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 ...
The Boolean Connectives
Logik f¨ur Informatiker Logic for computer scientists Till Mossakowski
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Logic
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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 Till Mossakowski
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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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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 Till Mossakowski
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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 Till Mossakowski
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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
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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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Logically possible
Logically impossible P ∧ ¬P a 6= a
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Logically and physically possible
Logically necessary P ∨ ¬P a=a
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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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