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EE1J2 – Discrete Maths Lecture 11 Introduction to Predicate Logic

Limitations of Propositional Logic Predicates, quantifiers and propositions NL interpretation of statements in Predicate

Logic Formalisation in Predicate Logic of statements

in NL Some equivalences

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Limitations of propositional logic Consider the following statements

All French people speak French Some French people speak French There are people in France who don’t speak

French Although there are clear relations between

these statements, these are not exposed by propositional logic

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Limitations of propositional logic (continued)

Consider also: ‘person x’ speaks French

This is not an elementary proposition because its truth or falsehood depends on x

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Examples from arithmetic There are many examples of this type in arithmetic:

7 3 = 21 7 3 = 23

are both elementary propositions, but: 7 x = 21

is not, because its truth or falsehood depends on the value of x

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Predicates

In each of the examples, truth or falsehood depends on whether or not a particular property is satisfied

In formal logic we use the term predicate to indicate a property which a variable may or may not have

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Example Predicates From the first examples we have the predicates:

P(x) = ‘x is French’ Q(x) = ‘x speaks French’

In the example from arithmetic R(x) = ‘7x = 21’ S(x) = ‘sin(x) 1’

These are examples of one-place, or unary predicates – I.e. they have just one argument

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N -place Predicates There are also two-place, or binary predicates, e.g:

P(x,y) = ‘3x + 5y = 25’ Q(x,y) = ‘sin(x) = sin(y)’

and 3-place predicates: R(x,y,z) = ‘6x + 3y2 + z = 21’ S(x,y,z) = ‘x2 + y2 z2

and, in general, N-place predicates

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Instantiation Once we choose values for its variables, a

predicate becomes a proposition For example:

R(x) = ‘7x = 21’ (from slide 6). So R(3) is the proposition ‘73=21’, which is true, but R(4) is the proposition ‘74=21’ which is false

P(x,y) = ‘3x + 5y = 25’ (from slide 7). So P(6,7) is the proposition ‘3 6 + 5 7 = 25’, which is false

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Quantifiers

Statements involving predicates can also be made into propositions using quantifiers: x ‘for all x’, ‘for every x’ x ‘there exists an x’, ‘for some x’

For example, if P(x) = ‘sin(x) 1’, then x:P(x)

is a proposition, which is true

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Example 1 As a second example, suppose

P(x,y) = ‘3x + 5y = 25’

then

x y:P(x,y)

is a proposition:

“There exists x and there exists y such that 3x+5y=25”

The proposition is true: e.g. take x=5 and y=2

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Example 2 Now consider:

xy: P(x,y) In ‘natural language’the proposition is:

“For every integer x there exists an integer y such that 3x+5y=25”

Assuming that the values of x and y are integers, the proposition is false (why?)

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Example 3

Now consider the 3-place predicate P(x,y,z) = ‘x2 + 5y + z3 = 27’

and consider the statement:

xy : P(x,y,z)

This is not a proposition (why?)

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NL interpretation of statements in Predicate Logic

Consider the following predicates: P(x) = ‘x is French’, Q(x) = ‘x was born in Paris’ R(x) = ‘x speaks French’, S(x,y) = ‘x is related to y’

Express the following as statements in ‘NL’ x : P(x) Q(x) “For every x, if x is French then x was born in Paris’ “Everyone who is French was born in Paris” (Better!) “All French people were born in Paris”

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NL interpretation (continued) P(x) = ‘x is French’, Q(x) = ‘x was born in Paris’ R(x) = ‘x speaks French’, S(x,y) = ‘x is related to y’

xy : P(x)P(y) S(x,y) “For every person x there is a person y such that x is

French and y is French and x and y are not related” “For every French person there is another French person

who they are not related to” “No French person is related to every other French

Person”

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NL Interpretation P(x) = ‘x is French’, Q(x) = ‘x was born in Paris’ R(x) = ‘x speaks French’, S(x,y) = ‘x is related to y’

x : Q(x) R(x) y : P(y) (x(Q(x) S(x,y))R(x)) x y : (P(x) P(y)) S(x,y)

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Formalisation of NL P(x) = ‘x is French’, Q(x) = ‘x was born in Paris’ R(x) = ‘x speaks French’, S(x,y) = ‘x is related to y’

“Everyone who was born in Paris speaks French” x : Q(x) R(x)

“There is at least one French person whose relatives all speak French” x : P(x) (y (S(x,y) R(y))

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Formalisation of NL (cont.) P(x) = ‘x is French’, Q(x) = ‘x was born in Paris’ R(x) = ‘x speaks French’, S(x,y) = ‘x is related to y’

“There is a French person all of whose relatives are also French”

“All French people who were not born in Paris cannot speak French”

“There is at least one French person who was neither born in Paris nor speaks French”

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Proofs involving To show that the statement xP(x) is true,

we have to show that P(a) is true for every possible a

Conversely, if P(a) is true for every possible a then xP(x) is true

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Proofs involving To show that the statement xP(x) is true, we

have to show that P(a) is true for at least one a Conversely, if P(a) is true for some a then

xP(x) is true

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Some equivalences

Let P be a predicate Suppose (xP(x))

Then it is not true that P(x) holds for every x So, there must be at least one x for which P(x)

is false So x(P(x)) is true

Hence (xP(x)) x(P(x))

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Equivalences (continued)

Suppose (xP(x)) Then it is not true that P(x) holds for at least

one x So, P(x) is false for every x So x(P(x)) is true

Hence ( xP(x)) x(P(x))

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More equivalences

x(P(x) Q(x)) xP(x) xQ(x) x(P(x) Q(x)) xP(x) xQ(x) If xP(x) xQ(x) then x(P(x) Q(x)) If x(P(x) Q(x)) then xP(x) xQ(x)

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Summary Introduction to predicate logic Introduction to N-place predicates Turning predicates into propositions using

instantiation or quantifiers NL interpretation of statements in predicate

logic (and vice versa) Some equivalences