Flashcards in TASK 8 - PROPOSITIONAL LOGIC Deck (32)

Loading flashcards...

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