lecture 7: propositional logic (1) - … 7 software engineering 1 what is a logic? when most people...

LECTURE 7: PROPOSITIONALLOGIC (1)

Software Engineering

Mike Wooldridge

Lecture 7 Software Engineering

1 What is a Logic?

•When most people say ‘logic’, they meaneither propositional logic or first-orderpredicate logic.

• However, the precise definition is quitebroad, and literally hundreds of logicshave been studied by philosophers,computer scientists and mathematicians.

• Any ‘formal system’ can be considered alogic if it has:

– a well-defined syntax;– a well-defined semantics; and– a well-defined proof-theory.

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• The syntax of a logic defines thesyntactically acceptable objects of thelanguage, which are properly calledwell-formed formulae (wff). (We shall justcall them formulae.)

• The semantics of a logic associate eachformula with a meaning.

• The proof theory is concerned withmanipulating formulae according tocertain rules.

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2 Propositional Logic

• The simplest, and most abstract logic wecan study is called propositional logic.

• Definition: A proposition is a statementthat can be either true or false; it must beone or the other, and it cannot be both.

• EXAMPLES. The following arepropositions:

– the reactor is on;– the wing-flaps are up;– John Major is prime minister.

whereas the following are not:

– are you going out somewhere?– 2+3

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• It is possible to determine whether anygiven statement is a proposition byprefixing it with:

It is true that . . .

and seeing whether the result makesgrammatical sense.

•We now define atomic propositions.Intuitively, these are the set of smallestpropositions.

• Definition: An atomic proposition is onewhose truth or falsity does not depend onthe truth or falsity of any otherproposition.

• So all the above propositions are atomic.

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• Now, rather than write out propositions infull, we will abbreviate them by usingpropositional variables.

• It is standard practice to use thelower-case roman letters

p, q, r, . . .

to stand for propositions.

• If we do this, we must define what wemean by writing something like:

Let p be John Major is prime Minister.

• Another alternative is to write somethinglike reactor is on, so that the interpretationof the propositional variable becomesobvious.

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2.1 The Connectives

• Now, the study of atomic propositions ispretty boring. We therefore now introducea number of connectives which will allowus to build up complex propositions.

• The connectives we introduce are:

∧ and (& or .)∨ or (| or +)¬ not (∼)⇒ implies (⊃ or→)⇔ iff

• (Some books use other notations; these aregiven in parentheses.)

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And

• Any two propositions can be combined toform a third proposition called theconjunction of the original propositions.

• Definition: If p and q are arbitrarypropositions, then the conjunction of p andq is written

p ∧ q

and will be true iff both p and q are true.

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•We can summarise the operation of ∧ in atruth table. The idea of a truth table forsome formula is that it describes thebehaviour of a formula under all possibleinterpretations of the primitivepropositions the are included in theformula.

• If there are n different atomic propositionsin some formula, then there are 2n differentlines in the truth table for that formula.(This is because each proposition can takeone 1 of 2 values — true or false.)

• Let us write T for truth, and F for falsity.Then the truth table for p ∧ q is:

p q p ∧ qF F FF T FT F FT T T

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Or

• Any two propositions can be combined bythe word ‘or’ to form a third propositioncalled the disjunction of the originals.

• Definition: If p and q are arbitrarypropositions, then the disjunction of p andq is written

p ∨ q

and will be true iff either p is true, or q istrue, or both p and q are true.

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• The operation of ∨ is summarised in thefollowing truth table:

p q p ∨ qF F FF T TT F TT T T

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If. . . Then. . .

•Many statements, particularly inmathematics, are of the form:

if p is true then q is true.

Another way of saying the same thing is towrite:

p implies q.

• In propositional logic, we have aconnective that combines two propositionsinto a new proposition called theconditional, or implication of the originals,that attempts to capture the sense of sucha statement.

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• Definition: If p and q are arbitrarypropositions, then the conditional of p and qis written

p⇒ q

and will be true iff either p is false or q istrue.

• The truth table for⇒ is:p q p⇒ qF F TF T TT F FT T T

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• The⇒ operator is the hardest tounderstand of the operators we haveconsidered so far, and yet it is extremelyimportant.

• If you find it difficult to understand, justremember that the p⇒ q means ‘if p istrue, then q is true’.If p is false, then we don’t care about q, andby default, make p⇒ q evaluate to T inthis case.

• Terminology: if φ is the formula p⇒ q,then p is the antecedent of φ and q is theconsequent.

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Iff

• Another common form of statement inmaths is:

p is true if, and only if, q is true.

• The sense of such statements is capturedusing the biconditional operator.

• Definition: If p and q are arbitrarypropositions, then the biconditional of pand q is written:

p⇔ q

and will be true iff either:

1. p and q are both true; or2. p and q are both false.

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• The truth table for⇔ is:p q p⇔ qF F TF T FT F FT T T

• If p⇔ q is true, then p and q are said to belogically equivalent. They will be true underexactly the same circumstances.

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Not

• All of the connectives we have consideredso far have been binary: they have takentwo arguments.

• The final connective we consider here isunary. It only takes one argument.

• Any proposition can be prefixed by theword ‘not’ to form a second propositioncalled the negation of the original.

• Definition: If p is an arbitrary propositionthen the negation of p is written

¬p

and will be true iff p is false.

• Truth table for ¬:p ¬pF TT F

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Comments

•We can nest complex formulae as deeply aswe want.

•We can use parentheses i.e., ),(, todisambiguate formulae.

• EXAMPLES. If p, q, r, s and t are atomicpropositions, then all of the following areformulae:

p ∧ q⇒ rp ∧ (q⇒ r)

(p ∧ (q⇒ r)) ∨ s((p ∧ (q⇒ r)) ∨ s) ∧ t

whereas none of the following is:

p ∧p ∧ q)

p¬

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3 Tautologies & Consistency

• Given a particular formula, can you tell ifit is true or not?

• No — you usually need to know the truthvalues of the component atomicpropositions in order to be able to tellwhether a formula is true.

• Definition: A valuation is a function whichassigns a truth value to each primitiveproposition.

• In Modula-2, we might write:

PROCEDURE Val(p : AtomicProp):BOOLEAN;

• Given a valuation, we can say for anyformula whether it is true or false.

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• EXAMPLE. Suppose we have a valuationv, such that:

v(p) = Fv(q) = Tv(r) = F

Then we truth value of (p ∨ q)⇒ r isevaluated by:

(v(p) ∨ v(q))⇒ v(r) (1)= (F ∨ T)⇒ F (2)

= T ⇒ F (3)= F (4)

Line (3) is justified since we know thatF ∨ T = T.Line (4) is justified since T ⇒ F = F.If you can’t see this, look at the truth tablesfor ∨ and⇒.

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•When we consider formulae in terms ofinterpretations, it turns out that some haveinteresting properties.

• Definition:

1. A formula is a tautology iff it is trueunder every valuation;

2. A formula is consistent iff it is trueunder at least one valuation;

3. A formula is inconsistent iff it is notmade true under any valuation.

• Now, each line in the truth table of aformula correponds to a valuation.

• So, we can use truth tables to determinewhether or not formulae are tautologies.

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lecture 7: propositional logic (1) - … 7 software engineering 1 what is a logic? when most people...

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