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

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lecture 12: propositional logic – part I

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propositions and connectives…

two-valued logic – every sentence is either true or false

some sentences are minimal – no proper part which is also a sentenceothers – can be taken apart into smaller partswe can build larger sentences from smaller ones by using connectives

ece 627, winter ‘13 2

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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.


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propositional logic…

can do …

logic that deals only with propositions (sentences, statements)

if it is raining then people carry their umbrellas raisedpeople are not carrying their umbrellas raised it is not raining


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cannot do …

if it is raining then everyone will have their umbrellas upJohn is a personJohn 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


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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)


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two stages of formalization

present a formal language specify a procedure for obtaining valid/true

propositions


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propositional logicHilbert 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)


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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


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language …

propositions names: p, q, r, …, p0, p1, p2, …

a name for “false”:


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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)


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propositional logicHilbert 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)


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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 ) …?


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answer – truth tables

A B A BT T TT F FF T TF F T


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a truth table of complex expressions

A B B A A (B A))T T T TT F T TF T F TF F T T


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negation -

A A T F FF F T


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disjunction

A B A A BT T F TT F F TF T T TF F T F


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conjunction

A B A B A B ( A B)T T F F F TT F F T T FF T T F T FF F T T T F


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propositional logicHilbert 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)


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+ 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


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a proof of a conclusion C from hypotheses H1, H2, …, Hk is a sequence of expressions E1, E2, …, En obeying the following conditions:

C is En

Each Ei is either an axiom, or one of the Hj or is a result of applying the inference rule MP to Ek and El where k and l are less then i


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proof of r from phypotheses: 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)


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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


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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)


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deduction theorem

if A1, …, An B then A1, …, An-1 An B


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proof of A A

1 A, A (MP)2 A (A ) (DT)3 A ((A ) (DT)4 A A (A A)


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proof of A

1 , A2 (A ) (DT)3 [(A ) ] A (PH3)4 A (MP on 2, 3)5 A (DT)


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propositional logicHilbert 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)


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propositional logicHilbert system – soundness definition

if A1, …, An A then A = T for any valuation function V for which all the Ai are T


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propositional logicHilbert 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


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propositional logicHilbert system – completeness definition

if A is valid then A


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propositional logicHilbert system – consistency definition

there is no proof of


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propositional logicHilbert system – decidability definition

there is a mechanical procedure which can decide for any A whether or not A


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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