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
Contradictory statements
Refer to 2 propositions that have opposite truth values on each line under their main operators
Consistent statements
Refer to 2 or more propositions where there is at least one line on which both of them turn out to be true
Inconsistent statements
Refer to 2 or more propositions where there is no line where both of them turn out to be true
How do you construct a truth table for arguments ?
- Symbolize arguments by using letters
- Write the argument out, using a single slash between the premises + double slash between last premise + conclusion
- Draw a truth table
- -> outline the columns who represent the P + C - Look for a line in which all of the premises are true + conclusion false
- -> invalid
Arguments corresponding conditional
Refers to the conditional segment having the conjunction of an arguments premises as its antecedent + the conclusion as its consequent
How do you check whether an argument is valid ?
- By assuming it is invalid, meaning the premises are true but conclusion false
- Then reconstructing the truth values
- If there is a contradiction the argument is valid
=> Same for consistency, assuming everything is true, if contradiction = inconsistent
Disjunctive syllogism
Valid Argument form
Consists of a disjunctive statement for one of its premises
ex.: H ∨ P ∼ H ------- P
Pure hypothetical syllogism
Valid Argument form
Consists of 2 premises + one conclusion, all of which are hypothetical statements
ex.: p⊃q q⊃r ------- p⊃r
Modus ponens
Valid Argument form
Consists of a
- conditional premise
- second premise that includes the antecedent of the conditional premise
- conclusion with the consequent
ex.: p⊃q p ------ q
Modus tollens
Argument form
Consists of
- conditional premise
- Second premise that denies the consequent of the conditional premise
- conclusion that denies the antecedent
ex.: p⊃q ∼q ------ ∼p
Affirming the consequent
Invalid Argument form
Consists of
- conditional premise
- Second premise that includes the consequent of the conditional
- conclusion that includes the antecedent
=> positive form of modus tollens
ex.: p⊃q q ------ p
Denying the antecedent
Invalid argument form
Consists of
- Conditional premise
- Second premise that denies the antecedent of the conditional
- Conclusion that denies the consequent
=> negative form of modus ponens
ex.: p⊃q ∼p ------- ∼q
Constructive dilemma (Valid argument form)
Is a valid argument form that consists of a conjunctive premise made up of 2 conditional statements
- Disjunctive premise including the antecedents of the conjunctive premise
- Disjunctive conclusion including the consequents of the conjunctive premise
ex.: (p ⊃ q) • (r ⊃ s) p∨r -------------------- q∨s
Destructive dilemma (Valid argument form)
Similar to constructive dilemma, but negative form
```
ex.:
p ⊃ q) • (r ⊃ s
∼q ∨ ∼s
——————–
∼p ∨ ∼r
~~~
What is the difference between the inclusive vs exclusive “or” ?
Inclusive
–> at least one has to be true, both can
Exclusive
–> only one can be true, not both
