Intro to Logic Homework 2
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Rutgers University, Newark *
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730:201:90
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Philosophy
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Jan 9, 2024
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Exercise 16 ... (For-Credit: 5 points)
[16-1] Explain why the truth-table for conditional is called
"Material" Conditional by explaining the difference between:
(1) using a conditional as non-truth-functional, and
(2) using it as truth-functional.
Answer: The truth table for conditional is called “Material” conditional because The truth-table for the conditional, models the material conditional statement in natural language, which is for practical implications. (1) Using
a conditional as non-truth-functional means to have an antecedent( something existing before or logically preceding another), followed by a consequent
(followed as a result of) in a “If, then” statement that makes up the conditional. When switching the statements in opposite order, it does not make sense any more. Individually, the statements do not provide enough information to determine whether it is true or not. The truth value on the conditional statement depends on both the truth value of the antecedent and consequent as well
as intentions of the speaker. Instead of logical implication, the conditional statement is treated as pragmatic meaning sensible based on practical rather than theoretical considerations. (2) When using it as a truth-functional, it means that the truth value of the conditional statement only depends on the truth value of the antecedent and consequent. The intention does not affect
the truth value. In a truth-functional statement the conditional statement is treated as a logical implication, rather than a pragmatic implicature.
[16-2] Determine the truth value of the following conditional
statement used as a material conditional:
If the earth is flat,
then the earth stops rotating around the sun
Answer: The truth value of the following conditional statement is considered true because the antecedent is false as the earth is not flat according to science, so according to the truth table in
a truth-functional statement, the truth value of the consequent does not matter it will be true regardless because the antecedent is false. P → Q is true in all cases except for when P is true and Q is false. If the statement “ the earth is flat” was alone without an after contention then it would not stand as true but since
it is given a restriction it will make it true.
Exercise 24 ... (For-Credit: 5 points)
[24-1] Explain what the three dichotomies of modality are; and
Explain why there are discordance as well as harmony
among them.
Answer: The 3 dichotomies of modality are metaphysical, epistemological, and linguistic. -
Metaphysical is necessary vs contingent: Contingent means that it is dependent on something else while necessary is that which must logically exist and is the foundation for others to exist. Metaphysical relates to the necessity of or possibility of propositions based on logical structure
-
Epistemological is A priori vs A posteriori: based on the theory of knowledge and understanding our beliefs and justifying what we claim to know. A priori knowledge means that a proposition is knowable independently of experience. The proposition shows logical reasoning and is not something that the person has to experience to know its true. A posteriori means to know based on the basis of experience.
-
Linguistic is Analytic vs Synthetic: Conveys the attitude or language of expression towards the truth of propositions. Analytic is based on deductive processes and synthetic is based on inductive. The interconnection and complexities between these 3 cause discordance as well as harmony. There is Discordance when there are different intuitions or perspectives of propositions and harmony
occurs when the model concepts support each other.
[24-2] Determine which of the following sentences are
tautologies, which are contradictions, and which are
contingent.
Show your results in Truth Tables.
(extracted from textbook exercise 3-6)
1. ~(A & ~A) -
=T
2. A V (A -> B)
-
=T
Answer: 1.)
It is tautologies because of principle of no contradiction
A
~A
(A&~A)
~(A&~A)
T
F
F
T
F
T
F
T
2.)
Tautology
A
B
A→B
A V (A→B)
T
T
T
T
T
F
F
T
F
T
T
T
F
F
T
T
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Exercise 27 ... (For-Credit: 15 points)
Do the three things with the argument given below:
1. Identify atomic sentences employed in the argument and assign
the sentence constants (P, Q, R) to those atomic sentences
(in their order of appearance in the argument).
2. Symbolize the whole argument by employing those sentence
constants and the proper connectives.
3. Determine whether the following argument is valid or not by
demonstrating how Truth Tables can be employed and
interpreted:
P1: If things are caused to exist,
then the infinite regress of existence is not possible.
P2: God is not the ultimate cause of existence,
'only if' the infinite regress of existence is possible.
P3: By the way, things are caused to exist.
C: Therefore, God is the ultimate cause of existence.
1.)
P= Things are caused to exist
Q= The infinite regress of existence is possible R= God is the ultimate cause of existence
2.)
P1: P→~Q
P2: ~R←>Q
P3: P
C: P→R
3.)
P
Q
R
P→~Q
~R <→Q
T
T
T
F
F
T
F
T
T
T
T
T
F
F
T
T
F
F
T
F
Because P is true, P→~Q and ~R←>Q is true
R is True because all statements ,( P, ~R←>Q, & P→~Q), are true at the same time Q is false so it is a valid statement since R (God is the ultimate cause of existence) is true.