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2/TRUTH-FUNCTIONS YolaFile-161107 ForclassDiscussionsOnly.Teacher.Armand.L.Tan.AssociateProfessor. PhilosophyDepartment.SillimanUniversity s6. S.variable: letter use to symbolize statements such as p, q, r, and s.

Statements are either simple such as `Roses are Red’ or compound: `Aristotle is Greek and Russell is English.’

Statement connectives: and, or, if...then, if and only if. When written in symbols they may be called logical operators. s7. Truth Values [TV]: statement either affirms/denies. Hence either true/false, but not both true/false. s8. Matrix Construction: truth table: systematic tabulation of all possible combination of truth values for any given variable.

General formula: 2n, where 2 refers to the number of truth values and n refers to the number of distinct statement variables. s9. Truth-Functions: Any expression whose truth value is defined in all cases by its logical operator.

9a. Negation: Not (-) curl -> -p Expressions: it is not true that/ it is false that/ it is not the case that Definition: p = p is true/ -p = p is false Example: it is false that `he who has a why to live for can bear with almost any how'(Nietzsche).

9b. Conjunction: And (.) dot -> p.q

Interpretation: p.q = both conjuncts are true -(p.q) = not both are true -p.-q = both are not true/both are false

`And' usually implies the idea of both, i.e. joined truth so that if either conjunct is false, then conjunction is false.

Other connectives: but/although/however/still/yet/whereas/moreover/neither...nor etc.

9c. Disjunction: Or (v) wedge -> pvq Sense of `Or’ a. Exclusive/Strict: @ most one: either one or the other but not both. b. Inclusive/Weak: @ least one: either one or the other but possibly both.

Example: a. here or there / for or against b. coffee or milk / classical or baroque Òr' implies alternative in which 1. only one alternative is possible both conceptually and factually; 2. that both alternatives are in fact possible. The exclusion of joined truth gives the idea of a strict disjunction. Its symbolic formulation: [(pvq).-(p.q)] In logic, as in mathematics, the òr’ is gen. used in its inclusive sense. This is preferred since it provides a minimal common meaning expressed in the assertion àt least one’ for both inclusive

/exclusive `or’ and not necessarily contrary to the idea of `possibly both’; neither is it incompatible with `@ most one’ of the disjuncts is true.

9d. Implication: If...then [>] ---> p>q Other terms: hypothetical/conditional

Interpretation: "if p then q" says nothing of the TV of the statements (otherwise we are asserting a conjunction of true statements). Rather it says that if the antecedent is true, then the consequent is also true. Implication implies logical relation between the antecedent/consequent. For any tf-conditional, q will be the logical consequence of p, though as a matter of fact, no real connection between the statements may be established. What is asserted in material implication is logical consistency rather than a necessary factual relation between the statements.

False Implication: antecedent true/consequent false. Means: no logical relation that exists between p and q.

Equivalence: (p>q)= -(p.-q), lg df.

Example: it is false: (-) that you got C: (p) but (.) did not pass: -q

Phrases: p entails q if p, q p implies q q if p p only if q q provided that p p is a sufficient q on the condition that p condition for q q is a necessary condition not p unless q for p

9e. Biconditional: if and only if (:) colon -> p:q Also known as: Material Equivalence interpretation: `if and only if' expresses a relationship of equivalence between the component statements. p:q is interpreted to mean that “ p is materially equivalent to q”

"if p then q and if q, then p" -> (p>q).(q>p) Equivalence: (p:q)=[(p>q).(q>p)] lg.df.

Phrases: when and only when/ If and only if /just exactly if/ is materially equivalent to s10. Á Summary of Truth-Functional Statements T-Functions Component Operator Symbolic Form ------Negation ... -p Conjunction conjunct . p.q Disjunction disjunct v pvq Implication antecedent/consequent ) p>q Biconditional .... : p:q

10b. Matrix for All Truth-Functions

GuideCol. Neg. Conj. Disj. Impli. Bicon. p q : -p p.q pvq p>q p:q ------:------1 1 : 0 1 1 1 1 1 0 : 0 0 1 0 0 0 1 : 1 0 1 1 0 0 0°: 1 0 0 1 1

Review Questions 1. Define a statement variable. 2. How do simple and compound statements differ? 3. How do strict and inclusive disjunctions differ? 4. What is a truth-functional statement? 5. What is a matrix/truth table? 6. State the other terms used for `truth-functional logic.' 7. What are the statement connectives/logical operator? 8. Explain the difference between `not both' and `both not.' 9. State the words/phrases that may express conjunction/negation /disjunction/implication /bicon 10. State symbolically the meaning of exclusive disjunction. s6/s9c EXERCISES

I. Construct a truth table for the following expressions.

1. (-pvq).(p:q) 2. [(pvq).-q]>p 3. -[p>(q.r)] 4. p>[r v(p:q)] 5. [(p>q).(q>p)]v r 6. [p>(q>r)]:(pvq) 7. (p:q)>(rvs) 8. [p:(q.r)]v s

II. Write CJ/DJ/and NO for neither of the two.

1. Either John Venn is logical or he is not. 2. Neither Venn nor Gödel was right or Wang was wrong. 3. It is not the case that either Venn or Gödel is logical. 4. The mathematics of distortion distorts one's thinking; so does Pierre de Fermat's last theorem. 5.`Every good mathematician is at least half a philosopher, And every good philosopher is at least half a mathematician.' III. Determine the TV the ff. assuming/exclusive disjunction.

1. 2+2= or Silliman University is in Dumaguete City. 2. Either Paris is in France or Manila is in RP. 3. All men are mortal or De Morgan is a man. 4. p:q is either a conjunction or a disjunction. 5. Aristotle was either the father of logic or music. 6. Either you are in Rm.3 or in Rm.4. 7. Either reasoning is thinking or thinking is reasoning.

IV. Library research project: submit a one-page paper about the

Aristotle Giuseppe Peano Willard V.O. Quine George Boole Bertrand Russell Alfred Tarski Gottlob Frege Alfred N. Whitehead John Venn Chryssippus Augustus De Morgan Charles Peirce Alonso Church Gottfried Leibniz Christine Ladd-Franklin -5-

Review Questions

1. State the logical definition of an implication. 2. When is a biconditional true? 3. Under what condition will an implication be false? 4. State the symbolic equivalent of implication/biconditional. 5. List some phrases that express implication/biconditional. s9d/s10 EXERCISES

I. Write MI/implication, BE/biconditional, and NOT for neither of the two.

1. R.Carnap's `logical involution' is false just in case the premises are true and the conclusion false. 2. “..a concept is clearer if and only if it is easier.” 3. Mathematics is not the `science of quantity', unless it distorts reality. 4. The reliability of Riemannian geometry is a sufficient condition for the falsity of Euclidian parallel postulate. 5. "p materially implies q" is true if either p is false or q is true. 6. Carnap's `this stone is thinking about Vienna' is meaningless just exactly if Brouwer's intuitionism is true. 7. X+Ù is commutative provided that x+y=y+x. 8. Either not (p)q) or q is true. 9. Einstein was not right, unless Plank's constant is not absolute. 10. Russell: Pure mathematics is the subject in which we do not Know what we are talking about, nor whether what we are saying is true (quoted, Nagel/Neumann "Godel's Proof"in Copi/Gould 1967:53).

S11. Computation of TV. Example: Granting that the TV of pq/1 while rs/0

-[(p.q)>(r:s)]v(-s>q) -[(1.1)>(0:0)]v(-0>1) -[( 1 )>( 1 )]v(1 >1) -[ 1 > 1 ]v( 1 ) -[ 1 ]v 1 0 v 1 1=true s12. Jan Lukaszewicks “Parenthesis-Free Notation”

N K A C E Common Sym.: PFN - . v > : -p : Np p.q : Kpq pvq : Apq p>q : Cpq P:q : Epq

Translation: (p.q)>(-p:q) -> CKpqENpq -(p.q):(-pv-q) -> ENKpqANpNq