Chapter 1: Truth Functional Logic

Question 1

Valid

Answer 1

A valid argument is one whose conclusion is true in every case in which the all its premises are true. Counterexamples are absent

Question 2

Counterexamples

Answer 2

Cases in which all premises are true but the conclusion is false

Question 3

A sound argument

Answer 3

Is one that’s valid and has no false premises.

Ensures truth in reality

Question 4

Logically equivalent

Answer 4

Sentences that have the same truth value in all cases

Question 5

How to use an inconsistency detector to test validity

Answer 5

Assuming the denial of the conclusion: if the set is inconsistent, the argument is valid. If the set is consistent, the argument is invalid

Question 6

How to use a validity detector to test inconsistency

Answer 6

Assume the conclusion is a truth-functionally inconsistent sentence: if the argument is valid, the set is inconsistent. If the argument is invalid, the set is consistent

Question 7

Tautology

Answer 7

A sentence with no f cases at all in its truth table

Question 8

truth functional

Answer 8

truth value depends on truth value of constituents only (ex. denial, conjunction, disjunction), regardless of meaning of sentence

Question 9

an interpretation I of a sentence S is an assignment of:

Answer 9

exactly one truth value (t or f) to each sentence letter that occurs in the sentence S

Question 10

a sentence is satisfiable jic…

Answer 10

it is true in at least one of its interpretations

Question 11

a sentence is valid jic…

Answer 11

it is true in all of its interpretations

Question 12

a satisfiable set of sentences is:

Answer 12

a set of sentences where all of its members (premises) are true JOINTLY in a single interpretation

Question 13

implication

Answer 13

a sentence alpha implies a sentence beta jic there is no interpretation of alpha and beta where alpha is true and beta is false

Question 14

another way of saying alpha implies beta is:

Answer 14

beta is a consequence of alpha, beta follows from alpha

Question 15

a sentence alpha is equivalent to a sentence beta jic…

Answer 15

alpha implies beta AND beta implies alpha

Question 16

argument

Answer 16

a collective of sentences, one of which is the conclusion of the argument and the rest are its premises

Question 17

counter-examples

Answer 17

cases where premises are all false but conclusion is true or premises are all true and conclusion is false (i.e. cases where conclusion does not follow from the premises)

Question 18

to test a (finite) set of sentences for satisfiability:

Answer 18

make a list of all members of the set and make a truth tree, apply rules, if all paths are closed then the initial list is unsatisfiable, if at least one path is open then the set is satisfiable

Question 19

to test a sentence S for validity:

Answer 19

make a truth tree starting with DENIAL of S for satisfiability, apply rules; if denial of S is satisfiable then S is invalid, if denial of S is unsatisfiable, then S is valid

Question 20

to test an argument for validity:

Answer 20

test the set of premises and the DENIAL of the CONCLUSION for satisfiability, apply rules: if satisfiable, then argument is invalid, if unsatisfiable, argument is valid

Question 21

to test two sentences for equivalence:

Answer 21

test DENIAL of BI-CONDITIONAL for satisfiability (if equivalent, bi-conditional must be valid); if satisfiable, not equivalent, if unsatisfiable, equivalent

Question 22

in a truth tree, if one list of the conclusions of a rule is true in an interpretation…

Answer 22

the premise of the rule is true in that interpretation

Question 23

“argument”

Answer 23

input

Question 24

“values”

Answer 24

output

Question 25

if there are n numbers of different inputs, what is the number of different outputs?

Answer 25

((2)^2)^n

Question 26

a set of logical connectives is EXPRESSIVELY COMPLETE just in case:

Answer 26

all truth functions can be expressed with those connectives (and sentence letters) - (all truth functions can be expressed using L1)

Question 27

algorithm to find function that produces truth function matching table:

Answer 27

locate where t occurs, conjoin the letters present in each true row. If there are no T's, then this is an unsatisfiable sentence

Question 28

to show that a set B is expressively complete given that a set A is expressively complete:

Answer 28

define each member of A in terms of B

Question 29

complete disjunctive normal form (CDNF)

Answer 29

1) sentence is a disjunction 2) every disjunct within the sentence is a conjunction 3) every conjunction in the sentence consists of sentence letters or denials of sentence letters 4) every sentence letter that appears in one disjunct appears in all the disjuncts

Question 30

if a sentence in complete disjunctive normal form has n sentence letters, how many disjuncts should it have?

Answer 30

2^n disjuncts