fall 2013 foundations of logic

Propositional Logic

Rosen 5Rosen 5thth ed., §§1.1-1.2 ed., §§1.1-1.2

Foundations of Logic

Mathematical Logic is a tool for handling compound statements. It includes:– A formal language for expressing them.– A methodology for reasoning about their truth

or falsity. (A “calculus”)

The ultimate foundation for proofs throughout mathematics

CS2013 Univ of Aberdeen 2

Two Logical Systems:

1.1. Propositional logicPropositional logic2.2. Predicate logicPredicate logic (extends (extends 1.1. ) )

Many other logical calculi exist, Many other logical calculi exist, but they all resemble these twobut they all resemble these two


Propositional Logic (§1.1)

Propositional LogicPropositional Logic is the logic of compound is the logic of compound statements built from simpler statements statements built from simpler statements using using BooleanBoolean connectives.connectives.

Some applications in computer science:Some applications in computer science:• Design of digital electronic circuits.Design of digital electronic circuits.• Expressing conditions in programs.Expressing conditions in programs.• Queries to databases & search engines.Queries to databases & search engines.

George Boole(1815-1864)

Chrysippus of Soli(ca. 281 B.C. – 205 B.C.)CS2013 Univ of Aberdeen 4

Propositions in natural language

In propositional logic, a In propositional logic, a propositionproposition is simply: is simply:• a a statement statement ((i.e.i.e., a declarative sentence), a declarative sentence)

– with some definite meaningwith some definite meaning• having a having a truth valuetruth value that that’’s either s either truetrue ( (TT) or ) or falsefalse

((FF). Only values statements can have.). Only values statements can have.– Never both, or somewhere Never both, or somewhere ““in betweenin between””. .

However, you might not However, you might not knowknow the the truth valuetruth value


Examples of NL Propositions

• ““It is raining.It is raining.”” (In a given situation.)(In a given situation.)• ““Beijing is the capital of China, and 1 + 2 = 2Beijing is the capital of China, and 1 + 2 = 2””

The following are The following are NOTNOT propositions: propositions:• ““WhoWho’’s there?s there?”” (interrogative: no truth value)(interrogative: no truth value)• ““x := x+1x := x+1”” (imperative: no truth value)(imperative: no truth value)• ““1 + 21 + 2”” (term: no truth value)(term: no truth value)


Propositions in Propositional Logic

• Atoms: Atoms: pp, , qq, , rr, …, …(Corresponds with simple English sentences, e.g.(Corresponds with simple English sentences, e.g.‘‘I had salad for lunchI had salad for lunch’’))

• Complex propositions : built up from atoms using Complex propositions : built up from atoms using operators: operators: ppqq (Corresponds with compound English sentences, (Corresponds with compound English sentences, e.g., e.g., ““I had salad for lunch I had salad for lunch andand I had steak for I had steak for dinner.dinner.””))


Defining Propositions

• Logic defines notions of atomic and Logic defines notions of atomic and complex propositions and what complex complex propositions and what complex propositions “propositions “mean”mean”


A A connectiveconnective combines with combines with nn arguments arguments to to form a larger expression.form a larger expression.

• UnaryUnary connectives take 1 operand; connectives take 1 operand;• Binary Binary connectives take 2 operands.connectives take 2 operands.

Connectives (also called operators)


Common Boolean Operators

Formal NameFormal Name NicknameNickname ArityArity SymbolSymbol

Negation operatorNegation operator NOTNOT UnaryUnary ¬¬Conjunction operatorConjunction operator ANDAND BinaryBinary Disjunction operatorDisjunction operator OROR BinaryBinary Exclusive-OR operatorExclusive-OR operator XORXOR BinaryBinary Implication operatorImplication operator IMPLIESIMPLIES BinaryBinary Biconditional operatorBiconditional operator IFFIFF BinaryBinary ↔↔


The syntax of propositional logic

• Atoms: p1, p2, p3, ..Atoms: p1, p2, p3, ..• Formulas:Formulas:

– All atoms are formulasAll atoms are formulas– For all For all , if , if is a formula then is a formula then ¬ ¬ is a is a

formulaformula– For all For all and and , if , if and and are formulas then the are formulas then the

following are formulas: (following are formulas: ( ), (), ( ), (), ( ) ) (etc.)(etc.)


The language of propositional logic defined more properly (i.e., as a formal language)

• Which of these are formulas, according to Which of these are formulas, according to this strict definition?this strict definition?

• p1 p1 ¬¬ p2 p2 • (p1 (p1 ¬¬ p2) p2) • ¬ ¬ ¬¬ ¬ ¬(p9 (p9 p8) p8)• (p(p11 p2 p2 p3) p3)• (p(p11 (p2 (p2 p3)) p3))



• Which of these are formulas, Which of these are formulas, according to according to this strict definitionthis strict definition??

• p1 p1 ¬¬ p2 p2 NoNo• (p1 (p1 ¬¬ p2) p2) YesYes• ¬ ¬ ¬¬ ¬ ¬(p9 (p9 p8) p8) YesYes• (p(p11 p2 p2 p3) p3) NoNo• (p(p11 (p2 (p2 p3)) p3)) YesYes


Simplifying conventions

• Convention 1: Convention 1: outermost brackets may be outermost brackets may be omitted,:omitted,:p1 p1 ¬¬ p2, p2, ¬ ¬ ¬¬ ¬ ¬(p9 (p9 p8), p p8), p11 (p2 (p2 p3) p3)

• Convention 2: Convention 2: associativity allows us to associativity allows us to omit even more brackets, e.g.:omit even more brackets, e.g.:pp11 p2 p2 p3, p3, pp1 1 p2 p2 p3 p3



• Which of these are formulas, Which of these are formulas, when using when using these two conventionsthese two conventions??

• p1 p1 ¬¬ p2 p2 YesYes• (p1 (p1 ¬¬ p2) p2) YesYes• ¬ ¬ ¬¬ ¬ ¬(p9 (p9 p8) p8) YesYes• (p(p11 p2 p2 p3) p3) NoNo• (p(p11 (p2 (p2 p3)) p3)) YesYes


The semantics of Propositional Logic


Propositional Logic is a very simplePropositional Logic is a very simpleformalism, becauseformalism, because

The meaning of a proposition can only be The meaning of a proposition can only be expressed in terms of the expressed in terms of the truth values truth values True (also: T, 1) and False (also: F, 0).True (also: T, 1) and False (also: F, 0).

This may not always always allow you toThis may not always always allow you tosay what you would want to say say what you would want to say

Remember when reading definitions of connectives


The Negation Operator

The unary The unary negation operatornegation operator ““¬¬”” ( (NOTNOT) ) combines with one prop, transforming the combines with one prop, transforming the prop into its prop into its negationnegation..

E.g.E.g. If If pp = = ““I have brown hair.I have brown hair.”” then ¬then ¬pp = = ““I do I do notnot have brown hair. have brown hair.””

The The truth tabletruth table for NOT: for NOT: p p T F F T

T :≡ True; F :≡ False“:≡” means “is defined as”

Operandcolumn

ResultcolumnCS2013 Univ of Aberdeen 18

Truth-functionality

• The meaning of a connective is a The meaning of a connective is a functionfunction from (one or more) truth values to truth from (one or more) truth values to truth values values

• Truth table expresses truth/falsity of Truth table expresses truth/falsity of ¬¬pp in in terms of truth/falsity of pterms of truth/falsity of p

• This not possible for the operator This not possible for the operator ‘tomorrow’, or `probably’: ‘tomorrow’, or `probably’: – ‘‘Tomorrow p’ is true iff p is ….’??Tomorrow p’ is true iff p is ….’??


Truth-functionality• Truth table expresses truth/falsity of Truth table expresses truth/falsity of ¬¬pp in terms of in terms of

truth/falsity of p.truth/falsity of p.• Each horizontal line of the table expresses some alternative Each horizontal line of the table expresses some alternative

context.context.• Truth-functional operator: an operator that is a Truth-functional operator: an operator that is a functionfunction

from the truth values of the component expressions to a from the truth values of the component expressions to a truth value.truth value.

• NOT is truth functional. Yesterday is not.NOT is truth functional. Yesterday is not.• Propositional logic is only about truth-functional Propositional logic is only about truth-functional

operatorsoperators..• We can compute the values of the complex expressions.We can compute the values of the complex expressions.


The Conjunction Operator

The binary The binary conjunction operatorconjunction operator ““”” ( (ANDAND) ) combines two propositions to form their combines two propositions to form their logical logical conjunctionconjunction..

E.g.E.g. If If pp==““I will have salad for lunch.I will have salad for lunch.”” and and q=q=““I will have steak for dinner.I will have steak for dinner.””, then , then ppqq==““I I will have salad for lunch will have salad for lunch andand I will have steak for dinner.I will have steak for dinner.””


• Note that aNote that aconjunctionconjunctionpp11 pp2 2 … … ppnn

of of nn propositions propositionswill have 2will have 2nn rows rowsin its truth table.in its truth table.

• Also: ¬ and Also: ¬ and operations together can express operations together can express anyany Boolean truth table! Boolean truth table!

Conjunction Truth TableOperand columns


more later

The Disjunction Operator

The binary The binary disjunction operatordisjunction operator ““”” ( (OROR) ) combines two propositions to form their combines two propositions to form their logical logical disjunctiondisjunction..

pp==““My car has a bad engine.My car has a bad engine.””q=q=““My car has a bad carburator.My car has a bad carburator.””ppqq==““Either my car has a bad engine, Either my car has a bad engine, oror

my car has a bad carburetor.my car has a bad carburetor.””


• Note that Note that ppq q meansmeansthat that pp is true, or is true, or qq is istrue, true, or bothor both are true! are true!

• So, this operation isSo, this operation isalso called also called inclusive or,inclusive or,because it because it includesincludes the thepossibility that both possibility that both pp and and qq are true. are true.

• ““¬¬”” and and ““”” together are also universal. together are also universal.

Disjunction Truth Table

p q p qF F FF T TT F TT T T

Notedifferencefrom AND


Nested Propositional Expressions

• Use parentheses to Use parentheses to group sub-expressionsgroup sub-expressions::““I just saw my old I just saw my old ffriendriend, and either , and either hehe’’s s ggrownrown or or II’’ve ve sshrunkhrunk..”” = = ff ( (gg ss))((ff gg) ) ss would mean something different would mean something differentff gg ss would be ambiguous would be ambiguous

• By convention, By convention, ““¬¬”” takes takes precedenceprecedence over over both both ““”” and and ““””..¬¬s s ff means (¬ means (¬ss)) f f rather than rather than ¬ (¬ (s s ff))


Logic as shorthand for NL

Let Let pp==““It rained last nightIt rained last night””, , qq==““The sprinklers came on last night,The sprinklers came on last night,”” rr==““The lawn was wet this morning.The lawn was wet this morning.””

¬¬pp = = rr ¬ ¬pp = = ¬ ¬ r r pp q =q =

It didn't rain last night.The lawn was wet this morning, andit didn’t rain last night.Either the lawn wasn't wet this morning, or it rained last night, or the sprinklers came on last night.


Some important ideas:

• Distinguishing between semantically Distinguishing between semantically different kinds of formulasdifferent kinds of formulas

• Some formulas that look different may Some formulas that look different may express the same informationexpress the same information


Tautologies

A A tautologytautology is a compound proposition that is is a compound proposition that is truetrue no matter whatno matter what the truth values of its the truth values of its atomic propositions are!atomic propositions are!

Ex.Ex. p p pp [What is its truth table?][What is its truth table?]


Tautologies

• When every row of the truth table gives When every row of the truth table gives T..

• Example: p Example: p pp T T TT FT FT F F TT TF TF


Contradictions

A A contradiction contradiction is a compound proposition is a compound proposition that is that is falsefalse no matter what! no matter what! Ex.Ex. p p p p [Truth table?][Truth table?]


Contradictions

• When every row of the truth table gives FWhen every row of the truth table gives F

• Example: p Example: p pp T T FF FT FT F F FF TF TF


Contingencies

All other props. are All other props. are contingenciescontingencies::

Some rows give T, others give FSome rows give T, others give F

Now: formulas that have the same meaningNow: formulas that have the same meaning


Propositional Equivalence

Two syntactically different propositions may be Two syntactically different propositions may be semantically semantically identical (have the same meaning). identical (have the same meaning). We call them We call them logically equivalentlogically equivalent. .

Notation: … Notation: … … …


Logical Equivalence

Compound proposition Compound proposition pp is is logically logically equivalent equivalent to compound proposition to compound proposition qq, , written written ppqq, , IFFIFF pp and and q q contain the same truth values contain the same truth values in in allall rows of their truth tables rows of their truth tablesThey express the same They express the same truth function truth function (= the same function (= the same function fromfrom values for atoms values for atoms toto values for the whole formula). values for the whole formula).


Ex.Ex. Prove that Prove that ppqq ((p p qq).).

p q pp qq pp qq pp qq ((pp qq))F FF TT FT T

Proving Equivalencevia Truth Tables

FT

TT

T

T

T

TTT

FF

F

F

FFF

F

TT


Shows that OR is equivalent to a combination of NOT and AND.

Before introducing more connectives

• … … let us step back and ask a few questions let us step back and ask a few questions about truth tablesabout truth tables


1.1. What does each line of the table "mean"?What does each line of the table "mean"?2.2. Consider a conjunction Consider a conjunction pp11 pp2 2 pp33

How many rows are there in its truth table?How many rows are there in its truth table?3.3. Consider a conjunctionConsider a conjunction

pp11 pp2 2 … … ppnn of of nn propositions. propositions.How many rows are there in its truth table?How many rows are there in its truth table?

4.4. Explain why ¬ and Explain why ¬ and together are sufficient to together are sufficient to express express anyany Boolean truth table Boolean truth table

Questions for you to think about


1.1. Consider a conjunction Consider a conjunction pp11 pp2 2 pp33How many rows are there in its truth table? 8How many rows are there in its truth table? 8 pp11 pp2 2 pp33 1 1 11 1 1 1 1 0 1 1 0 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 0 1 1 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0



Two truth values (0,1) and three propositions:

23 = 8.

2. Consider 2. Consider pp11 pp2 2 … … ppnn How many rows are there in its truth How many rows are there in its truth table?table?

2*2*2* … *2 (n factors)2*2*2* … *2 (n factors)Hence 2Hence 2n n (This grows exponentially!)(This grows exponentially!)



3. Explain why ¬ and 3. Explain why ¬ and together are sufficient together are sufficient to express to express anyany other complex expression other complex expression in propositional logic.in propositional logic.



3.3. Explain why ¬ and Explain why ¬ and together are sufficient together are sufficient to express to express anyany other complex expression in other complex expression in propositional logic.propositional logic.

• Obviously, if we add new connectivesObviously, if we add new connectives(like (like ) ) we can write new formulas.we can write new formulas.

• CLAIM: these formulas would always be CLAIM: these formulas would always be equivalent with ones that only use equivalent with ones that only use ¬ and ¬ and (This is what we need to prove).(This is what we need to prove).



• Saying this in a different way: if we add new Saying this in a different way: if we add new connectives, we can write new formulas, but connectives, we can write new formulas, but these formulas will always only express truth these formulas will always only express truth functions that can already be expressed by functions that can already be expressed by formulas that only use formulas that only use ¬ and ¬ and ..

• That is, they will be equivalent.That is, they will be equivalent.• Example of writing a disjunction in another Example of writing a disjunction in another

form (equivalence shown before): form (equivalence shown before):

p p q q ¬(¬p ¬(¬p ¬q) ¬q)

Relating AND and OR


Mystery Operator

PQRPQR Formula (containing P,Q,R)Formula (containing P,Q,R)1 1 1 01 1 1 01 1 0 11 1 0 11 0 1 11 0 1 11 0 0 01 0 0 00 1 1 00 1 1 00 1 0 00 1 0 00 0 1 00 0 1 00 0 0 0 0 0 0 0


Suppose, given the truth values of P, Q, and R, we construct a Formula with the given resulting truth value. This is our 'mystery' operator . Can it be written equivalently with NOT and AND?

3. Explain why ¬ and 3. Explain why ¬ and together are sufficient together are sufficient to express to express anyany Boolean truth table Boolean truth table

• Suppose precisely two rows give T.Suppose precisely two rows give T.For exampleFor example, the rows where, the rows where

– P=T, Q=T, R=F. This is P P=T, Q=T, R=F. This is P Q Q ¬R ¬R– P=T, Q=F, R=T. This is P P=T, Q=F, R=T. This is P ¬Q ¬Q R R

T-values in Conjunction


Suppose precisely two rows give T.Suppose precisely two rows give T.For exampleFor example, the rows where, the rows where

– P=T, Q=T, R=F. This is P P=T, Q=T, R=F. This is P Q Q ¬R ¬R– P=T, Q=F, R=T. This is P P=T, Q=F, R=T. This is P ¬Q ¬Q R R

• We’ve proven our claim if we can express We’ve proven our claim if we can express the disjunction of these two rows: the disjunction of these two rows: (P (P Q Q ¬R) ¬R) (P (P ¬Q ¬Q R) R)

Table as a disjunction of T-rows


• We’ve arrived if we can express their We’ve arrived if we can express their disjunction: disjunction: (P (P Q Q ¬R) ¬R) (P (P ¬Q ¬Q R) R)

• But we’ve seen that disjunction can be expressed But we’ve seen that disjunction can be expressed using using and ¬: and ¬: A A B B ¬(¬A ¬(¬A ¬B) ¬B)

• So: So: (P (P Q Q ¬R) ¬R) (P (P ¬Q ¬Q R) R) ¬(¬ (P ¬(¬ (P Q Q ¬R) ¬R) (P (P ¬Q ¬Q R)) R))

• We’ve only used We’ve only used and and ..

Disjoining rows of the table


(P (P Q Q ¬R) ¬R) (P (P ¬Q ¬Q R) R) ¬(¬ (P ¬(¬ (P Q Q ¬R) ¬R) (P (P ¬Q ¬Q R)) R)) 1 1 0 01 0 1 01 0 1 0 1 1 1 0 01 1 1 1 01 0 11 1 0 01 0 1 01 0 1 0 1 1 1 0 01 1 1 1 01 0 1 1 1 1 10 1 1 01 0 0 1 0 1 1 1 10 0 1 1 01 0 01 1 1 10 1 1 01 0 0 1 0 1 1 1 10 0 1 1 01 0 0 1 0 0 01 1 1 10 1 1 1 1 1 0 0 01 0 0 1 10 1 11 0 0 01 1 1 10 1 1 1 1 1 0 0 01 0 0 1 10 1 1 1 0 0 10 0 1 10 0 0 0 1 1 0 0 10 1 1 1 10 0 01 0 0 10 0 1 10 0 0 0 1 1 0 0 10 1 1 1 10 0 0 0 1 0 01 0 0 01 0 1 0 1 0 1 0 01 1 1 0 01 0 10 1 0 01 0 0 01 0 1 0 1 0 1 0 01 1 1 0 01 0 1 0 1 0 10 0 0 01 0 0 0 1 0 1 0 10 1 1 0 01 0 00 1 0 10 0 0 01 0 0 0 1 0 1 0 10 1 1 0 01 0 0 0 0 0 01 0 0 10 0 1 0 1 0 0 0 01 1 1 0 10 0 10 0 0 01 0 0 10 0 1 0 1 0 0 0 01 1 1 0 10 0 1 0 0 0 10 0 0 10 0 0 0 1 0 0 0 10 1 1 0 10 0 00 0 0 10 0 0 10 0 0 0 1 0 0 0 10 1 1 0 10 0 0

Check


About this proof …

• We’ve made our task a bit easier, assuming We’ve made our task a bit easier, assuming that there were only 2 rows resulting in Tthat there were only 2 rows resulting in T

• But the case with 1 or 3 or 4 or …. rows is But the case with 1 or 3 or 4 or …. rows is analogous (and there are always only analogous (and there are always only finitely many rows.)finitely many rows.)

• So, the proof can be made preciseSo, the proof can be made precise


Having proved this …

• We can express every possible truth-functional operator in We can express every possible truth-functional operator in propositional logic in terms of AND and NOTpropositional logic in terms of AND and NOT

• This is sometimes called This is sometimes called functional completeness.functional completeness. Also Also universality.universality.

• Reduce other operators to other more basic operators.Reduce other operators to other more basic operators.


Let’s introduce some additional connectives

• A variant of disjunctionA variant of disjunction• The conditionalThe conditional• The biconditionalThe biconditional


The Exclusive Or Operator

The binary The binary exclusive-or operatorexclusive-or operator ““”” ( (XORXOR) ) combines two propositions to form their combines two propositions to form their logical logical ““exclusive orexclusive or””..

pp = = ““I will earn an A in this course,I will earn an A in this course,””qq = = ““I will drop this course,I will drop this course,””pp q q = = ““I will either earn an A in this course, I will either earn an A in this course,

or I will drop it (but not both!)or I will drop it (but not both!)””


• Note that Note that ppq q meansmeansthat that pp is true, or is true, or qq is istrue, but true, but not bothnot both!!

• This operation isThis operation iscalled called exclusive or,exclusive or,because it because it excludesexcludes the thepossibility that both possibility that both pp and and qq are true. are true.

• ““¬¬”” and and ““”” together are together are notnot universal. universal.

Exclusive-Or Truth Table

p q pqF F FF T TT F TT T F Note

differencefrom OR.


Note that Note that EnglishEnglish ““oror”” can be can be ambiguousambiguous regarding the regarding the ““bothboth”” case! case!

Need context to disambiguate the meaning!Need context to disambiguate the meaning!For this class, assume For this class, assume ““oror”” means means inclusiveinclusive..

Natural Language is Ambiguous

p q p "or" qF F FF T TT F TT T ?


Test your understanding of the two types of disjunction

1.1. Suppose p Suppose p q is true. q is true.Does it follow that pDoes it follow that pqq is true?is true?

2.2. Suppose Suppose ppqq is true.is true.Does it follow that Does it follow that p p q is true?q is true?


Test your understanding of the two types of disjunction

1.1. Suppose p Suppose p q is true. q is true.Does it follow that pDoes it follow that pqq is true?is true?NoNo: consider p TRUE, q TRUE: consider p TRUE, q TRUE

2.2. Suppose Suppose ppqq is true. Does it follow is true. Does it follow that that p p q is true? q is true? YesYes. Check each . Check each of the two assignments that make pof the two assignments that make pqq true: true: a) p TRUE, q FALSE a) p TRUE, q FALSE (makes (makes p p q true) q true) b) p FALSE, q TRUE b) p FALSE, q TRUE (makes (makes p p q true) q true)


The Implication Operator

The The implicationimplication p p qq states that states that pp implies implies q.q.I.e.I.e., If , If pp is true, then is true, then qq is true; but if is true; but if pp is not is not

true, then true, then qq could be either true or false. could be either true or false.E.g.E.g., let , let p p = = ““You study hard.You study hard.””

q q = = ““You will get a good grade.You will get a good grade.””p p q = q = ““If you study hard, then you will get If you study hard, then you will get

a good grade.a good grade.””

antecedent consequent


Implication Truth Table

• p p q q is is falsefalse onlyonly when when((pp is true but is true but qq is is notnot true) true)

• p p q q does does not not saysaythat that pp causescauses qq!!

• p p q q does does not not requirerequirethat that pp or or qq are trueare true!!

• E.g.E.g. ““(1=0) (1=0) pigs can fly pigs can fly”” is TRUE! is TRUE!

p q p q F F T F T T T F F T T T

The onlyFalsecase!



• Suppose you know Suppose you know that that qq is is TT. What . What do you know aboutdo you know aboutppqq ??




• Suppose you know Suppose you know that that qq is is TT. What . What do you know aboutdo you know aboutppqq ??

• The conditional The conditional must be must be TT




• Suppose you knowSuppose you knowthat that pp is is FF. What. Whatdo you know aboutdo you know aboutppqq ??

• The conditionalThe conditionalmust be must be TT




• Suppose you knowSuppose you knowthat that pp is is TT..

• What do you know aboutWhat do you know aboutppqq ? ? T or FT or F..

• What do you know aboutWhat do you know about q? q? T or FT or F..



Implications between real sentencs

• ““If this lecture ever ends, then the sun has If this lecture ever ends, then the sun has risen this morning.risen this morning.”” TrueTrue or or FalseFalse??

• ““If Tuesday is a day of the week, then I am a If Tuesday is a day of the week, then I am a penguin.penguin.”” TrueTrue or or FalseFalse??

• ““If 1+1=6, then Bush is president.If 1+1=6, then Bush is president.”” TrueTrue or or FalseFalse??

• ““If the moon is made of green cheese, then If the moon is made of green cheese, then 1+1=7.1+1=7.”” True True oror False False??


Why does this seem wrong?

• Recall Recall ““If [you study hard] then [youIf [you study hard] then [you’’ll get ll get a good grade]a good grade]””

• In normal English, this asserts a causal In normal English, this asserts a causal connection between the two propositions. connection between the two propositions. The connective The connective does not capture this does not capture this connection.connection.


Biconditional Truth Table

• p p q q means that means that pp and and qqhave the have the samesame truth value. truth value.

• Note this truth table is theNote this truth table is theexact exact oppositeopposite of of ’’s!s!Thus, Thus, p p q q means ¬(means ¬(p p qq))

• p p q q does does not not implyimplythat that pp and and qq are true, are true, or that either of them causes the other.or that either of them causes the other.

p q p qF F TF T FT F FT T T


Consider ...

The truth of p The truth of p q, where q, where1.1. p= p= Scotland is in the UKScotland is in the UK

q= q= 2+2 =42+2 =42.2. p= p= Scotland is not in the UKScotland is not in the UK

q= q= 2+2 =52+2 =53.3. p= p= Scotland is in the UKScotland is in the UK

q= q= Wales is not in the UKWales is not in the UK


Consider ...

The truth of p The truth of p q, where q, where• p= p= Scotland is in the UKScotland is in the UK

q= q= 2+2 =4 2+2 =4 TRUETRUE• p= p= Scotland is not in the UKScotland is not in the UK

q= q= 2+2 =5 2+2 =5 TRUETRUE• p= p= Scotland is in the UKScotland is in the UK

q= q= Wales is not in the UK Wales is not in the UK FALSEFALSE


Contrapositive

Some terminology, for an implication Some terminology, for an implication p p qq::• Its Its converseconverse is: is: q q pp..• Its Its contrapositivecontrapositive:: ¬¬q q ¬ ¬ p.p.

• Which of these two has/have the Which of these two has/have the same same meaningmeaning (express same truth function) (express same truth function) as as pp q q? Prove it.? Prove it.


Contrapositive

Proving the equivalence of Proving the equivalence of p p q q and its and its contrapositive, using truth tables:contrapositive, using truth tables:p q q p p q q p F F T T T T F T F T T T T F T F F F T T F F T T


For you to think about:

1.1. Can you think of yet another 2-place Can you think of yet another 2-place connective?connective?How many possible connectives do there How many possible connectives do there exist? exist?


For you to think about:

1.1. How many possible connectives do there exist? How many possible connectives do there exist? p connective qp connective q

T ? T T ? T T ? F T ? F F ? T F ? T F ? F F ? FEach question mark can be T or F, henceEach question mark can be T or F, hence2*2*2*2=16 connectives 2*2*2*2=16 connectives


Example of another connective

p connective q p connective q compare: compare: p and q p and q T F T TT F T T

T T F FT T F FF T T FF T T FF T F FF T F F

Names: NAND, Sheffer strokeNames: NAND, Sheffer stroke


Some Alternative Notations

Name: notandorxorimplies iffPropositional logic: Boolean algebra: ppq+C/C++/Java (wordwise):!&&||!= ==C/C++/Java (bitwise): ~&|^Logic gates:


Tautologies revisited

• We’ve introduced the notion of a tautology We’ve introduced the notion of a tautology using the example using the example p p pp

• Now, you know more operators, so can Now, you know more operators, so can formulate many more tautologies, e.g.,formulate many more tautologies, e.g.,the following are tautologies:the following are tautologies:((ppq)q) ((p p qq))((ppq) q) ( (¬¬q q ¬ ¬ p), and so onp), and so on


Equivalence Laws

• Similar to arithmetic identities in algebraSimilar to arithmetic identities in algebra• Patterns that can be used to match (part of) Patterns that can be used to match (part of)

another propositionanother proposition• Abbreviation: Abbreviation: TT stands for an arbitrary stands for an arbitrary

tautology; tautology; FF an arbitrary contradiction an arbitrary contradiction


Equivalence Laws - Examples

• IdentityIdentity: : ppT T p pp pF F pp• DominationDomination: : ppT T T T ppF F FF• IdempotenceIdempotence: : ppp p p pp pp p pp• Double negation: Double negation: p p pp• Commutativity: Commutativity: ppq q qqp pp pq q qqpp• Associativity: Associativity: ((ppqq))rr pp((qqrr))

( (ppqq))rr pp((qqrr))


More Equivalence Laws

• DistributiveDistributive: : pp((qqrr) ) ( (ppqq))((pprr)) pp((qqrr) ) ( (ppqq))((pprr))

• De MorganDe Morgan’’ss::((ppqq) ) p p qq

((ppqq) ) p p qq • Trivial tautology/contradictionTrivial tautology/contradiction::

pp pp TT pp pp FFAugustus

De Morgan(1806-1871)


Defining Operators via Equivalences

Some equivalences can be thought of as Some equivalences can be thought of as definitionsdefinitions of one operator in terms of others:of one operator in terms of others:

• Exclusive or: Exclusive or: ppqq ( (ppqq))((ppqq)) ppqq ( (ppqq))((qqpp))

• Implies: Implies: ppq q p p qq• Biconditional: Biconditional: ppq q ( (ppqq)) ( (qqpp))

ppq q ((ppqq))


How you may use equivalence laws: Example (1)

• Use equivalences to prove that Use equivalences to prove that ((r r ss) ) ss rr..


How you may use equivalence laws: Example (1)

((r r ss) ) (De Morgan) (De Morgan)r r s s (Commutativity) (Commutativity)ss r r ( (2x2x Double Negation) Double Negation)ss rr..


Summary

• In practice, Propositional Logic In practice, Propositional Logic equivalences are seldom strung together equivalences are seldom strung together into long proofs: using truth tables is into long proofs: using truth tables is usually easier.usually easier.


fall 2013 foundations of logic

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