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Flashcards in Problem 9 Deck (39)
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1
Q

Operator/Connective

A

Are used to “translate” english expressions

–> but as translation might not be identical, a certain distortion of the meaning will occur

2
Q

Compound statement

A

Contains at least one simple statement as a component

ex.: Either people get serious about conservation or energy prices will skyrocket

3
Q

Name the 5 Operators.

A
  1. Tilde
  2. Dot
  3. Wedge
  4. Horseshoe
  5. Triple bar
4
Q

Tilde

Operator

A

Are used as a negation

ex.: not, it is not the case that

5
Q

Tilde

Negation

A

Are used as a negation

–> ∼

ex.: “not”, “it is not the case that”

BUT: Only operator to be placed in front of the proposition

6
Q

Dot

Conjunctive statement

A

Are used as a conjunction/addition

–> •

ex.: “and”, “also”, “moreover”

7
Q

Wedge

Disjunctive statement

A

Are used as a disjunction

–> ∨

ex.: “or”, “unless/either or”

8
Q

Horseshoe

Conditional statement

A

Are used to implicate something, containing a antecedent + consequent

–> ⊃

ex.: “if … then”, “only if/implies that”

REMEMBER: “if” follows antecedent, “only if” follows consequent

9
Q

Triple bar

Biconditional Statement

A

Expresses the relation of material equivalence

–> ≡

ex.: “if and only if”

10
Q

Main operator

A

Has its scope everything else in the statement

ex.: H • (J ∨ K) –> Dot is the main operator

11
Q

Sufficient condition

Horseshoe

A

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

12
Q

Necessary condition

Horseshoe

A

When B cannot occur without the occurrence of A

ex.: Having air to breathe (A) is necessary to survive (B)

–> consequent

13
Q

Well-formed formulas

WWF

A

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

14
Q

Truth function

A

Refers to any compound proposition whose truth value is completely determined by the truth values of its components

15
Q

Statement variables

A

Refer to lowercase letters that can stand for the statements

–> used to construct statement forms

ex.: p,q,r,s
p = A • B

16
Q

Statement form

A

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

17
Q

According to the truth table, when is any of the 5 operator statements false or true?

A
  1. Tilde
    - -> If P is false, its complement must be true
  2. Dot
    - -> both conjuncts must be true and false in all other cases
  3. Wedge
    - -> the disjunction is true when at least one of the disjuncts is true, otherwise is false
  4. Horseshoe
    - -> conditional statement is false when the antecedent is true + consequent is false
  5. Triple bar
    - -> biconditional is true when its 2 components have the same truth value
18
Q

In which case is it inappropriate to use the triple bar operator ?

A

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

19
Q

How do you construct a truth table ?

A
  1. Determine number of lines/rows
  2. Total number of lines is equal to the number of possible conventions of truth values

=> L = 2^n

20
Q

Tautologous statement/

Logically true compound statement

A

Refers to a statement that is true regardless of the truth values of its components

21
Q

Self-contradictory statement/ Logically false compound statement

A

Refers to a statement that is false regardless of the truth values of tis components

22
Q

Contingent statement

A

Refers to a statement whose truth value varies depending on the truth values of its components

23
Q

How do we determine whether a compound statement is self-contradictory, tautologous or contingent ?

A

By inspecting the column of truth values under the main operator

  1. Tautologous
    - -> all true
  2. Self-contradictory
    - -> all false
  3. Contingent
    - -> at least one true or at least one false
24
Q

Logically equivalent statements

A

Refer to 2 propositions that have the same truth value on each line under their main operators

25
Q

Contradictory statements

A

Refer to 2 propositions that have opposite truth values on each line under their main operators

26
Q

Consistent statements

A

Refer to 2 or more propositions where there is at least one line on which both of them turn out to be true

27
Q

Inconsistent statements

A

Refer to 2 or more propositions where there is no line where both of them turn out to be true

28
Q

How do you construct a truth table for arguments ?

A
  1. Symbolize arguments by using letters
  2. Write the argument out, using a single slash between the premises + double slash between last premise + conclusion
  3. Draw a truth table
    - -> outline the columns who represent the P + C
  4. Look for a line in which all of the premises are true + conclusion false
    - -> invalid
29
Q

Arguments corresponding conditional

A

Refers to the conditional segment having the conjunction of an arguments premises as its antecedent + the conclusion as its consequent

30
Q

How do you check whether an argument is valid ?

A
  1. By assuming it is invalid, meaning the premises are true but conclusion false
  2. Then reconstructing the truth values
  3. If there is a contradiction the argument is valid

=> Same for consistency, assuming everything is true, if contradiction = inconsistent

31
Q

Disjunctive syllogism

Valid Argument form

A

Consists of a disjunctive statement for one of its premises

ex.: 
H ∨ P
∼ H
-------
P
32
Q

Pure hypothetical syllogism

Valid Argument form

A

Consists of 2 premises + one conclusion, all of which are hypothetical statements

ex.: 
p⊃q
q⊃r 
-------
p⊃r
33
Q

Modus ponens

Valid Argument form

A

Consists of a

  1. conditional premise
  2. second premise that includes the antecedent of the conditional premise
  3. conclusion with the consequent
ex.: 
p⊃q 
p
------
q
34
Q

Modus tollens

Argument form

A

Consists of

  1. conditional premise
  2. Second premise that denies the consequent of the conditional premise
  3. conclusion that denies the antecedent
ex.:
p⊃q 
∼q 
------
∼p
35
Q

Affirming the consequent

Invalid Argument form

A

Consists of

  1. conditional premise
  2. Second premise that includes the consequent of the conditional
  3. conclusion that includes the antecedent

=> positive form of modus tollens

ex.:
p⊃q
q
------
p
36
Q

Denying the antecedent

Invalid argument form

A

Consists of

  1. Conditional premise
  2. Second premise that denies the antecedent of the conditional
  3. Conclusion that denies the consequent

=> negative form of modus ponens

ex.:
p⊃q 
∼p 
-------
∼q
37
Q
Constructive dilemma 
(Valid argument form)
A

Is a valid argument form that consists of a conjunctive premise made up of 2 conditional statements

  1. Disjunctive premise including the antecedents of the conjunctive premise
  2. Disjunctive conclusion including the consequents of the conjunctive premise
ex.: 
(p ⊃ q) • (r ⊃ s) 
p∨r
--------------------
q∨s
38
Q
Destructive dilemma 
(Valid argument form)
A

Similar to constructive dilemma, but negative form

```
ex.:
p ⊃ q) • (r ⊃ s
∼q ∨ ∼s
——————–
∼p ∨ ∼r
~~~

39
Q

What is the difference between the inclusive vs exclusive “or” ?

A

Inclusive
–> at least one has to be true, both can

Exclusive
–> only one can be true, not both