28. predicate logic lidity - helsingin yliopisto · 2011-04-01 · last viewed deductions! en a...

28. Predicate LogicValidity

The Lecture

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Validity

Jouko Väänänen: Predicate logic

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ValidityIn everyday language a person utters a validity if he or she says something which is true but only because of its form, like every day is rainy or else some days are not rainy.


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ValidityIn everyday language a person utters a validity if he or she says something which is true but only because of its form, like every day is rainy or else some days are not rainy.A formula of predicate logic is valid, or a validity, if it is satisfied by every assignment in every structure.


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ValidityIn everyday language a person utters a validity if he or she says something which is true but only because of its form, like every day is rainy or else some days are not rainy.A formula of predicate logic is valid, or a validity, if it is satisfied by every assignment in every structure. Examples

xA ¬ x¬A xA ¬ x¬Ax(A B) xA xBx(A B) xA xBx x=x (identity axioms have the special role that they are

always assumed when �”=�” is part of the formula.)


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Satisfiable


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Satisfiable

A formula is satisfiable if it is satisfied by some assignment in some structure.


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Satisfiable

A formula is satisfiable if it is satisfied by some assignment in some structure.Examples


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Satisfiable

A formula is satisfiable if it is satisfied by some assignment in some structure.Examples

�• x yR0(x,y) ¬ y xR0(x,y)

�• x(P0(x) P1(x)) (¬ xP0(x) ¬ xP1(x))

�•¬( xP0(x) xP0(x))


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Contingent


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ContingentA formula is contingent if it is both consistent and refutable. A person utters a contingency, like �”It is raining�” if what he or she says can be true but can also be false.


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ContingentA formula is contingent if it is both consistent and refutable. A person utters a contingency, like �”It is raining�” if what he or she says can be true but can also be false.Examples

xP0(x)

x yR0(x,y)

x(R0(x,y) P0(x))

x=y (depending on the assignment this can be true or false)


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Refutable


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Refutable

A formula is refutable if there are an assignment s and a structure M that refute it, i.e. The assignment s does not satisfy the formula in M. It is like someone says �”Every day in August is rainy�” and you refute it by pointing out that in the year 1996 there was a day in August when it did not rain.


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Refutable

A formula is refutable if there are an assignment s and a structure M that refute it, i.e. The assignment s does not satisfy the formula in M. It is like someone says �”Every day in August is rainy�” and you refute it by pointing out that in the year 1996 there was a day in August when it did not rain.

ExamplesxP0(x) xP0(x)

x yR0(x,y) y xR0(x,y)

x=y


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Contradiction


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ContradictionIn everyday language a person utters a contradiction if he or she says something which is false merely because of its form, like Every day is sunny but August 15 is not sunny.


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ContradictionIn everyday language a person utters a contradiction if he or she says something which is false merely because of its form, like Every day is sunny but August 15 is not sunny.A formula of predicate logic is a contradiction if it is not satisfied by any assignment in any structure.


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ContradictionIn everyday language a person utters a contradiction if he or she says something which is false merely because of its form, like Every day is sunny but August 15 is not sunny.A formula of predicate logic is a contradiction if it is not satisfied by any assignment in any structure.Examples

xA x¬A xA x¬Ax(A B) x(A ¬B)

x=y x=z ¬y=z (Identity has a special role. We always assume the identity axioms if = is present. Therefore this formula is a contradiction.)


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Categories of formulas of predicate logic

Contradictions

Validities

Contingencies


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Every formula of predicate logic is either a validity, a contradiction or a contingency.

Contradictions

Validities

Contingencies


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Every formula of predicate logic is either a validity, a contradiction or a contingency.Every satisfiable formula is either valid or contingent.

Contradictions

Validities

Contingencies


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Every formula of predicate logic is either a validity, a contradiction or a contingency.Every satisfiable formula is either valid or contingent.Every refutable formula is either a contradiction or a contingency

Contradictions

Validities

Contingencies


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Hard question


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Hard question

Given a formula, can you decide mechanically whether it is a validity, a contradiction or a contingency?


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Hard question

Given a formula, can you decide mechanically whether it is a validity, a contradiction or a contingency?It can be proved that this is not possible (if mechanically is interpreted as is now common).


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The method of deductions


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�•Given a potential deduction for a formula A it is not difficult to check whether the deduction is correct or not.


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�•Given a potential deduction for a formula A it is not difficult to check whether the deduction is correct or not.�•This can be done mechanically. Such computer programs are called proof checkers.


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�•Given a potential deduction for a formula A it is not difficult to check whether the deduction is correct or not.�•This can be done mechanically. Such computer programs are called proof checkers.�•One can make a list of all possible deductions and check them one by one.


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�•Given a potential deduction for a formula A it is not difficult to check whether the deduction is correct or not.�•This can be done mechanically. Such computer programs are called proof checkers.�•One can make a list of all possible deductions and check them one by one.�•The hard case is when there is no deduction, so it takes infinite time to find it out.


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Equivalence of formulas


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Two formulas of predicate logic, A and B, are called (logically) equivalent if A B is a validity.


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Two formulas of predicate logic, A and B, are called (logically) equivalent if A B is a validity.

Equivalence of formulas is used in everyday language and in science all the time, often without paying much attention to it.


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Formula Equivalent formula

Condition

xA ¬ x¬A

xA ¬ x¬A

x(A B) xA xB

x(A B) xA xB

x yA y xA

x yA y xA

x(A B) xA B x not free in B

x(A B) A xB x not free in A

x(A B) xA B x not free in B

Equivalent formulas of predicate logic


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The method

Jouko Väänänen: Propositional logic

28. predicate logic lidity - helsingin yliopisto · 2011-04-01 · last viewed deductions! en a...

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