TASK 8 - PROPOSITIONAL LOGIC Flashcards
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
simple statement
= one that does not contain another statement as a component (e.g. fastfood is unhealthy’)
- statement is represented by an uppercase letter
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
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
logical operators
- tilde (∼)/¬
= negation
= not, it is not the case that
- always in front of the proposition it negates
- true if: false
logical operators
- dot (⋅)/∧
= conjunction
= and, also, moreover
- true if: both true
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
logical operators
- horseshoe (⊃)/–>
= implication; conditionals
= if…then, only if
- true if: the second is true OR the first one is false
conditionals
= expresses the relation of material implication
- antecedent = first letter
- -> statement following ‘if’
- consequent = second letter
- -> statement following ‘only if’
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
conditionals
- necessary condition
= A is necessary for B because B CANNOT OCCUR WITHOUT occurrence of A
- placed in the consequent
logical operators
- triple bar (≡)/
= equivalence; biconditionals
= if and only if; then and only then
- true if: both true OR both false
propositions
= statements that can be either true or false
truth value
= function of the truth value of its components
truth function
= any compound propositions whose truth value is completely determined by the truth values of its components
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
statement variables
= lowercase letters that can stand for any compound statement (truth value of combination)
- if P and Q true: P ∧ Q also true
compute truth value of longer propositions
- enter truth values of simple components directly beneath the letters
- then use these truth values to compute the truth values of the compound components
- the truth value of a compound statement is written beneath the operator representing it
tautology
= logically true = tautologous statement
= statement which is always true
contradiction
= logically false
= proposition which is always false
contingency
= proposition which is sometimes true and sometimes false
equivalence
= two statements are logically equivalent if they have the same truth value on each line under their main operators
consistency
= if there is at least one line on which both (or all) of them turn out to be true
inconsistency
= no line on which both (or all) are true
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
valid argument forms
- pure hypothetical syllogism
= two premises and one conclusion, all of which are hypothetical (conditional) statements P ⊃/--> Q Q ⊃/--> R ----- P ⊃/--> R
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
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
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
valid argument forms
- destructive dilemma
(P ⊃/–> Q) ⋅/∧ (R ⊃/–> S)
∼/¬Q ∨ ∼/¬S
—–
∼/¬P ∨ ∼/¬R
fallacies/invalid argument forms
- affirming the consequent
= a conditional premise, a second premise that asserts the consequent of the conditional and a conclusion that asserts the antecedent P ⊃/--> Q Q ----- P
fallacies/invalid argument forms
- denying the antecedent
P ⊃/–> Q
∼/¬ P
—–
∼/¬ Q
