1. ece 720 intelligent web: ontology and beyond
lecture 12: propositional logic – part I

2. 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
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3. 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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4. 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
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5. 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
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6. 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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7. propositional logic … two stages of formalization
present a formal language
specify a procedure for obtaining valid/true propositions
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8. 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)
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9. 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
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10. propositional logic Hilbert system
language … propositions names: p, q, r, …, p0, p1, p2, … a name for “false”:
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11. 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)
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12. 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)
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13. 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 ) …?
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14. propositional logic Hilbert system - semantics
answer – truth tables A B A B T T T T F F F T T F F T
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15. propositional logic Hilbert system - semantics
a truth table of complex expressions A B B A A (B A)) T T T T T F T T F T F T F F T T
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16. propositional logic Hilbert system - semantics
negation - A A T F F F F T
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17. propositional logic Hilbert system - semantics
disjunction A B A A B T T F T T F F T F T T T F F T F
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18. propositional logic Hilbert system - semantics
conjunction A B A B A B ( A B) T T F F F T T F F T T F F T T F T F F F T T T F
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19. 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)
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20. 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
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21. propositional logic Hilbert 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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22. 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)
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23. 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
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24. 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)
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25. propositional logic Hilbert system – inference rules
deduction theorem if A1, …, An B then A1, …, An-1 An B
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26. 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)
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27. 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)
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28. 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)
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29. propositional logic Hilbert 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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30. 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
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31. propositional logic Hilbert system – completeness definition
if A is valid then A
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32. propositional logic Hilbert system – consistency definition
there is no proof of
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33. propositional logic Hilbert system – decidability definition
there is a mechanical procedure which can decide for any A whether or not A
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