Problem 9 Flashcards
Operator/Connective
Are used to “translate” english expressions
–> but as translation might not be identical, a certain distortion of the meaning will occur
Compound statement
Contains at least one simple statement as a component
ex.: Either people get serious about conservation or energy prices will skyrocket
Name the 5 Operators.
- Tilde
- Dot
- Wedge
- Horseshoe
- Triple bar
Tilde
Operator
Are used as a negation
ex.: not, it is not the case that
Tilde
Negation
Are used as a negation
–> ∼
ex.: “not”, “it is not the case that”
BUT: Only operator to be placed in front of the proposition
Dot
Conjunctive statement
Are used as a conjunction/addition
–> •
ex.: “and”, “also”, “moreover”
Wedge
Disjunctive statement
Are used as a disjunction
–> ∨
ex.: “or”, “unless/either or”
Horseshoe
Conditional statement
Are used to implicate something, containing a antecedent + consequent
–> ⊃
ex.: “if … then”, “only if/implies that”
REMEMBER: “if” follows antecedent, “only if” follows consequent
Triple bar
Biconditional Statement
Expresses the relation of material equivalence
–> ≡
ex.: “if and only if”
Main operator
Has its scope everything else in the statement
ex.: H • (J ∨ K) –> Dot is the main operator
Sufficient condition
Horseshoe
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
Necessary condition
Horseshoe
When B cannot occur without the occurrence of A
ex.: Having air to breathe (A) is necessary to survive (B)
–> consequent
Well-formed formulas
WWF
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
Truth function
Refers to any compound proposition whose truth value is completely determined by the truth values of its components
Statement variables
Refer to lowercase letters that can stand for the statements
–> used to construct statement forms
ex.: p,q,r,s
p = A • B
Statement form
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
According to the truth table, when is any of the 5 operator statements false or true?
- Tilde
- -> If P is false, its complement must be true - Dot
- -> both conjuncts must be true and false in all other cases - Wedge
- -> the disjunction is true when at least one of the disjuncts is true, otherwise is false - Horseshoe
- -> conditional statement is false when the antecedent is true + consequent is false - Triple bar
- -> biconditional is true when its 2 components have the same truth value
In which case is it inappropriate to use the triple bar operator ?
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
How do you construct a truth table ?
- Determine number of lines/rows
- Total number of lines is equal to the number of possible conventions of truth values
=> L = 2^n
Tautologous statement/
Logically true compound statement
Refers to a statement that is true regardless of the truth values of its components
Self-contradictory statement/ Logically false compound statement
Refers to a statement that is false regardless of the truth values of tis components
Contingent statement
Refers to a statement whose truth value varies depending on the truth values of its components
How do we determine whether a compound statement is self-contradictory, tautologous or contingent ?
By inspecting the column of truth values under the main operator
- Tautologous
- -> all true - Self-contradictory
- -> all false - Contingent
- -> at least one true or at least one false
Logically equivalent statements
Refer to 2 propositions that have the same truth value on each line under their main operators