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**INTRODUCTION TO SYMBOLIC LOGIC**

Propositional Logic & Truth Functional Analysis

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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.” [(p q) (q r)] (p r)

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**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.

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**GOALS Learn the elements of the new language**

Learn how to translate ordinary language grammar into symbolic notation Represent arguments in this new language

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**The 2 Subfields of Symbolic Logic…**

Propositional Logic Originally called propositional calculus Studies the properties of propositions formed from “constants” and logical “operators” Predicate Logic Originally 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).

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**Propositional Logic: Propositions and Operators**

Propositions are considered as “ATOMS” of propositional logic. What are the two types of propositions? Simple Propositions Compound propositions

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**Propositional Logic: Propositions and Operators**

What 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”

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Propositional Logic But “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.

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**Propositional Logic “John and Leah is a couple.” = A**

“Juanita and Juanito are diligent students.” =A • B The symbol “•” is used to represent the logical operator “and” or “conjunction”

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**Constants and Variables**

What 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.” = A What 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.” = a “Chimps and Men are apes” = a

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Simple vs Compound Tuguegarao 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.

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**Propositional Logic What 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 propositions” All you need to know is the definition of the “operator” and the “truth value of the propositions used.”

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**Propositional Logic: Truth Functionality**

Any argument’s 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 argument Understanding how the component sentences are connected to one another

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**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.

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**Conditional / Implication**

The Operators Connective Symbol Formal Name Not ~ Negation And • Conjunction Or V Disjunction If… then… Conditional / Implication … if and only if… = Biconditional

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NEGATION The 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 ~p T F

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CONJUNCTION Truth-functional connective similar to “and” in English and is represented in symbolic logic with the dot “•” A connective forming compound propositions Expressed in the following truth table: p q p.q T F

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CONJUNCTION The 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. p q p.q T F

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CONJUNCTION Some characteristics of conjunction (in mathematical jargon) include: associative—internal grouping is immaterial I. e., “[(p • q) • r]” is equivalent to “[p • (q • r)]”. communicative—order is immaterial I. e., “p • q” is an equivalent expression to “q • p”. idempotent—reduction of repetition I. e., “p • p” is an equivalent expression to “p”.

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**DISJUNCTION p q p v q Sometimes called alternation**

A connective which forms compound propositions which are false only if both statements (disjuncts) are false. Expressed in the following truth table: p q p v q T F

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DISJUNCTION The 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.”

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DISJUNCTION E.g “Either Gloria or Erap is the greatest Filipino president.” G v E “Neither Gloria nor Erap is the greatest Filipino president.” ~(G v E) or (~G) • (~E)

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DISJUNCTION The 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)

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**DISJUNCTION How should we understand this? ~ W • J**

It is understood in the 2nd sense. The original notation is accepted for brevity purposes.

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**IMPLICATION “material implication”**

The falsity of such a statement is established only when the antecedent is true and the consequent if false. p q p q T F

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IMPLICATION E.g. If the Miko lives morally (p), then he will go to heaven (q). p q p q T F

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**BICONDITIONAL p q q p F T Material Equivalence “If and only if”**

Two 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: F T p q q p

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Summary of Operators T F F F F T T F T T p q p • q p v q p q

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Punctuation When 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.

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Punctuation In 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.

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Punctuation Examples Either 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)

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**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)

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**Translate the following molecular statements in symbolic form**

It 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.

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TRUTH TABLE p q r q v r p v (q v r) 1 T 2 F 3 4 5 6 7 8

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

F T T F F F T T F F F F T F

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**Forms of Propositions Inconsistent Proposition**

Two or more beliefs are said to be inconsistent when they both cannot be true at the same time for any possible situation. E.g. p q p q ~(~p v q) T F

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**Inconsistent Proposition**

If a belief cannot be true in any possible occasion Often it is referred to as self-contradictory or simply contradiction E.g. p ~p p • ~p T F

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**Consistent Proposition**

If two or more beliefs can be true together in some possible situation. p q p . q p v q T F

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**Consistent Proposition**

If a belief can be true in some possible occasion. p q p q T F

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**Tautologous Proposition**

Beliefs that are always true no matter what E.g. If it rains then it rains. R R Equivalent Proposition Propositions that are perfectly consistent. p q p q ~p v q T F

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**Determine what type of Proposition**

R v ~R R • ~R (R v S) • T R ~R R S

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Compare the following pairs of proposition and determine whether they are consistent, inconsistent, contradictory 1. 2. 3. 4. 5. 6. 7. R S ~S ~R D E D • ~E O v P P • O ~D v T ~(D • ~T) ~K L K ~L (O P) Q O (P Q) O • (P v Q) (O • P) v (O • Q)

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Replacement Equivalent 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

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**Rules of Replacement Material Implication (Impl.) Association (Assoc.)**

(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)]

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**Rules of Replacement Commutation (Com.) De Morgan’s Rule (De M.)**

(p v q) (q v p) (p • q) (q • p) De Morgan’s Rule (De M.) ~(p • q) (~p v ~q) ~(p v q) (~p • ~q) Double Negation (D.N.) p ~~p

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**Rules of Replacement Transposition (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)

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Formal Proof The following are said to be the Elemtary Rules of Inference for Modern Symbolic Logic: 2. Modus Tolens (M.P.) p q ~q ~p 1. Modus Ponens (M.P.) p q p q 3. Addition (Add.) p p v q

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**Formal Proof 6. Disjunctive 4. 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 q q r p r 8. Constructive Dilemma (C.D.) p q r s p v r q v s 9. Destructive Dilemma (D.D.) p q r s ~q v ~s ~p v ~r 10. Absorption (Abs.) p q p (p • q)

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**Exercise. Determine what argument form was used in the following arguments**

1. 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 Mike’s house is in Cagayan. If Tuguegarao is in Cagayan, then Sir Mike’s house is in Cagayan. 3. If this flashlight works, then the batteries are good. This flashlight does work. Therefore, the batteries are good.

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**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.

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**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.

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**Proving the validity of an argument**

If 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.

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**Proving the validity of an argument**

1. 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

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**Proving the validity of an argument**

1. L P 2. ~L (J v B) 3. ~P 4. ~B / J 5. ~L , 3, MT 6. B v J , 5, MP 7. J , 6, DS

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**Proving the validity of an argument**

EX. 1 1. S T 2. T U 3. R S / R U 4. R T , 3, HS 5. R U , 4, HS

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**Proving the validity of an argument**

EX. 2 1. A v B 2. ~C ~A 3. C D 4. ~D / B 5. ~C , 4, MT 6. ~A , 5, MP 7. B , 6, DS

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**Proving the validity of an argument**

EX. 3 1. E (K L) 2. F (L M) 3. G v E 4. ~G 5. F / K M 6. E , 4, DS 7. K L , 6, MP 8. L M , 5, MP 7. K M , 8, HS

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**Exercise. Supply the required justification for the derived steps in the following proofs:**

(Ans) 1. J (K L) 2. L v J 3. ~L /~K 4. J __ 5. K L _______ 6. ~K _______ (1) 1. J (K L) 2. L v J 3. ~L /~K 4. J _______ 5. K L _______ 6. ~K _______

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**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 _

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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

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Logic Chapter 2. Proposition "Proposition" can be defined as a declarative statement having a specific truth-value, true or false. Examples: 2 is a odd.

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