The Boolean Connectives
Logik für Informatiker
Logic for computer scientists
Till Mossakowski
WiSe 2013/14
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The Boolean Connectives
The Boolean Connectives
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The Boolean Connectives
Negation — Truth table
P
true
false
¬P
false
true
e for negation is very simple, since you
ou commit yourself to the truth of ¬P th
self to the falsity of P. Similarly, if you co
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The Boolean Connectives
The Henkin-Hintikka game
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The Boolean Connectives
The Henkin-Hintikka game
Is a sentence true in a given world?
Players: you and the computer (Tarski’s world)
You claim that a sentence is true (or false), Tarski’s world will
claim the opposite
In each round, the sentence is reduced to a simpler one
When an atomic sentence is reached, its truth can be directly
inspected in the given world
You have a winning strategy exactly in those cases where your
claim is correct.
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The Boolean Connectives
Negation — Game rule
Form
¬P
Your commitment
either
Player to move
—
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Goal
Replace ¬P by P and
switch commitment
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A sentence P ∧ Q is true if and only if both
— Truth table
e ifConjunction
either or
both of P or Q is false. This ca
ruth table.
The Boolean Connectives
P
true
true
false
false
Q
true
false
true
false
P∧Q
true
false
false
false
World game is more interesting for conjunc
e game proceeds depends on whether you
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The Boolean Connectives
Conjunction — Game rule
Form
P ∧Q
Your commitment
TRUE
Player to move
Tarski’s World
you
FALSE
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Goal
Choose one of P,
Q that is false.
Logic
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tences P and Q of fol, atomic or not, we c
Disjunction
— Truth
new
sentence
P table
∨ Q. The sentence P ∨ Q
rue. Otherwise it is false. Here is the truth
The Boolean Connectives
P
true
true
false
false
Q
true
false
true
false
P∨Q
true
true
true
false
s for ∨ are the “duals” of those for ∧. If you
∨ Q, then Tarski’s World will make you l
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The Boolean Connectives
Disjunction — Game rule
Form
P ∨Q
Your commitment
TRUE
FALSE
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Player to move
you
Goal
Choose one of P,
Q that is true.
Tarski’s World
Logic
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The Boolean Connectives
Formalisation
Sometimes, natural language double negation means logical
single negation
The English expression and sometimes suggests a temporal
ordering; the FOL expression ∧ never does.
The English expressions but, however, yet, nonetheless, and
moreover are all stylistic variants of and.
Natural language disjunction can mean invlusive-or (∨) or
exclusive-or: A xor B ⇔ (A ∨ B) ∧ (¬A ∨ ¬B)
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The Boolean Connectives
Logical necessity
A sentence is
logically necessary, or logically valid, if it is true in all
circumstances (worlds),
logically possible, or satisfiable, if it is true in some
circumstances (worlds),
logically impossible, or unsatisfiable, if it is true in no
circumstances (worlds).
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The Boolean Connectives
Logically possible
Logically impossible
P ∧ ¬P
a 6= a
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Logically and physically possible
Logically necessary
P ∨ ¬P
a=a
Logic
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The Boolean Connectives
Logic, Boolean logic and Tarski’s world
A sentence is
logically necessary, or logically valid, if it is true in all
circumstances (worlds),
TW-necessary, if it is true in all worlds of Tarski’s world,
a tautology, if it is true in all valuations of the atomic
sentences with {TRUE, FALSE}.
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The Boolean Connectives
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