TASK 8 - PROPOSITIONAL LOGIC

Question 1

propositional logic

Answer 1

= fundamental elements are whole statements (propositions)

• statements are represented by letters
• statements are combined by means of the operators to represent more complex statements

Question 2

simple statement

Answer 2

= one that does not contain another statement as a component (e.g. fastfood is unhealthy’)
- statement is represented by an uppercase letter

Question 3

compound statement

Answer 3

= one that contains at least one simple statement as a component (e.g either people get serious about conversation (1) or energy prices will rise (2))
- each statement is represented by an uppercase letter

Question 4

logical operators

- main operator

Answer 4

= operator that has as its scope everything else in the statement

1) either the only one
2) if there are NO parentheses: the only one that is not a tilde ∼/¬
3) if there are parentheses: the one that lies outside of them

Question 5

logical operators

- tilde (∼)/¬

Answer 5

= negation
= not, it is not the case that
- always in front of the proposition it negates
- true if: false

Question 6

logical operators

- dot (⋅)/∧

Answer 6

= conjunction
= and, also, moreover
- true if: both true

Question 7

logical operators

- wedge (∨)/I

Answer 7

= disjunction
= or, unless
- inclusive: both possibilities are allowed to happen at the same point
- true if: one of the two OR both true

Question 8

logical operators

- horseshoe (⊃)/–>

Answer 8

= implication; conditionals
= if…then, only if
- true if: the second is true OR the first one is false

Question 9

conditionals

Answer 9

= expresses the relation of material implication

• antecedent = first letter
• -> statement following ‘if’
• consequent = second letter
• -> statement following ‘only if’

Question 10

conditionals

- sufficient condition

Answer 10

= A is sufficient for event B whenever the occurrence of A is ALL THAT IS REQUIRED for the occurrence of B
- placed in the antecedent of the conditional

Question 11

conditionals

- necessary condition

Answer 11

= A is necessary for B because B CANNOT OCCUR WITHOUT occurrence of A
- placed in the consequent

Question 12

logical operators

- triple bar (≡)/

Answer 12

= equivalence; biconditionals
= if and only if; then and only then
- true if: both true OR both false

Question 13

propositions

Answer 13

= statements that can be either true or false

Question 14

truth value

Answer 14

= function of the truth value of its components

Question 15

truth function

Answer 15

= any compound propositions whose truth value is completely determined by the truth values of its components

Question 16

truth tables

Answer 16

= arrangement of truth values that show every possible case how the truth value of a compound proposition is determined by the truth values of its simple components

Question 17

statement variables

Answer 17

= lowercase letters that can stand for any compound statement (truth value of combination)
- if P and Q true: P ∧ Q also true

Question 18

compute truth value of longer propositions

Answer 18

1. enter truth values of simple components directly beneath the letters
2. then use these truth values to compute the truth values of the compound components
3. the truth value of a compound statement is written beneath the operator representing it

Question 19

tautology

Answer 19

= logically true = tautologous statement

= statement which is always true

Question 20

contradiction

Answer 20

= logically false

= proposition which is always false

Question 21

contingency

Answer 21

= proposition which is sometimes true and sometimes false

Question 22

equivalence

Answer 22

= two statements are logically equivalent if they have the same truth value on each line under their main operators

Question 23

consistency

Answer 23

= if there is at least one line on which both (or all) of them turn out to be true

Question 24

inconsistency

Answer 24

= no line on which both (or all) are true

Question 25

valid argument forms

- disjunctive syllogism

Answer 25

= one of the premises presents two alternatives and the other eliminates one of them (method of elimination) 
P ∨ Q
∼/¬ P
-----
Q

Question 26

valid argument forms

- pure hypothetical syllogism

Answer 26

= two premises and one conclusion, all of which are hypothetical (conditional) statements
P ⊃/--> Q
Q ⊃/--> R
-----
P ⊃/--> R

Question 27

valid argument forms

- modus ponens (MP)

Answer 27

= a conditional premise, a second premise that asserts the antecedent of the conditional premise and a conclusion that asserts the consequent
P ⊃/--> Q
P
-----
Q

Question 28

valid argument forms

- mous tollens (MT)

Answer 28

= a conditional premise, a second premise that denies the consequent of the conditional premise and a conclusion that denies the antecedent
P ⊃/--> Q
∼/¬ Q
-----
∼/¬ P

Question 29

valid argument forms

- constructive dilemma

Answer 29

= a conjunctive premise made up of two conditional statements, a disjunctive premise that asserts the antecedents in the conjunctive premise (like MP) and a disjunctive conclusion that asserts the consequence of the conjunctive premise 
(P ⊃/--> Q) ⋅/∧ (R ⊃/--> S)
P ∨ R
-----
Q ∨ S

Question 30

valid argument forms

- destructive dilemma

Answer 30

(P ⊃/–> Q) ⋅/∧ (R ⊃/–> S)
∼/¬Q ∨ ∼/¬S
—–
∼/¬P ∨ ∼/¬R

Question 31

fallacies/invalid argument forms

- affirming the consequent

Answer 31

= a conditional premise, a second premise that asserts the consequent of the conditional and a conclusion that asserts the antecedent 
P ⊃/--> Q
Q
-----
P

Question 32

fallacies/invalid argument forms

- denying the antecedent

Answer 32

P ⊃/–> Q
∼/¬ P
—–
∼/¬ Q