Automated Reasoning Lecture 2: Propositional Logic and Natural Deduction
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1 Automated Reasoning Lecture 2: Propositional Logic and Natural Deduction Jacques Fleuriot
2 Logic Puzzles 1. Tomorrow will be sunny or rainy. Tomorrow will not be sunny. What will the weather be tomorrow? 2. I like classical or pop music. If I like classical music, then I am sophisticated. I don't like pop music. Am I sophisticated? 3. Fred bought milk or Fred bought lemonade. Fred bought milk or Fred bought water. Fred did not buy both water and lemonade. What did Fred buy?
3 Syntax of Propositional Logic Propositional Logic represents the problems we have just seen by using symbols to represent (atomic) propositions. These can be combined using the following connectives: Name symbol usage not P and P Q or P Q implies P Q if and only if P Q precedence Treat all binary connectives as right associative (following Isabelle) Example 1. (SunnyTomorrow RainyTomorrow) ( SunnyTomorrow) 2. (Class Pop) (Class Soph) Prop 3. (M L) (M W) (L W)
4 Syntax and Ambiguity The meanings of some statements can (appear to) be ambiguous: Class Pop Class Soph Pop We can use brackets (parentheses) to disambiguate a statement: (Class Pop) (Class Soph Pop) Note that, based on our choice of precedence (on the previous slide), A B C denotes A (B C) Also note that implication is right associative, so: P Q R denotes P (Q R)
5 Formal Syntax A syntactically correct formula is called a well-formed formula (wff) Given a (possible infinite) alphabet of propositional symbols L, the set of wffs is the smallest set such that any symbol A L is a wff; if P and Q are wffs, so are P, P Q, P Q, P Q, and P Q; if P is a wff, then (P) is a wff. When we are interested in abstract syntax (tree-structure of formulas) rather than concrete syntax, we forget the last clause.
6 Semantics Each wff is assigned a meaning or semantics, T or F, depending on whether its constituent wffs are assigned T or F. Truth tables are one way to assign truth values to wffs. P Q P Q T T T T F F F T F F F F P P T F F T P Q P Q T T T T F T F T T F F F P Q P Q T T T T F F F T T F F T
7 Semantics of the weather problem 1 Tomorrow will be sunny or rainy 2 Tomorrow will not be sunny What will the weather be tomorrow? SunnyTomorrow T T F F RainyTomorrow T F T F S R S (S R) S
8 Semantics of the weather problem 1 Tomorrow will be sunny or rainy 2 Tomorrow will not be sunny What will the weather be tomorrow? SunnyTomorrow RainyTomorrow S R S (S R) S T T T T F F T F F
9 Semantics of the weather problem 1 Tomorrow will be sunny or rainy 2 Tomorrow will not be sunny What will the weather be tomorrow? SunnyTomorrow RainyTomorrow S R S (S R) S T T T T F T F T F F
10 Semantics of the weather problem 1 Tomorrow will be sunny or rainy 2 Tomorrow will not be sunny What will the weather be tomorrow? SunnyTomorrow RainyTomorrow S R S (S R) S T T T T F T F T T F F
11 Semantics of the weather problem 1 Tomorrow will be sunny or rainy 2 Tomorrow will not be sunny What will the weather be tomorrow? SunnyTomorrow RainyTomorrow S R S (S R) S T T T T F T F T T F F F
12 Semantics of the weather problem 1 Tomorrow will be sunny or rainy 2 Tomorrow will not be sunny What will the weather be tomorrow? SunnyTomorrow RainyTomorrow S R S (S R) S T T T F T F T F T T F F F
13 Semantics of the weather problem 1 Tomorrow will be sunny or rainy 2 Tomorrow will not be sunny What will the weather be tomorrow? SunnyTomorrow RainyTomorrow S R S (S R) S T T T F T F T F F T T F F F
14 Semantics of the weather problem 1 Tomorrow will be sunny or rainy 2 Tomorrow will not be sunny What will the weather be tomorrow? SunnyTomorrow RainyTomorrow S R S (S R) S T T T F T F T F F T T T F F F
15 Semantics of the weather problem 1 Tomorrow will be sunny or rainy 2 Tomorrow will not be sunny What will the weather be tomorrow? SunnyTomorrow RainyTomorrow S R S (S R) S T T T F T F T F F T T T F F F T
16 Semantics of the weather problem 1 Tomorrow will be sunny or rainy 2 Tomorrow will not be sunny What will the weather be tomorrow? SunnyTomorrow RainyTomorrow S R S (S R) S T T T F F T F T F F T T T F F F T
17 Semantics of the weather problem 1 Tomorrow will be sunny or rainy 2 Tomorrow will not be sunny What will the weather be tomorrow? SunnyTomorrow RainyTomorrow S R S (S R) S T T T F F T F T F F F T T T F F F T
18 Semantics of the weather problem 1 Tomorrow will be sunny or rainy 2 Tomorrow will not be sunny What will the weather be tomorrow? SunnyTomorrow RainyTomorrow S R S (S R) S T T T F F T F T F F F T T T T F F F T
19 Semantics of the weather problem 1 Tomorrow will be sunny or rainy 2 Tomorrow will not be sunny What will the weather be tomorrow? SunnyTomorrow RainyTomorrow S R S (S R) S T T T F F T F T F F F T T T T F F F T F
20 Semantics of the weather problem 1 Tomorrow will be sunny or rainy 2 Tomorrow will not be sunny What will the weather be tomorrow? SunnyTomorrow RainyTomorrow S R S (S R) S T T T F F T F T F F F T T T T F F F T F So it will rain tomorrow.
21 Semantics: Definitions Definition (Interpretation) An interpretation (or valuation) is a truth assignment to the symbols in the alphabet L: it is a function V from L to {T, F}.
22 Semantics: Definitions Definition (Interpretation) An interpretation (or valuation) is a truth assignment to the symbols in the alphabet L: it is a function V from L to {T, F}. An interpretation V of L is extended to an interpretation of a wff P by induction on its structure: A V = V(A) P Q V = P V and Q V P Q V = P V or Q V P V = not P V P Q V = P V implies Q V This is the Tarski definition of truth: the truth value of a compound sentence is established by breaking it down until we get to atom propositions.
23 Semantics: Definitions Definition (Satisfaction) An interpretation V satisfies a wff P if P V = T Definition (Satisfiable) A wff is satisfiable is there exists an interpretation that satisfies it. A wff is unsatisfiable if it is not satisfiable. Definition (Valid or Tautology) A wff is valid or is a tautology if every interpretation satisfies it. Example (S R) S is satisfiable (there is a state of affairs that makes it true) ((S R) S) R is valid (it is always true, no matter what the state of affairs)
24 Semantics: Definitions Definition (Entailment) The wffs P 1, P 2,..., P n entail Q if for any interpretation which satisfies all of P 1, P 2,..., P n also satisfies Q. We then write P 1, P 2,..., P n Q. Note If there is no interpretation which satisfies all of P 1, P 2,..., P n then P 1, P 2,..., P n Q for any Q. Contradictory assumptions entail everything! Note Everything entails a tautology. If Q is a tautology, then P 1, P 2,..., P n Q holds not matter what P 1, P 2,... P n are. We write Q when Q is a tautology. Example Is P, Q Q (P Q) a valid entailment?
25 Proof, Inference Rules and Deductive Systems For propositional logic, it is possible to reason on a computer directly using the semantics: Satisfiability, validity and entailment are decidable So, in theory, we make conjectures and the computer checks them. But this is not always possible: Propositional logic is not very expressive; but Expressive logics like FOL and HOL are not decidable; and Even for propositional logic, checking satisfiability, validity, entailment is not always feasible. So we encode a notion of proof : A formal deductive system is a set of valid inference rules that tell us what conclusions we can draw from some premises. Inference rules can be applied manually or automatically. We will look at Natural Deduction, developed by Gentzen and Prawitz.
26 Inference Rules An inference rule tells us how one wff can be derived in one step from zero, one, or more other wffs. We write P 1 P 2... P n Q if wff Q is derived from wffs P 1, P 2,..., P n using the rule R. Example inference rules (with their corresponding Isabelle names): (R) P Q P Q (conji) P P Q (mp) Q The P and Q here are meta-variables (denoted by?p and?q in Isabelle) and mp is the modus ponens rule of inference. This rule schema characterises an infinite number of rule instances, obtained by substituting wffs for the P and Q. Example of an instance of mp is: P {}}{ A B P Q {}}{{}}{ (A B) C C }{{} Q
27 Validity Inference rules must be valid. They must preserve truth. Formally, for all instances of P 1 P 2... P n Q (R) of the rule R, we must have P 1, P 2,..., P n Q. Inference is transitive. If we can infer R from Q and we can infer Q from P, then we can infer R from P. This means we can chain deductions together to form a deduction ``tree''.
28 Introduction and Elimination In Natural Deduction (ND), rules are split into two groups: Introduction rules : how to derive P Q P Q P Q (conji) P P Q (disji1) Q P Q (disji2) Elimination rules : what can be derived from P Q? P Q P (conjunct1) P Q Q (conjunct2) [P] [Q] P Q. R R. R (disje)
29 A proof: distributivity of and A proof that P (Q R) (P Q) (P R). P (Q R) Q R P (Q R) P [Q] P Q (P Q) (P R) (P Q) (P R) P (Q R) P [R] P R (P Q) (P R) Note Each proof step will normally be annotated with the name of its associated inference rule (e.g. disje for the bottom most step).
30 Ways of applying rules Inference rules are applied in two basic ways. Forward proof if we derive new wffs from existing wffs by applying rules top down. Backward proof if we conjecture some wff true and apply rules bottom-up to produce new wffs from which the original wff is derived. In general, interactive proof (as we will do in Isabelle) proceeds backwards, from the goal. Forwards proof is also possible, though.
31 Summary Propositional logic Syntax (atomic propositions, wffs) (H&R ) Semantics (interpretations, satisfication, satisfiability, validity, entailment) (H&R 1.4.1, first part of 1.4.2) Natural deduction (H&R 1.2) Introduction and Elimination rules; Proofs are trees, with assumptions at the leaves. Next time More on Natural Deduction (,, ); Natural deduction in Isabelle/HOL.
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