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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 sentence
others – can be taken apart into smaller parts
we 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 raised
people 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 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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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 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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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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propositional logic
Hilbert system
language …
propositions names: p, q, r, …, p0, p1, p2, …
a name for “false”:
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^
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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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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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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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propositional logic
Hilbert system - semantics
answer – truth tables
A
T
T
F
F
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B
T
F
T
F
A
B
T
F
T
T
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propositional logic
Hilbert system - semantics
a truth table of complex expressions
A
T
T
F
F
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B
T
F
T
F
B
A
T
T
F
T
A (B
A))
T
T
T
T
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propositional logic
Hilbert system - semantics
negation - Ø
A
T
F
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^
F
F
^
A
F
T
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propositional logic
Hilbert system - semantics
disjunction Ú
A
T
T
F
F
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B
T
F
T
F
ØA
F
F
T
T
ØA
B
T
T
T
F
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propositional logic
Hilbert system - semantics
conjunction Ù
A
T
T
F
F
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B
T
F
T
F
ØA
ØB
F
F
T
T
F
T
F
T
ØA ÚØB
F
T
T
T
Ø(ØA ÚØ B)
T
F
F
F
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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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propositional logic
Hilbert system – inference rules
+ three axioms (expressions that are accepted as
being true under any evaluation)
PH1: A (B
PH2: [A (B
PH3: Ø ( Ø A)
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A)
C)]
A
[(A
B)
(A
C)]
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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:
(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
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propositional logic
Hilbert system – inference rules
proof of r from p
hypotheses: p q, q
1p
2p q
3q
4q r
5r
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r
(hypothesis)
(hypothesis)
(from 1 and 2 by MP)
(hypothesis)
(from 3 and 4 by MP)
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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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propositional logic
Hilbert system – inference rules
proof of A
1
2
3
4
5
[A
[A
[A
(A
(A
A
((A A)
((A A)
(A A)]
(A A))
A)
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A)] [(A
A)]
(A A)
(A
A))
(A
A)] (PH2)
(PH1)
(MP on 1, 2)
(PH1)
(MP on 3, 4)
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propositional logic
Hilbert system – inference rules
deduction theorem
if A1, …, An
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B then A1, …, An-1
An
B
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propositional logic
Hilbert system – inference rules
proof of A
ØØ A
1 A, A ^ ^
2 A (A ^ )
^
3 A ((A
^)
^
4 A Ø ØA
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(A
(MP)
(DT)
(DT)
A)
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propositional logic
Hilbert system – inference rules
proof of
1
2
3
4
5
^,
^
^
^
A ^ ^
^)
(A
[(A ^ )
A
A
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A
^
^]
A
(DT)
(PH3)
(MP on 2, 3)
(DT)
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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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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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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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propositional logic
Hilbert system – completeness definition
if A is valid then
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A
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propositional logic
Hilbert system – consistency definition
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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