, ϕ2, . ., ϕn ⊢ψ ✤ They are given these names because the proofs are very different, and in general the proof of the completeness theorem is much harder. ✤ You can kind of see why: to prove soundness, you must simply check that our rules for our proof system preserve truth. To prove completeness you must show that all valid arguments have proofs. Initially it’s not obvious how you would show this. 9 Philosophical Significance 1/3 ✤ After we covered the introduction and elimination rules for the propositional connectives, we kept on adding rules to our deductive system like EFSQ and ￢￢ Introduction. ✤ Is there something non-arbitrary about this specific list of rules? It seems that the soundness and completeness proofs give us a reason to think so. ✤ Kleene: “[...] we have listed eleven postulates for the propositional calculus [...]. Can we give a reason why we stop with just these? Might we with advantage attempt to discover others which could be added to the list to give more provable formulas?'' (Introduction to Metamathematics p. 131). 10 Philosophical Significance 2/3 ✤ A. Seems plausible that: if ϕ1, ϕ2, . ., ϕn ⊢ψ, then the truth of the premises ϕ1, ϕ2, . ., ϕn guarantees the truth of the conclusion ψ. ✤ B. Further, seems plausible that: if the truth of the premises ϕ1, ϕ2, . ., ϕn guarantees the truth of the conclusion ψ, then whenever the premises ϕ1, ϕ2, . ., ϕn are true, the conclusion ψ is true. ✤ C. Moreover, seems plausible that: if whenever the premises ϕ1, ϕ2, . ., ϕn are true, the conclusion ψ is true, then ϕ1, ϕ2,…, ϕn ⊨ ψ. ✤ D. Finally, completeness theorem tells us: if ϕ1, ϕ2,…, ϕn ⊨ ψ then ϕ1, ϕ2, . ., ϕn ⊢ψ. 11 Philosophical Significance 3/3 ✤ So the following are equivalent: ✤ 1. ϕ1, ϕ2, . ., ϕn ⊢ψ ✤ 2. the truth of premises ϕ1, ϕ2, . ., ϕn guarantees the truth of the conclusion ψ. ✤ 3. whenever premises ϕ1, ϕ2, . ., ϕn are true, the conclusion ψ is true. ✤ 4. ϕ1, ϕ2,…, ϕn ⊨ ψ ✤ This is significant because we suspected but did not know this equivalence beforehand. This argument is called the Kreisel squeezing argument (from his “Informal Rigour and Completeness Proofs”). 12 Applications of the Major Result ✤ Suppose that someone asked ✤ Show: p, p➝q,q➝r, r➝s, s➝t ⊢t you to show the following: ✤ 1. p (assumption) ✤ p, p➝q,q➝r, r➝s, s➝t ⊨t ✤ 2. p➝q (assumption) ✤ 3. q➝r (assumption) ✤ This would be a really really big ✤ 4. r➝s (assumption) 5 truth table! It would have 2 =32 ✤ 5. s➝t (assumption) different rows! ✤ 6. q (E➝ 1,2) ✤ 7. r (E➝ 2,6) ✤ However, once we know our ✤ 8. s (E➝ 4,7) major result, we know it ✤ 9. t (E➝5, 8) suffices to show that there’s a deduction. 13 From Lec 17, Handout 2: Example 8 (Pierce’s Law) ✤ Show: ⊢ ((p➝q)➝p)➝p ✤ What’s the strategy here? Well, ✤ 1. (p➝q)➝p (assumption) we want to go from ((p➝q)➝p) ✤ 2. ¬p (assumption) to p inside a box. ✤ 3. p (assumption) ✤ How in the world are we going ✤ 4. ⊥ (E¬ 2,3) to get p? The assumption that ✤ 5. q (EFSQ 4) ((p➝q)➝p) doesn’t give us a lot ✤ 6. p➝q (I➝ 3-5) to work with. So we try to first ✤ 7. p (E➝ 1,6) to obtain ¬¬p. ✤ 8. ⊥ (E¬ 2,7) ✤ ✤ 9. ¬¬p (I¬ 2-8) So we try to go from ¬p to ⊥ ✤ 10. p (¬¬ 9) inside a box. We do this by trying to obtain p as well. ✤ 11. ((p➝q)➝p)➝p (I➝ 1-10) 14 Applications of the Major Result ✤ We just did Pierce’s law: ⊢ ((p➝q)➝p)➝p. Conceptually, this proof was complicated, even though it has only two letters. By the major result, we could have instead done the following truth-table. p q p➝q p (p➝q)➝p p ((p➝q)➝p)➝p T T T T T T T T F F T T T T F T T F F F T F F T F F F T 15 Two R u l e s o f T h u m b ✤ So I suppose that these two ✤ If your argument seems like it applications suggest some has a lot of negations in it, or general “rules of thumb” for seems like it might require deciding when to try to show EFSQ or ¬¬-Rule to prove, you provability vs. validity: might want to try to show that it’s valid with e.g. a truth-table. ✤ provability: ϕ1, ϕ2, . ., ϕn ⊢ ψ ✤ There’s might well be situations ✤ validity: ϕ1, ϕ2,…, ϕn ⊨ ψ in which neither of these rules would apply: imagine for ✤ If you argument has more than instance something like Pierce’s three letters, you might want to law but which had 5 give a proof. propositional letters in it! 16 Ω 17.
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