Download propositions and connectives propositions and connectives

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
lecture 12: propositional logic – part I
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propositions and connectives
…
connectives – each has one or more meanings in
natural language – need for precise, formal language
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propositional logic
…
can do …
logic that deals only with propositions (sentences,
statements)
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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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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propositional logic
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propositional logic
…
…
cannot do …
we will look at three different versions
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)
!  Hilbert system (the view that logic is about proofs)
!  Gentzen system and tableau (or Beth) system (the
view that is about consistency and entailment)
no way to substitute the name of the individual John into the general
rule about everyone
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propositional logic
propositional logic
…
Hilbert system
two stages of formalization
PROP_H
!  present a formal language
the syntax is extremely simple
!  a set of names for prepositions
!  specify a procedure for obtaining valid/true
propositions
!  single logical operator
!  a single rule for combining simple expressions via
logical operator (using brackets to avoid ambiguity)
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propositional logic
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propositional logic
Hilbert system
Hilbert system
PROP_H
language …
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
propositions names: p, q, r, …, p0, p1, p2, …
a name for false :
⊥
€
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propositional logic
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propositional logic
Hilbert system
Hilbert system - semantics
language …
is given by explaining the meaning of the basic
formulae - the proposition letters and ⊥
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)
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 logic
propositional logic
Hilbert system - semantics
Hilbert system - semantics
V(P) – maps P (basic proposition letters and ⊥ ) to
{T, F}
answer – truth tables
A
T
T
F
F
€ to expr
expr - denotes the meaning assigned
so, how to find out the value of complex expressions
€
€ (built from simple ones using → , i.e., A → B ) …?
€ €
€
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€
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propositional logic
B →A
T
T
€
F
T
negation - ¬
A →(B → A))
T
T
€
T
T
€
€
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propositional logic
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⊥
F
F
€€
A →⊥
F
T
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Hilbert system - semantics
disjunction ∨
A
T
T
F
F
A
T
F
propositional logic
Hilbert system - semantics
€
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Hilbert system - semantics
a truth table of complex expressions
B
T
F
€
T
F
A→B
T
F
T
T
propositional logic
Hilbert system - semantics
A
T
T
F
F
B
T
F
€
T
F
conjunction ∧
B
T
F
€
T
F
¬A
F
F
T
T
¬ A →B
€
€
A
T
T
€
F
F
€
T
T
T
F
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¬B
B ¬A
T
F
F
F
F
T
€
€ €
T
T
F
F
T
T
¬ A ∨¬B
F
T
€€ €
T
T
¬ (¬A ∨¬ B)
T
F
F
F
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propositional logic
propositional logic
Hilbert system – inference rules
Hilbert system – inference rules
only one inference rule
+ three axioms (expressions that are accepted as
being true under any evaluation)
for any expressions A and B:
from A and A →B we can infer B
PH1: A → (B → A)
PH2: [A → (B → C)] → [(A → B) → (A → C)]
PH3: ¬ ( ¬ A) → A
it is known as MODUS PONENS (MP)
€
€
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€
€
€
€ €
€
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€
€
€
€
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propositional logic
propositional logic
Hilbert system – inference rules
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:
(i) C is En
(ii) 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
proof of r from p
hypotheses: p → q, q → r
1p
2 p →q
€
€
3q
4 q →r
€
5r
(hypothesis)
(hypothesis)
(from 1 and 2 by MP)
(hypothesis)
(from 3 and 4 by MP)
€
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propositional logic
propositional logic
Hilbert system – inference rules
Hilbert system – inference rules
proof of A → A
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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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 logic
propositional logic
Hilbert system – inference rules
Hilbert system – inference rules
deduction theorem
if A1, …, An
proof of A → ¬ ¬ A
B then A1, …, An-1
An → B
1 A, A → ⊥ ⊥
€ €€
2 A (A → ⊥ ) → ⊥
3 A → ((A → ⊥) → ⊥
€ € €
4 A → ¬ ¬A
€
€ € € €
€
€ € € €
€ €€
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propositional logic
€
€ € € € €
€
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Hilbert system – soundness
⊥→ A
1 ⊥, A → ⊥ ⊥
2 ⊥€ (A → ⊥ ) → ⊥
3 ⊥ [(A → ⊥ ) → ⊥] → A
€ € €
4⊥ A
€ € € €
5 ⊥→ A
€
€
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propositional logic
Hilbert system – inference rules
proof of
(MP)
(DT)
(DT)
(A →⊥≡¬A)
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)
(DT)
(PH3)
(MP on 2, 3)
(DT)
€
€
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propositional logic
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propositional logic
Hilbert system – soundness definition
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
if A1, …, An A then A = T for any valuation
function V for which all the Ai are T
€ €
€ €
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propositional logic
propositional logic
Hilbert system – completeness definition
if A is valid then
Hilbert system – consistency definition
A
there is no proof of ⊥
€
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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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