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2/TRUTH-FUNCTIONS
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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
`Or' 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 `or’ is gen. used in its inclusive sense. This is preferred since it
provides a minimal common meaning expressed in the assertion `at least one’ for both inclusive
--2/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
p implies q
p only if q
p is a sufficient
condition for q
not p unless q
if p, q
q if p
q provided that p
q on the condition that p
q is a necessary condition
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
-310b. 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)
4. p>[r v(p:q)]
7. (p:q)>(rvs)
2. [(pvq).-q]>p
5. [(p>q).(q>p)]v r
8. [p:(q.r)]v s
3. -[p>(q.r)]
6. [p>(q>r)]:(pvq)
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
George Boole
Gottlob Frege
Chryssippus
Giuseppe Peano
Willard V.O. Quine
Bertrand Russell
Alfred Tarski
Alfred N. Whitehead John Venn
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
v
C
>
E
:
Common Sym.: PFN
-p
p.q
pvq
p>q
P:q
:
:
:
:
:
Np
Kpq
Apq
Cpq
Epq
Translation: (p.q)>(-p:q) -> CKpqENpq
-(p.q):(-pv-q) -> ENKpqANpNq