of 33 lecture 12: propositional logic – part I

of 33 propositions and connectives … two-valued logic – every sentence is either true or false some sentences are minimal – no proper part which is also a sentence others – can be taken apart into smaller parts we can build larger sentences from smaller ones by using connectives ece 627, winter ‘132

of 33 propositions and connectives … connectives – each has one or more meanings in natural language – need for precise, formal language If I open the window then 1+3=4. John is working or he is at home. Euclid was a Greek or a mathematician. ece 627, winter ‘133

of 33 propositional logic … can do … logic that deals only with propositions (sentences, statements) if it is raining then people carry their umbrellas raised people are not carrying their umbrellas raised it is not raining ece 627, winter ‘134

of 33 propositional logic … cannot do … if it is raining then everyone will have their umbrellas up John is a person John does not have his umbrella up (it is not raining) no way to substitute the name of the individual John into the general rule about everyone ece 627, winter ‘135

of 33 propositional logic … we will look at three different versions  Hilbert system (the view that logic is about proofs)  Gentzen system and tableau (or Beth) system (the view that is about consistency and entailment) ece 627, winter ‘136

of 33 propositional logic … two stages of formalization  present a formal language  specify a procedure for obtaining valid/true propositions ece 627, winter ‘137

of 33 propositional logic Hilbert system PROP_H the syntax is extremely simple  a set of names for prepositions  single logical operator  a single rule for combining simple expressions via logical operator (using brackets to avoid ambiguity) ece 627, winter ‘138

of 33 propositional logic Hilbert system PROP_H is a system which is tailored for talking about what can and what cannot be proved within the language, rather than for actually saying things and exploring entailments ece 627, winter ‘139

of 33 propositional logic Hilbert system language … propositions names: p, q, r, …, p 0, p 1, p 2, … a name for “false”: ece 627, winter ‘1310

of 33 propositional logic Hilbert system language … the connective: (intended to be read “implies”) single combining rule: if A and B are expressions then so is A B (A, B stand for arbitrary expressions of the language) ece 627, winter ‘1311

of 33 propositional logic Hilbert system - semantics is given by explaining the meaning of the basic formulae - the proposition letters and is given in terms of two objects T and F is given by providing a function V called valuation (it defines meaning of expressions) ece 627, winter ‘1312

of 33 propositional logic Hilbert system - semantics V(P) – maps P (basic proposition letters and ) to {T, F} expr - denotes the meaning assigned to expr so, how to find out the value of complex expressions (built from simple ones using, i.e., A B ) …? ece 627, winter ‘1313

of 33 propositional logic Hilbert system - semantics answer – truth tables ABA B TT T TF F FT T FF T ece 627, winter ‘1314

of 33 propositional logic Hilbert system - semantics a truth table of complex expressions ABB AA (B A)) TT TF TT FT FF TT ece 627, winter ‘1315

of 33 propositional logic Hilbert system - semantics negation -A TF F FF T ece 627, winter ‘1316

of 33 propositional logic Hilbert system - semantics disjunction AB A A B TT FT TF FT FT TT FF TF ece 627, winter ‘1317

of 33 propositional logic Hilbert system - semantics conjunction ABABA B (A B) TTFF F T TFFT T F FTTF T F FFTT T F ece 627, winter ‘1318

of 33 propositional logic Hilbert system – inference rules only one inference rule for any expressions A and B: from A and A B we can infer B it is known as MODUS PONENS (MP) ece 627, winter ‘1319

of 33 propositional logic Hilbert system – inference rules + three axioms (expressions that are accepted as being true under any evaluation) PH1: A (B A) PH2: [A (B C)] [(A B) (A C)] PH3: ( A) A ece 627, winter ‘1320

of 33 propositional logic Hilbert system – inference rules a proof of a conclusion C from hypotheses H 1, H 2, …, H k is a sequence of expressions E 1, E 2, …, E n obeying the following conditions: (i) C is E n (ii) Each E i is either an axiom, or one of the H j or is a result of applying the inference rule MP to E k and E l where k and l are less then i ece 627, winter ‘1321

of 33 propositional logic Hilbert system – inference rules proof of r from p hypotheses: p q, q r 1 p(hypothesis) 2 p q(hypothesis) 3 q(from 1 and 2 by MP) 4 q r(hypothesis) 5 r(from 3 and 4 by MP) ece 627, winter ‘1322

of 33 propositional logic Hilbert system – inference rules p, p q, q r r is a meta-symbol indicating that proof exists if we have a proof of a conclusion from an empty set of hypotheses – a formula which can be proved in such a way is called a theorem ece 627, winter ‘1323

of 33 propositional logic Hilbert system – inference rules proof of A A 1 [A ((A A) A)] [(A (A A)) (A A)] (PH2) 2 [A ((A A) A)] (PH1) 3 [A (A A)] (A A) (MP on 1, 2) 4 (A (A A)) (PH1) 5 (A A) (MP on 3, 4) ece 627, winter ‘1324

of 33 propositional logic Hilbert system – inference rules deduction theorem if A 1, …, A n B then A 1, …, A n-1 A n B ece 627, winter ‘1325

of 33 propositional logic Hilbert system – inference rules proof of A A 1 A, A(MP) 2 A (A ) (DT) 3 A ((A ) (DT) 4 A A (A A) ece 627, winter ‘1326

of 33 propositional logic Hilbert system – inference rules proof of A 1, A 2 (A ) (DT) 3 [(A ) ] A (PH3) 4 A (MP on 2, 3) 5 A (DT) ece 627, winter ‘1327

of 33 propositional logic Hilbert system – soundness would like to be reassured that Prop_H will not permit us to infer conclusions which can be false when their supporting hypotheses are true (otherwise, we would be able to prove things which are not true) ece 627, winter ‘1328

of 33 propositional logic Hilbert system – soundness definition if A 1, …, A n A then A = T for any valuation function V for which all the A i are T ece 627, winter ‘1329

of 33 propositional logic Hilbert system – completeness would like to know that if some conclusion is always true whenever every member of a given set of hypotheses is true, then there is a proof of the conclusion from the hypotheses ece 627, winter ‘1330

of 33 propositional logic Hilbert system – completeness definition if A is valid then A ece 627, winter ‘1331

of 33 propositional logic Hilbert system – consistency definition there is no proof of ece 627, winter ‘1332

of 33 propositional logic Hilbert system – decidability definition there is a mechanical procedure which can decide for any A whether or not A ece 627, winter ‘1333