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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
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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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propositional logic…
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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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)
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propositional logic…
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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propositional logicHilbert 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
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propositional logicHilbert system
language …
propositions names: p, q, r, …, p0, p1, p2, …
a name for “false”:
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propositional logicHilbert 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)
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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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propositional logicHilbert 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 ) …?
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propositional logicHilbert system - semantics
answer – truth tables
A B A BT T TT F FF T TF F T
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propositional logicHilbert system - semantics
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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propositional logicHilbert system - semantics
negation -
A A T F FF F T
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propositional logicHilbert system - semantics
disjunction
A B A A BT T F TT F F TF T T TF F T F
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propositional logicHilbert system - semantics
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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propositional logicHilbert 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
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propositional logicHilbert system – inference rules
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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propositional logicHilbert system – inference rules
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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propositional logicHilbert 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
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propositional logicHilbert 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)
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propositional logicHilbert system – inference rules
deduction theorem
if A1, …, An B then A1, …, An-1 An B
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propositional logicHilbert system – inference rules
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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propositional logicHilbert system – inference rules
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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