Predicate Logic18. Free and bound variables

The Lecture

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Free and bound

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Free and bound

! Variables play two different roles in predicate logic.

! The meaning of !x(xEy) is that y has a neighbor. This is a property of y and may be true or false depending on what y is.

! The role of x in !x(xEy) is to bind the quantifier !x and the formula xEy together.

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Bound occurrence 1

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Bound occurrence 1

! Every occurrence of a variable x in a formula of the form !xB or of the form "xB is called a bound occurrence. Occurrences which are not bound are called free.

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Bound occurrence 1

! Every occurrence of a variable x in a formula of the form !xB or of the form "xB is called a bound occurrence. Occurrences which are not bound are called free.

!x(xEy & "z(zEy!z=x))

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Bound occurrence 1

! Every occurrence of a variable x in a formula of the form !xB or of the form "xB is called a bound occurrence. Occurrences which are not bound are called free.

!x(xEy & "z(zEy!z=x))

Bound occurrence

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Bound occurrence 2

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Bound occurrence 2

!x(xEy & !y(¬yEx))

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Bound occurrence 2

!x(xEy & !y(¬yEx))

y boundy free

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Assignments and free variables

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Assignments and free variables

! Whether an assignment s satisfies a formula in a model or not, depends only on the values of s on variables that occur free in the formula.

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Assignments and free variables

! Whether an assignment s satisfies a formula in a model or not, depends only on the values of s on variables that occur free in the formula.

! Whether s satisfes !x(xEy & !y(¬yEx)) or not, depends only on s(y), not on s(x).

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Sentences

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Sentences

! Some formulas have no free variables. They are called sentences.

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Sentences

! Some formulas have no free variables. They are called sentences.

! "y!x(xEy & !z(¬zEx)) is a sentence.

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Sentences

! Some formulas have no free variables. They are called sentences.

! "y!x(xEy & !z(¬zEx)) is a sentence.! "y!x(xEy & !z(¬zEx)) says of a graph

that every vertex has a neighbor with a non-neighbor.

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Truth

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Truth

! Sentences are true or false in a structure, according to whether some (equivalently, all) assignments satisfy them.

Last viewedJouko Väänänen: Propositional logic

Truth

! Sentences are true or false in a structure, according to whether some (equivalently, all) assignments satisfy them.

! If a sentence A is true in a structure M, the structure M is called a model of the sentence A.

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Truth

! Sentences are true or false in a structure, according to whether some (equivalently, all) assignments satisfy them.

! If a sentence A is true in a structure M, the structure M is called a model of the sentence A.

! This is denoted M A.