TASK 8 - PROPOSITIONAL LOGIC Flashcards Preview

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Flashcards in TASK 8 - PROPOSITIONAL LOGIC Deck (32)
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1

propositional logic

= fundamental elements are whole statements (propositions)
- statements are represented by letters
- statements are combined by means of the operators to represent more complex statements

2

simple statement

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

3

compound statement

= 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

4

logical operators
- main operator

= 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

5

logical operators
- tilde (∼)/¬

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

6

logical operators
- dot (⋅)/∧

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

7

logical operators
- wedge (∨)/I

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

8

logical operators
- horseshoe (⊃)/-->

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

9

conditionals

= expresses the relation of material implication
- antecedent = first letter
--> statement following ‘if’
- consequent = second letter
--> statement following ‘only if’

10

conditionals
- sufficient condition

= 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

11

conditionals
- necessary condition

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

12

logical operators
- triple bar (≡)/

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

13

propositions

= statements that can be either true or false

14

truth value

= function of the truth value of its components

15

truth function

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

16

truth tables

= 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

17

statement variables

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

18

compute truth value of longer propositions

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

19

tautology

= logically true = tautologous statement
= statement which is always true

20

contradiction

= logically false
= proposition which is always false

21

contingency

= proposition which is sometimes true and sometimes false

22

equivalence

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

23

consistency

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

24

inconsistency

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

25

valid argument forms
- disjunctive syllogism

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

26

valid argument forms
- pure hypothetical syllogism

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

27

valid argument forms
- modus ponens (MP)

= 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

28

valid argument forms
- mous tollens (MT)

= 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

29

valid argument forms
- constructive dilemma

= 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

30

valid argument forms
- destructive dilemma

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