INTRODUCTION TO SYMBOLIC LOGICPropositional Logic & Truth Functional Analysis

What is Symbolic Logic?It uses symbolic notation in expressing propositions and arguments.

Concentrates in the form.Simplest kind of logic.Its modern development began with George Boole in the 19th century. If, if the first then the second and if the second then the third, then, if the first then the third.

What prompted the creation of symbolic logic?Ordinary everyday language is flowery and ambiguous (because of equivocation, amphiboly, accent, vagueness, & confusion in emotive significances)In order to avoid the common difficulties of ordinary language, we need an artificial language that substitutes it using minimal characters, and, consequently, represents it with high degree of clarity and simplicity.There is also economy of space and time.

GOALSLearn the elements of the new languageLearn how to translate ordinary language grammar into symbolic notationRepresent arguments in this new language

The 2 Subfields of Symbolic LogicPropositional LogicOriginally called propositional calculusStudies the properties of propositions formed from constants and logical operatorsPredicate LogicOriginally called predicate calculus expands on propositional logic by introducing variables, usually denoted by x, y, z, or other lowercase letters, It also introduces sentences containing variables, called predicates, usually denoted by an uppercase letter followed by a list of variables, such as P(x) or Q(y,z).

Propositional Logic:Propositions and OperatorsPropositions are considered as ATOMS of propositional logic.What are the two types of propositions?Simple PropositionsCompound propositions

Propositional Logic:Propositions and OperatorsWhat are Simple Propositions?Statements which cannot be broken down without a loss in meaning.E.g. John and Leah is a couple cannot be broken down without a change in meaning of the statement. Note what happens if we break it down to John is a couple and Leah is a couple

Propositional LogicBut Juanita and Juanito are diligent students is not a simple sentence because it can be broken down without a change in meaning. Juanita is a diligent student. Juanito is a diligent student.This is an example of a Compound Proposition.How do we represent (simple) propositions in propositional logic?Conventionally, capital letters (usually towards the beginning of the alphabet) may be used as abbreviations for propositions.

Propositional LogicJohn and Leah is a couple. = AJuanita and Juanito are diligent students. =A BThe symbol is used to represent the logical operator and or conjunction

Constants and VariablesWhat are constants?Capital letters that represent the actual statements. Since it represents an definite statement, it does not vary.E.g. John and Leah is a couple. = AWhat are variables?It is not a proposition, but is a place holder for any proposition. Used for making templates of forms.E.g. John and Leah is a couple. = aChimps and Men are apes = a

Simple vs CompoundTuguegarao is the capital of Cagayan and Basco is the capital of Batanes.Either I will be forgotten or I will forever be remembered.If I will be remembered, then my soul will shout for joy.Noli De Castro is the future president if and only if Manny Villar is not a presidential candidate.

Propositional LogicWhat are Logical Operators?Another basic element of propositional logic.They connect propositions.Compound Proposition = proposition + proposition + What then is the basic skill needed in studying elementary propositional logic?Knowing the truth value of propositionsAll you need to know is the definition of the operator and the truth value of the propositions used.

Propositional Logic: Truth Functionality Any arguments worth is quite dependent on the combination of the truth values of the component sentences.Understanding arguments is basically: Understanding how truth values of the component sentences are distributed in the compound or complex sentence used to express the argumentUnderstanding how the component sentences are connected to one another

Propositional Logic: Truth Functionality In order to know the truth value of the proposition which results from applying an operator to propositions, all that need be known is the definition of the operator and the truth value of the propositions used.The basic symbolic conventions of Propositional Logic are thus about propositions and about the operators used to connect them.

The Operators

ConnectiveSymbolFormal NameNot~NegationAndConjunctionOrVDisjunctionIf thenConditional / Implication if and only if=Biconditional

NEGATIONThe phrase It is false that or not inserted in the appropriate place in a statement.E.g., It is not the case that Bugoy is ugly can be represented by ~B.It is represented by the following truth table:

p~pTFFT

CONJUNCTIONTruth-functional connective similar to and in English and is represented in symbolic logic with the dot A connective forming compound propositionsExpressed in the following truth table:

pqp.qTTTTFFFTFFFF

CONJUNCTIONThe statement Miko is cute and Popo is hideous can be represented as M P There are four possible states of affairs which might have occurred with respect to Miko as cute and Popo as hideous.

pqp.qTTTTFFFTFFFF

CONJUNCTIONSome characteristics of conjunction (in mathematical jargon) include:associativeinternal grouping is immaterial I. e., [(p q) r] is equivalent to [p (q r)].communicativeorder is immaterial I. e., p q is an equivalent expression to q p.idempotentreduction of repetition I. e., p p is an equivalent expression to p.

DISJUNCTIONSometimes called alternation A connective which forms compound propositions which are false only if both statements (disjuncts) are false. Expressed in the following truth table:

pqp v qTTTTFTFTTFFF

DISJUNCTIONThe connective or in English is quite different from disjunction. Or in English has two quite distinctly different senses.The exclusive sense of or is Either A or B (but not both) as in You may go to the left or to the right. In Latin, the word is aut.The inclusive sense of or is Either A or B {or both). as in John is at the library or John is studying. In Latin, the word is vel.We use the second sense of Or.

DISJUNCTIONE.g Either Gloria or Erap is the greatest Filipino president.G v ENeither Gloria nor Erap is the greatest Filipino president.~(G v E) or (~G) (~E)

DISJUNCTIONThe order of the words both and not is very important in translating propositions connected by disjunction.Eg.Willy and Joey will not both be elected.~ ( W J ) Willy and Joey will both not be elected.(~W) (~J)

DISJUNCTIONHow should we understand this?~ W J ~ ( W J )~ (W) JIt is understood in the 2nd sense. The original notation is accepted for brevity purposes.

IMPLICATIONmaterial implicationThe falsity of such a statement is established only when the antecedent is true and the consequent if false.

pqp qTTTTFFFTFFFF

IMPLICATIONE.g. If the Miko lives morally (p), then he will go to heaven (q).

pqp qTTTTFFFTTFFT

BICONDITIONALMaterial Equivalence If and only ifTwo statements are materially equivalent if they have the same truth value, i.e. if both is true or false.Expressed in the following truth table:

Summary of Operators

PunctuationWhen sentences involved are complex enough to have more than two sentences, we use parentheses, brackets, or braces.To avoid ambiguities in understanding complex sentences like P Q R S, we use the grouping apparatuses in this one possible fashion (P Q) (R S)Mistake in the punctuation means a mistake in determining the truth value of arguments.

PunctuationIn using punctuations, the following rules are to be followed:The major truth functional connective representing the entire compound statement must be identified.If negation symbol is found immediately outside a set of punctuated symbols then it must be interpreted to mean that the whole punctuated symbols are negated.If negation symbol is attached to a symbol outside a set of punctuated symbols then the negation symbol negates only that particular symbol.

PunctuationExamplesEither philosophers are intelligent and innovative thinkers or they are simply insane and weird.(I N) v (S W)If mountains are high and majestic then they are either not small nor plain.(H M) ~(S v P)

Determine the main operator of the following statements~(A v M) (C E)(G ~P) ~(H v ~W)~[P (S K)]~(K ~O) ~(R v ~B)(M B) v ~ [E ~(C v I)]~[(P ~R) (~E v F)]~[(S v L) M] (C v N)

Translate the following molecular statements in symbolic formIt is not the case that Hitler killed millions of innocent Jews.Either Caesar married Cleopatra or FJP became the President of the Philippines.Alexander the Great conquered America if Napoleon rules France.Arabs are treated as terrorists only if Bin laden bombed the Twin Towers.

TRUTH TABLE

pqrq v rp v (q v r)1TTTTT2TTFTT3TFTTT4TFFFT5FTTTT6FTFTT7FFTTT8FFFFF

TRUTH TABLE 2 p v ( q v r ) T T T T T T T F T F T T T F F F F T T T F T T F F F T T F F F FTTTTTTTF

Forms of PropositionsInconsistent PropositionTwo or more beliefs are said to be inconsistent when they both cannot be true at the same time for any possible situation.E.g.

pqp q~(~p v q)TTTFTFFTFTTFFFTF

Inconsistent PropositionIf a belief cannot be true in any possible occasionOften it is referred to as self-contradictory or simply contradictionE.g.

p~pp ~pTFFTFFFTFFTF

Consistent PropositionIf two or more beliefs can be true together in some possible situation.

pqp . qp v qTTTTTFFTFTFTFFFF

Consistent PropositionIf a belief can be true in some possible occasion.

pqp qTTTTFFFTTFFT

Tautologous PropositionBeliefs that are always true no matter whatE.g. If it rains then it rains.R REquivalent PropositionPropositions that are perfectly consistent.

pqp q~p v qTTTTTFFFFTTTFFTT

Determine what type of PropositionR v ~RR ~R(R v S) TR ~RR S

Compare the following pairs of proposition and determine whether they are consistent, inconsistent, contradictory1.2.3.4.5.6.7.D ~E P O O v P ~D v T ~(D ~T)O (P Q) O (P v Q) (O P) v (O Q)

ReplacementEquivalent sentences can replace each other within any complex proposition salva veritate, meaning without gain or loss of truth values.Equivalent sentences are represented using material equivalence ( ) and all forms are summarized in the

Rules of ReplacementMaterial Implication (Impl.)(p q) (~p v q)Association (Assoc.)[p v ( q v r)] [(p v q) v r][p ( q r)] [(p q) r]Distribution (Dist.)[p ( q v r)] [(p q) v (p r)][p v ( q r)] [(p v q) (p v q)]

Rules of ReplacementCommutation (Com.)(p v q) (q v p)(p q) (q p)De Morgans Rule (De M.) ~(p q) (~p v ~q)~(p v q) (~p ~q)Double Negation (D.N.)p ~~p

Rules of ReplacementTransposition (Trans.)(p v q) (q v p)Material Equivalence (Equiv.) (p q) (p q) (q p) (p q) (p q) v (~p ~q)Exportation (Exp.)[(p q) r] [p (q r)]Tautology (Taut.)p (p v p)p (p p)

Formal ProofThe following are said to be the Elemtary Rules of Inference for Modern Symbolic Logic:

1. Modus Ponens (M.P.)p qp q

2. Modus Tolens (M.P.)p q~q~p

3. Addition (Add.)p p v q

Formal Proof4. Simplification (Simp.)p q p

5. Conjunction (Conj.)p q p q

6. Disjunctive Syllogism (D.S.)p v q~p ~q

7. Hypothetical Syllogism (H.S.)p qq rp r

8. Constructive Dilemma (C.D.)p qr sp v rq v s

9. Destructive Dilemma (D.D.)p qr s~q v ~s~p v ~r

10. Absorption (Abs.)p qp (p q)

Exercise. Determine what argument form was used in the following arguments1. Thomas will apologize or Michelle will be angry. Thomas will not apologize. Therefore, Michelle will be angry.

2. If Tuguegarao is in Cagayan, then San Gabriel is in Cagayan. If San Gabriel is in Cagayan, then Sir Mikes house is in Cagayan. If Tuguegarao is in Cagayan, then Sir Mikes house is in Cagayan.

3. If this flashlight works, then the batteries are good. This flashlight does work. Therefore, the batteries are good.

4. If this flashlight works, then the batteries are good. The batteries are not good. Therefore, the flashlight does not work.

5. If this punch contains gin, then Emil will like it, and if it contains Matador, then Loid will like it. This punch contains either gin or Matador. Therefore, either Emil will like it or Loid will like it.

6. If this punch contains gin, then Emil will like it, and if it contains Matador, then Loid will like it. Either Emil will not like it or Loid will not like it. Therefore, either this does not contain gin or it does not contain Matador.

7. If Napoleon was killed in a plane crash, then Napoleon is dead. Napoleon was not killed in a plane crash. Therefore, Napoleon is not dead.

Proving the validity of an argumentIf the Lakers win the playoff, then the Pistons will lose the championship. If the Lakers do not win the playoff, then either Jackson or Bryant will be fired. The Pistons will not lose the championship. Furthermore, Bryant will not be fires. Therefore, Jackson will be fired.

Proving the validity of an argument1. If the Lakers win the playoff, then the Pistons will lose the championship. 2. If the Lakers do not win the playoff, then either Jackson or Bryant will be fired. The Pistons will not lose the championship. Furthermore, Bryant will not be fired. Therefore, Jackson will be fired. 1. L P

2. ~L (J v B)

3. ~P

4. ~B

/ J

Proving the validity of an argument1. L P2. ~L (J v B)3. ~P 4. ~B / J5. ~L 1, 3, MT6. B v J 2, 5, MP7. J 4, 6, DS

Proving the validity of an argumentEX. 11. S T2. T U3. R S / R U4. R T 1, 3, HS5. R U 2, 4, HS

Proving the validity of an argumentEX. 21. A v B2. ~C ~A3. C D4. ~D / B5. ~C 3, 4, MT6. ~A 2, 5, MP7. B 1, 6, DS

Proving the validity of an argumentEX. 31. E (K L)2. F (L M)3. G v E4. ~G5. F / K M6. E 3, 4, DS7. K L 1, 6, MP8. L M 2, 5, MP7. K M 7, 8, HS

Exercise. Supply the required justification for the derived steps in the following proofs: (1) 1. J (K L) 2. L v J 3. ~L /~K 4. J _______ 5. K L _______ 6. ~K _______(Ans) 1. J (K L) 2. L v J 3. ~L /~K 4. J __ 5. K L _______ 6. ~K _______

Exercise. Supply the required justification for the derived steps in the following proofs: (2) 1. ~(S T) (~P Q) 2. (S T) P 3. ~P /Q 4. ~(S T) _______ 5. ~P Q _______ 6. Q _______(Ans) 1. ~(S T) (~P Q) 2. (S T) P 3. ~P /Q 4. ~(S T) _ 5. ~P Q _ 6. Q _

Exercise. Use the first four rules of inference to derive the conclusions of the following symbolized arguments:(1) 1. (G J) v (B P) 2. ~(G J) / B P

(2) 1. (K O) (N v T) 2. K O / N v T