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EE1J2 – Discrete Maths Lecture 6 Limitations of propositional logic Introduction to predicate logic Symbols, terms and formulae, Parse trees Formalising NL statements in predicate logic Standard equivalences in predicate logic Propositional Logic revisited The Completeness Theorem

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DNF - Example

Let p, q and r be atomic propositions

Consider f = (p(q r)) ((p q) r)

How do we put this in disjunctive normal form?

Use the construction from the proof of the theorem.

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Truth table for f(p (q r)) ((p q) r)

T T T T T T T T T T T

T F T F F T T T T F F

T T F T T T T F F T T

T T F T F T T F F T F

F T T T T T F T T T T

F T T F F F F T T F F

F T F T T T F T F T T

F T F T F F F T F F F

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Example (continued) From row 1: (pq r) From row 2: (pq r) From row 3: (pqr) From row 4: (pqr) From row 5: (pqr) From row 7: (pqr) Hence the desired formula is:(pq r)(pq r)(pqr)(pqr)

(pqr)(pqr)

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Switching Circuits Connections between propositional logic and

switching circuits Can think of a truth table as indicating the ‘output’

of a particular circuit once its inputs have been set to ‘On’ or ‘Off’

Now know that any desired behaviour can be obtained provided that the gates of the circuit can instantiate the connectives , and

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nand and nor gates Most common gates are nand gates and nor

gates. Their truth tables are given by Truth tables for nand and nor

p q p nand q p nor q

T T F F

T F T F

F T T F

F F T T

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Theorem 3Adequacy of nand and nor Theorem: The sets {nand} and {nor} are both

adequate Proof

{nand}: Since {, } is adequate, enough to show that and can be expressed in terms of nand.

Let p and q be atomic propositions. Then:

p p nand p

and

p q (p nand q) nand (p nand q)

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Proof (continued) For {nor}: It is enough to notice that:

p p nor p

p q (p nor p) nor (q nor q)

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Arithmetic Semantics of a system like arithmetic

cannot be expressed in propositional logic. We’ve already discussed formulae such as

(x < 5). The truth or falsehood of this formula

obviously depends on the value of x

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The qualifiers and A more powerful language for describing

arithmetic is provided by the introduction of the qualifiers and .

(x) (x < 5) is clearly true (there exists an x such that x < 5)

(x) (x < 5) is clearly false (for all x, x < 5)

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The Predicates = and < Formally, the symbols = and < are (binary)

predicates, since they express properties of pairs of terms in arithmetic

So, instead of x < 5 could write P(x)where P(x) denotes the property x < 5

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The functions + and The symbols + and are (binary)

functions A (binary) function associates pairs of

elements in the domain with single elements in the domain

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Predicate Logic: Syntax Predicate logic is defined in terms three

types of ingredient: Symbols, Terms, and Formulae

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Symbols of Predicate Logic The symbols of predicate logic are

Logical symbols: , , , , and Brackets: (, ) Variables: x, y, z, x0, x1,…

Constants: a, b, c,… Functions: f, g (number of arguments specified) Predicate symbols: P, Q, R,… (number of arguments

specified) Contradiction formula:

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Terms of Predicate Logic The terms of predicate logic are defined as

follows: Any variable or constant is a term If f is an n place function and t1,…,tn are terms,

then f(t1,…,tn) is a term

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Formulae of Predicate Logic If R is an n place predicate and t1,…,tn are

terms, then R(t1,…,tn) is a formula (called an

atomic formula is a formula If A and B are formulae, then so are A,

AB, AB, AB, (x)(A), (x)(A)

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Parse Tree: Formulae Suppose

P is a unary predicate (i.e. a predicate that has one argument),

S is the successor function from arithmetic, which associates each natural number with the next biggest natural number (I.e. S(n)=n+1 )

n is a variable. Consider the formula

P(0) {(n)(P(n) P(S(n)))} (n)P(n)

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Parse Tree for Formula

n

P(n)P(0) n

P(n) P(sn) Parse tree for the predicate logic formula

P(0) {(n)(P(n) P(S(n)))} (n)P(n).

The tree shows how the formula is built up from atomic formulae

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Formalising ‘NL’ statements in Predicate Logic

“Everyone in the room spoke French or German” In propositional logic look for atomic

propositions In predicate logic look for predicates

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Identification of predicates

Everyone in the room spoke French or German

Three predicates: R(x) : x was a person in the room F(x) : x spoke French G(x) : x spoke German

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Formalisation Re-state natural language statement in a

form closer to predicate logic:

For every x, if x was a person in the room, then x spoke French or x spoke German

Formally:

(x)(P(x) (F(x) G(x)))

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Another example:

“Everyone in the room spoke French or German” becomes:

(x)(P(x) (F(x) G(x)))

“Someone in the room spoke French or German” becomes:

(x)(P(x) (F(x) G(x)))

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The Scope of a Quantifier In a formula (x)() or (x)(), the formula is

called the scope of the quantifier x or x The variable x is bound if it falls immediately to

the right of or , or within the scope of the qualifier x or x

If a variable is not bound, then it is free A formula with no free variables is called a

sentence

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Example Consider the formula

{(x)(P(x,y))}Q(x,y) First two occurrences of the variable x are

bound and the third is free All occurrences of y are free This formula is not a sentence

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Example(x)(x = y x2+x < 5 x < 5) x = 5y2

The variable x occurs 6 times:- First 5 are bound, last is free- All occurrences of y and z are free- Bound occurrences of x can be substituted by

other symbols (not y or z, though)- y and z cannot be substitutedHence, original formula is equivalent to(a)(a = y a2+a < 5 a < 5) x = 5y2

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Semantics in Predicate Logic Two formulae f and g in predicate logic are

said to be logically equivalent if for any interpretation, f is true if and only g is true.

In this case we write f g. This is not rigorous, but will do for now!

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Some equivalences f and g are formulae in predicate logic x and y are variables such that

x does not occur bound in f or at all in g y does not occur at all in f or g

Then: x(f(x)) x(f(x)), x(f(x)) x(f(x)) (xf ) g (x fg), (x f)g x (f g)

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More equivalences… (xf ) g (x f g), (x f) g x (f g)

(xf ) g x( f g), and (xf ) g x( f g)

f (xg) x( f g), and f (xg) x( f g)

xf(x) yf(y), xf(x) yf(y)

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Proofs of equivalences x(f(x)) x(f(x))

Suppose x(f(x)) Then it is not true that f(x) holds for all x Therefore there must be at least one x for which

it does not hold, i.e. for which f(x) is false Or, formally, x(f(x))

Conversely, suppose x(f(x)).

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Proof (continued) x(f(x)) x(f(x))

suppose x(f(x)) Then there exists an x such that f(x) is false So it is not true that f(x) holds for all x …which is the meaning of x(f(x))

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Analysis of Arguments in Propositional Logic So far, our analysis of the ‘soundness’ of

an argument is based on logical consequence:

Given an argument in NL, First identify the set of formulae which are

premises Then identify the formula f which is the

consequence Then demonstrate that ⊨f (or not)

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Method We have two techniques to prove or disprove logical

consequence: Proof by contradiction:

To show that ⊨f is true, assume there is a truth function such that each formula in is true and f is false. Deduce that one of the formulae in must be false – contradiction!

Proof by counter-example: To show that ⊨f is false, find a truth function such

that each formula in is true but f is false

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‘Syntactic’ Proof ‘Logical Consequence’ is defined in terms

of the semantics of Propositional Logic An alternative definition of the soundness

of an argument might be based solely on the syntax of Propositional Logic

Closer to the intuitive notion of proof

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Proofs in Propositional Logic In mathematics, a proof is a sequence of

steps, each of which is: Self-evident (or axiomatic), or An explicitly stated assumption, or Deduced from previous steps by sound logical

reasoning

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Completeness Theorem for Propositional Logic If is a set of formulae in Propositional

Logic, and f is a formula in Propositional Logic, then ⊨f if and only if ⊢f

f is a logical consequence of

if and only if

f is provable from

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Summary of Lecture 6 Limitations of propositional logic Introduction to predicate logic Formalising NL statements in predicate

logic Standard equivalences in predicate logic Propositional Logic revisited The Completeness Theorem