Semantic Entailment and Natural Deduction Alice Gao Lecture 6, September 26, 2017 Entailment 1/55
Learning goals Semantic entailment valuation trees, and/or logical identities. Natural deduction in propositional logic inference rules. conclusion formula using natural deduction. Entailment 2/55 • Defjne semantic entailment. • Explain subtleties of semantic entailment. • Determine whether a semantic entailment holds by using truth tables, • Prove semantic entailment using truth tables and/or valuation trees. • Describe rules of inference for natural deduction. • Prove a conclusion from given premises using natural deduction • Describe strategies for applying each inference rule when proving a
A review of the conditional The following two statements are equivalent: Entailment 3/55 Consider the formulas 𝑞 1 ∧ 𝑞 2 ∧ 𝑞 3 and 𝑑 . • for any truth valuation 𝑢 , if (𝑞 1 ∧ 𝑞 2 ∧ 𝑞 3 ) is true, then 𝑑 is true. • (𝑞 1 ∧ 𝑞 2 ∧ 𝑞 3 ) → 𝑑 is a tautology.
Subtleties about the conditional statements are true? d. Two of (a), (b), and (c) are true. e. All of (a), (b), and (c) are true. Entailment 4/55 Consider the formulas 𝑞 1 ∧ 𝑞 2 ∧ 𝑞 3 and 𝑑 . How many of the following a. If 𝑞 1 is false, then (𝑞 1 ∧ 𝑞 2 ∧ 𝑞 3 ) → 𝑑 is true. b. If 𝑞 1 = 𝑦 and 𝑞 2 = (¬𝑦) , then (𝑞 1 ∧ 𝑞 2 ∧ 𝑞 3 ) → 𝑑 is false. c. If 𝑑 is a tautology, then (𝑞 1 ∧ 𝑞 2 ∧ 𝑞 3 ) → 𝑑 is true.
Proving arguments valid Recall that logic is the science of reasoning. One important goal of logic is to infer that a conclusion is true based on a set of premises. A logical argument: Premise 1 Premise 2 ... Premise n ——— Conclusion A common problem is to prove that an argument is valid, that is the set of premises semantically entails the conclusion. Entailment 5/55
Formalizing argument validity: Semantic Entailment Let Σ = {𝑞 1 , 𝑞 2 , ..., 𝑞 𝑜 } be a set of premises and let 𝛽 be the conclusion that we want to derive. the conclusion 𝛽 is true under 𝑢 . If Σ semantically entails 𝛽 , then we say that the argument (with the premises in Σ and the conclusion 𝛽 ) is valid. Entailment 6/55 Σ semantically entails 𝛽 , denoted Σ ⊨ 𝛽 , if and only if • Whenever all the premises in Σ are true, then the conclusion 𝛽 is true. • For any truth valuation 𝑢 , if every premise in Σ is true under 𝑢 , then • For any truth valuation 𝑢 , if 𝑢 satisfjes Σ (denoted Σ 𝑢 = T ), then 𝑢 satisfjes 𝛽 ( 𝛽 𝑢 = T ). • (𝑞 1 ∧ 𝑞 2 ∧ ... ∧ 𝑞 𝑜 ) → 𝛽 is a tautology. What does Σ 𝑢 = T ( 𝑢 satisfjes Σ ) mean? See the next slide.
If Σ is the empty set ∅ , then any valuation satisfjes Σ . Why? The defjnition of “ 𝑢 satisfjes Σ ” says There is no formula in ∅ , so the premise of the above statement is false, which means the statement is vacuously true. Thus, any valuation satisfjes the empty set ∅ . Entailment 7/55 What does Σ 𝑢 = T mean? Σ 𝑢 = T ( 𝑢 satisfjes Σ ) means ... • Every formula in Σ is true under the valuation 𝑢 . • If a formula 𝛾 is in Σ , then 𝛾 is true under 𝑢 . • If a formula 𝛾 is in Σ , then 𝛾 is true under 𝑢 .
Subtleties about entailment Consider a set of formulas Σ and the formula 𝛽 . How many of the following statements are true? b. If Σ = {𝑦, (¬𝑦)} , then Σ ⊨ 𝛽 is true. c. If ∅ ⊨ 𝛽 is true, then 𝛽 is a tautology ( ∅ is the empty set). d. Two of (a), (b), and (c) are true. e. All of (a), (b), and (c) are true. Entailment 8/55 a. If 𝑞 1 in Σ is false, then Σ ⊨ 𝛽 is false.
Proving or disproving entailment Proving that Σ entails 𝛽 , denoted Σ ⊨ 𝛽 : the formulas in Σ are true. Verify that 𝛽 is true in all of these rows. are true, show that the conclusion is also true under this valuation. which means that there is a truth valuation under which all of the premises are true and the conclusion is false. Derive a contradiction. Proving that Σ does not entail 𝛽 , denoted Σ ⊭ 𝛽 : true and the conclusion 𝛽 is false. Entailment 9/55 • Using a truth table: Consider all rows of the truth table in which all of • Direct proof: For every truth valuation under which all of the premises • Proof by contradiction: Assume that the entailment does not hold, • Find one truth valuation 𝑢 under which all of the premises in Σ are
Proving entailment using a truth table 0 1 1 0 1 0 1 0 0 1 1 1 0 1 0 1 Entailment 1 0 Let Σ = {(¬(𝑞 ∧ 𝑟)), (𝑞 → 𝑟)} , 𝑦 = (¬𝑞) , and 𝑧 = (𝑞 ↔ 𝑟) . Based on the (¬(𝑞 ∧ 𝑟)) truth table, which of the following statements is true? a. Σ ⊨ 𝑦 and Σ ⊨ 𝑧 . b. Σ ⊨ 𝑦 and Σ ⊭ 𝑧 . c. Σ ⊭ 𝑦 and Σ ⊨ 𝑧 . d. Σ ⊭ 𝑦 and Σ ⊭ 𝑧 . 𝑞 𝑟 (𝑞 → 𝑟) 1 𝑦 = (¬𝑞) 𝑧 = (𝑞 ↔ 𝑟) 0 0 1 1 1 10/55
Proving entailment 1 Entailment 1 1 0 1 1 0 0 1 0 1 0 0 1 What is {(¬(𝑞 ∧ 𝑟)), (𝑞 ∧ 𝑟)} ⊨ (𝑞 ↔ 𝑟) ? 0 1 0 1 0 0 (𝑞 ↔ 𝑟) (𝑞 ∧ 𝑟) (¬(𝑞 ∧ 𝑟)) 𝑟 𝑞 b. False a. True 11/55
Equivalence and Entailment Equivalence can be expressed using the notion of entailment. Lemma . 𝛽 ≡ 𝛾 if and only if both {𝛽} ⊨ 𝛾 and {𝛾} ⊨ 𝛽 . Entailment 12/55
Proofs in Propositional Logic: Natural Deduction Natural Deduction 13/55
Solution to the previous puzzle A very special island is inhabited only by knights and knaves. Knights always tell the truth, and knaves always lie. You meet three inhabitants: Alice, Rex and Bob. 2. Rex says, “it’s false that Bob is a knave (or Bob is a knight).” This means Rex and Bob are the same. 3. Bob claims, “I am a knight or Alice is a knight.” Bob is a knight, or Bob and Alice are both knaves. Based on 1 and 2, Alice and Bob are difgerent, so they cannot both be knaves (2nd option in 3). Thus, the only possibility left is Alice is a knave, and Rex and Bob are knights. Natural Deduction Overview 14/55 1. Alice says, “Rex is a knave.” This means Alice and Rex are difgerent.
Labyrinth Puzzle Natural Deduction Overview 15/55
Learning goals Natural deduction in propositional logic inference rules. conclusion formula using natural deduction. Natural Deduction Overview 16/55 • Describe rules of inference for natural deduction. • Prove a conclusion from given premises using natural deduction • Describe strategies for applying each inference rule when proving a
The Natural Deduction Proof System We will consider a proof system called Natural Deduction. formal arguments. Natural Deduction Overview 17/55 • It closely follows how people (mathematicians, at least) normally make • It extends easily to more-powerful forms of logic.
Why would you want to study natural deduction proofs? mysterious symbols as justifjcations. that males should play female characters in Japanese kabuki theatres. creative and a scientifjc endeavour. situations. Natural Deduction Overview 18/55 • It is impressive to be able to write proofs with nested boxes and • Be able to prove or disprove that Superman exists (on Tuesday). • Be able to prove or disprove that the onnagata are correct to insist • To realize that writing proofs and problem solving in general is both a • To develop problem solving strategies that can be used in many other
A proof is syntactic First, we think about proofs in a purely syntactic way. A proof matching), We write or simply Σ ⊢ 𝜒 if we can fjnd such a proof that starts with a set of premises Σ and ends with the conclusion 𝜒 . Natural Deduction Overview 19/55 • starts with a set of premises, • transforms the premises based on a set of inference rules (by pattern • and reaches a conclusion. Σ ⊢ ND 𝜒
Goal is to show semantic entailment Next, we think about connecting proofs to semantic entailment. We will answer these questions: If I can fjnd a proof from Σ to 𝜒 , can I conclude that Σ semantically entails 𝜒 ? Does Σ ⊢ 𝜒 imply Σ ⊨ 𝜒 ? it? If I know that Σ semantically entails 𝜒 , can I fjnd a proof from Σ to 𝜒 ? Does Σ ⊨ 𝜒 imply Σ ⊢ 𝜒 ? Natural Deduction Overview 20/55 • (Soundness) Does every proof establish a semantic entailment? • (Completeness) For every semantic entailment, can I fjnd a proof for
Refmexivity / Premise 𝛽 Basic Rules Natural Deduction Deduction. Other rules will make more use of it. The version in the center reminds us of the role of assumptions in Natural the formula below the line. The notation on the right: Given the formulas above the line, we can infer 𝛽 If you want to write down a previous formula in the proof again, you can Σ, 𝛽 ⊢ 𝛽 or Premise Refmexivity, inference notation ⊢ -notation Name 21/55 do it by refmexivity .
An example using refmexivity Refmexivity: 1 Basic Rules Natural Deduction and be done. Premise 𝑞 1. Alternatively, we could simply write 𝑞 Here is a proof of {𝑞, 𝑟} ⊢ 𝑞 . 3. Premise 𝑟 2. Premise 𝑞 1. 22/55
For each symbol, the rules come in pairs. Natural Deduction Basic Rules 23/55 • An “introduction rule” adds the symbol to the formula. • An “elimination rule” removes the symbol from the formula.
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