Problem 9

Question 1

Operator/Connective

Answer 1

Are used to “translate” english expressions

–> but as translation might not be identical, a certain distortion of the meaning will occur

Question 2

Compound statement

Answer 2

Contains at least one simple statement as a component

ex.: Either people get serious about conservation or energy prices will skyrocket

Question 3

Name the 5 Operators.

Answer 3

1. Tilde
2. Dot
3. Wedge
4. Horseshoe
5. Triple bar

Question 4

Tilde

Operator

Answer 4

Are used as a negation

ex.: not, it is not the case that

Question 5

Tilde

Negation

Answer 5

Are used as a negation

–> ∼

ex.: “not”, “it is not the case that”

BUT: Only operator to be placed in front of the proposition

Question 6

Dot

Conjunctive statement

Answer 6

Are used as a conjunction/addition

–> •

ex.: “and”, “also”, “moreover”

Question 7

Wedge

Disjunctive statement

Answer 7

Are used as a disjunction

–> ∨

ex.: “or”, “unless/either or”

Question 8

Horseshoe

Conditional statement

Answer 8

Are used to implicate something, containing a antecedent + consequent

–> ⊃

ex.: “if … then”, “only if/implies that”

REMEMBER: “if” follows antecedent, “only if” follows consequent

Question 9

Triple bar

Biconditional Statement

Answer 9

Expresses the relation of material equivalence

–> ≡

ex.: “if and only if”

Question 10

Main operator

Answer 10

Has its scope everything else in the statement

ex.: H • (J ∨ K) –> Dot is the main operator

Question 11

Sufficient condition

Horseshoe

Answer 11

Occurs when the occurrence of A is all that is required for the occurrence of B

ex.: Having the flu (A) is sufficient to feel miserable (B)

–> antedecent

Question 12

Necessary condition

Horseshoe

Answer 12

When B cannot occur without the occurrence of A

ex.: Having air to breathe (A) is necessary to survive (B)

–> consequent

Question 13

Well-formed formulas

WWF

Answer 13

Refer to syntactically correct arrangements of symbols

ex. in english: there is a cat on the porch vs porch on the cat is a there

Question 14

Truth function

Answer 14

Refers to any compound proposition whose truth value is completely determined by the truth values of its components

Question 15

Statement variables

Answer 15

Refer to lowercase letters that can stand for the statements

–> used to construct statement forms

ex.: p,q,r,s
p = A • B

Question 16

Statement form

Answer 16

Is an arrangement of statement variables + operators so that the uniform substitution of statements in place of the variables results in a statement

ex.: ∼p and p ⊃ q –> ∼A and A ⊃ B

Question 17

According to the truth table, when is any of the 5 operator statements false or true?

Answer 17

1. Tilde
   - -> If P is false, its complement must be true
2. Dot
   - -> both conjuncts must be true and false in all other cases
3. Wedge
   - -> the disjunction is true when at least one of the disjuncts is true, otherwise is false
4. Horseshoe
   - -> conditional statement is false when the antecedent is true + consequent is false
5. Triple bar
   - -> biconditional is true when its 2 components have the same truth value

Question 18

In which case is it inappropriate to use the triple bar operator ?

Answer 18

When the ordinary meaning of a biconditional conflicts with the truth-functional meaning

ex.: Ministry building is a hexagon if and only if it has eight sides

–> component is false but content is also false

Question 19

How do you construct a truth table ?

Answer 19

1. Determine number of lines/rows
2. Total number of lines is equal to the number of possible conventions of truth values

=> L = 2^n

Question 20

Tautologous statement/

Logically true compound statement

Answer 20

Refers to a statement that is true regardless of the truth values of its components

Question 21

Self-contradictory statement/ Logically false compound statement

Answer 21

Refers to a statement that is false regardless of the truth values of tis components

Question 22

Contingent statement

Answer 22

Refers to a statement whose truth value varies depending on the truth values of its components

Question 23

How do we determine whether a compound statement is self-contradictory, tautologous or contingent ?

Answer 23

By inspecting the column of truth values under the main operator

1. Tautologous
   - -> all true
2. Self-contradictory
   - -> all false
3. Contingent
   - -> at least one true or at least one false

Question 24

Logically equivalent statements

Answer 24

Refer to 2 propositions that have the same truth value on each line under their main operators

Question 25

Contradictory statements

Answer 25

Refer to 2 propositions that have opposite truth values on each line under their main operators

Question 26

Consistent statements

Answer 26

Refer to 2 or more propositions where there is at least one line on which both of them turn out to be true

Question 27

Inconsistent statements

Answer 27

Refer to 2 or more propositions where there is no line where both of them turn out to be true

Question 28

How do you construct a truth table for arguments ?

Answer 28

1. Symbolize arguments by using letters
2. Write the argument out, using a single slash between the premises + double slash between last premise + conclusion
3. Draw a truth table
   - -> outline the columns who represent the P + C
4. Look for a line in which all of the premises are true + conclusion false
   - -> invalid

Question 29

Arguments corresponding conditional

Answer 29

Refers to the conditional segment having the conjunction of an arguments premises as its antecedent + the conclusion as its consequent

Question 30

How do you check whether an argument is valid ?

Answer 30

1. By assuming it is invalid, meaning the premises are true but conclusion false
2. Then reconstructing the truth values
3. If there is a contradiction the argument is valid

=> Same for consistency, assuming everything is true, if contradiction = inconsistent

Question 31

Disjunctive syllogism

Valid Argument form

Answer 31

Consists of a disjunctive statement for one of its premises

ex.: 
H ∨ P
∼ H
-------
P

Question 32

Pure hypothetical syllogism

Valid Argument form

Answer 32

Consists of 2 premises + one conclusion, all of which are hypothetical statements

ex.: 
p⊃q
q⊃r 
-------
p⊃r

Question 33

Modus ponens

Valid Argument form

Answer 33

Consists of a

1. conditional premise
2. second premise that includes the antecedent of the conditional premise
3. conclusion with the consequent

ex.: 
p⊃q 
p
------
q

Question 34

Modus tollens

Argument form

Answer 34

Consists of

1. conditional premise
2. Second premise that denies the consequent of the conditional premise
3. conclusion that denies the antecedent

ex.:
p⊃q 
∼q 
------
∼p

Question 35

Affirming the consequent

Invalid Argument form

Answer 35

Consists of

1. conditional premise
2. Second premise that includes the consequent of the conditional
3. conclusion that includes the antecedent

=> positive form of modus tollens

ex.:
p⊃q
q
------
p

Question 36

Denying the antecedent

Invalid argument form

Answer 36

Consists of

1. Conditional premise
2. Second premise that denies the antecedent of the conditional
3. Conclusion that denies the consequent

=> negative form of modus ponens

ex.:
p⊃q 
∼p 
-------
∼q

Question 37

Constructive dilemma 
(Valid argument form)

Answer 37

Is a valid argument form that consists of a conjunctive premise made up of 2 conditional statements

1. Disjunctive premise including the antecedents of the conjunctive premise
2. Disjunctive conclusion including the consequents of the conjunctive premise

ex.: 
(p ⊃ q) • (r ⊃ s) 
p∨r
--------------------
q∨s

Question 38

Destructive dilemma 
(Valid argument form)

Answer 38

Similar to constructive dilemma, but negative form

```
ex.:
p ⊃ q) • (r ⊃ s
∼q ∨ ∼s
——————–
∼p ∨ ∼r
~~~

Question 39

What is the difference between the inclusive vs exclusive “or” ?

Answer 39

Inclusive
–> at least one has to be true, both can

Exclusive
–> only one can be true, not both