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Philosophy Ch. 1, 2, & 3 > Unit 3 Definitions > Flashcards

Unit 3 Definitions Flashcards

1
Q

Define Logically Equivalent

A

When two statement forms have truth-values that agree in all instance. AKA, their truth tables agree in every row.

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2
Q

What is a Subformula?

A

A grammatically correct sentence that is a part of another grammatically correct sentence.

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3
Q

Identify the Subformula in this equation: (AvB) -> C

A

(AvB).

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4
Q

What is a Proof?

A

A proof is a sequence of statements, each of which is either a premise or a statement that is obtained from one or more earlier statements by applying one of the rules of inference.

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5
Q

What are the 10 Basic Inference Rules? (abbreviations)

A

~I, &E, vI, <->E, ->I, ~E, &I, vE, <->I, and ->E

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6
Q

What are the 3 Derived Rules? (abbreviations)

A

Modus Tollens, Hypothetical Syllogism, and Disjunctive Syllogism.

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7
Q

What are the 5 Equivalence Rules? (abbreviations)

A

DeMorgans, Double Negation, Transportaion, Material Implication, and Exportation.

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8
Q

What is the form for [~I]?

A

|p
|…
|q&~q
———
.: ~P

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9
Q

What are the two forms for [&E]?

A

.: p

.: q

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10
Q

What are the two forms for [vI]?

A

p
.: p v q

q
.: p v q

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11
Q

What are the two forms for [<->E]?

A

p <-> q
.: p -> q

p <-> q
.: q -> p

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12
Q

What is the form for [->I]?

A

|p
|…
|q
———
.: p -> q

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13
Q

What is the form for [~E]?

A

~~p
.: p

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14
Q

True or False: this is a valid form of the inference rule [~E]…
p
——
.: ~~p

A

FALSE. Inference rules do not go both ways. The only correct form for [~E] is:
~~p
——-
.: p

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15
Q

What is the form for [&I]?

A

p
q
——-
.: p&q

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16
Q

What is the form for [vE]?

A

p v q
p -> r
q -> r
———
r

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17
Q

What is the form for [<->I]?

A

p -> q
q -> p
———
.: p <-> q

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18
Q

What is the form for [->E]?

A

p -> q
p
——–
q

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19
Q

What is the form for Modus Tollens?

A

p -> q
~q
——–
~p

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20
Q

What is the form for Hypothetical Syllogism?

A

p -> q
q -> r
———
p -> r

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21
Q

What are the two forms for Disjunctive Syllogism?

A

p v q
~p
———
q

p v q
~q
———
p

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22
Q

What are the two forms for DeMorgans?

A

~(p&q) = ~p v ~q

~(pvq) = ~p & ~q

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23
Q

What is the form for [DN]?

A

p = ~~p

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24
Q

True or False: This is a valid form for Double Negation…
~~p
=
p

A

Yes. It is an equivalence rule so it can be used both ways. The order doesn’t matter because they are still able to be substituted for each other.

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25
Q

What is the form for Transportation?

A

p -> q = ~q -> ~p

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26
Q

What is the form for Material Implication?

A

p -> q = ~p v q

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27
Q

What is the form for [EXP]?

A

(p&q) -> r = p -> (q -> r)

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28
Q

Of the following, identify which represents an Instance:
1) D -> K, D, .: K
2) p -> q, p, .: q
3) p : D, q : k

A

1 is an instance. Sentence letters (representing statements) replaced the form’s variables.

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29
Q

Of the following, identify which represents a Substitution:
1) D -> K, D, .: K
2) p -> q, p, .: q
3) p : D, q : k

A

3 is a substitution. Sentence D takes the place of p and sentence K takes the place of q in the form structure.

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30
Q

Of the following, identify which represents a Form:
1) D -> K, D, .: K
2) p -> q, p, .: q
3) p : D, q : k

A

2 is a form. These are variables and don’t represent anything. They are meant to display a form’s structure so that they can later be replaced with sentence letters in proofs/truth tables/etc.

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31
Q

Are these sentence letters or variables:
A, B, C, D

A

Sentence letters, they represent statements!

32
Q

Are these sentence letters or variables:
p, q, r

A

Variables, they stand as placeholders in form structures for sentence letters!

33
Q

Why can’t Logicians just continue to use truth tables?

A
  1. truth tables don’t represent actual process (which are usually in steps)
  2. they don’t work for every deductively valid argument
  3. they can be too long
34
Q

Is (~(A <-> E) v C) -> ~(D&E) an instance of the form p -> q?

A

Yes, substitue!
p: (~(A <-> E) v C)
q: ~(D&E)

35
Q

True or False: All proofs are valid because each step must be verified by a rule.

A

True.

36
Q

True or False: you can hypothesis whatever, whenever in proofs.

A

False. You can can (or rather should) only hypothesize if you are trying to prove [~I] or [->I].

37
Q

Fill in the blank: Every _______ begins a new vertical line.

A

Hypothesis.

38
Q

Can a line in a hypothesis’ scope be used/cited after it is has been discharged?

A

No. Any citing or usage must be done within the hypothesis’ scope, otherwise we would be wrongfully combining the real and hypothetical world.

39
Q

True or False: Hypotheses can overlap.

A

False. A hypothesis made in the scope of another hypothesis needs to be discharged before the first one is.

40
Q

What is Modus Tollens?

A

MT. From any statement p -> q and ~q, we may infer ~p.

p -> q
~q
——–
~p

41
Q

What is Hypothetical Syllogism?

A

HS. From any statement p -> q and q -> r, we can infer p -> r.

p -> q
q -> r
———
p -> r

42
Q

What is Disjunctive Syllogism?

A

DS. From any statement p v q and ~q, we may infer p.

p v q
~p
———
q

p v q
~q
———
p

43
Q

Fill in the blank: The given statement for a theorem should placed in the ______ line of the proof.

A

Last, always. It is the conclusion!

44
Q

Fill in the blank: A statement form is a theorem iff it is a _____.

A

Tautology.

45
Q

True or False: All theorems tautologous.

A

True. Tautology <-> Theorem.

46
Q

True or False: If you can prove p1…p2 .: C, then the related formula (p1…p2) -> C is a theorem.

A

True, it is also a tautology. And also vice versa.

47
Q

What are the 3 features of Proofs of Theorems?

A
  1. always uses either [~I] or [->I]
  2. the first step is always a hypothesis
  3. reasoning always goes from bottom up
48
Q

True or False: Basic and Derived rules can be applied to subformulas.

A

False. Basic and Derived rules can only be applied to whole lines; Equivalence rules can be applied to both subformulas and whole lines.

49
Q

What is the difference between [DN] and [~E]?

A

~E only works on way: if you have ~~p then you can assume p. DN works both ways, if you have p you can assume ~~p and vice versa.

50
Q

Fill in the blank: If you need to switch a wedge to an arrow, use _____.

A

MI.

51
Q

True or False: Two statement forms are Logically Equivalent if they can be substituted for one another, even if the truth value changes.

A

False. They are logically equivalent if they can be substituted for each other WITHOUT the truth values changing.

52
Q

True or False: Two statement forms, p and q, are Logically Equivalent because p <-> q is a tautology.

A

True.

53
Q

Fill in the Blank: To show that p is a contradiction, you need to prove that ~p is a ______.

A

Tautology.

54
Q

True or False: You can use proofs to prove that a statement form, like p, is a contingency.

A

False.

55
Q

True or False: You can prove that an argument is invalid using a proof.

A

False, proofs only prove validity because it needs the approval/guidelines of the rules, never invalidity.

56
Q

What is the proof strategy for a premise with a ~ as the major operator?

A

Try driving the ~ inwards using DeMorgans, MI, etc.

57
Q

What is the proof strategy for a premise with a <-> as the major operator?

A

Try <->E.

58
Q

What is the proof strategy for a premise with a -> as the major operator?

A

Look for ->E or Modus Tollens.

59
Q

What is the proof strategy for a premise with a v as the major operator?

A

Look for Disjunctive Syllogism or vE.

60
Q

What is the proof strategy for a premise with a & as the major operator?

A

Try &E.

61
Q

What is the proof strategy for a conclusion with a <-> as the major operator?

A

Try <->I.

62
Q

What is the proof strategy for a conclusion with a -> as the major operator?

A

Try ->I.

63
Q

What is the proof strategy for a conclusion with a v as the major operator?

A

Try ->I, Material Implication, or Double Negation to get it.

64
Q

What is the proof strategy for a conclusion with a & as the major operator?

A

Try &I to get.

65
Q

What is the proof strategy for a conclusion with a ~ as the major operator?

A

1) Try ~I
2) if it is a complex statement (has subformulas) try driving ~ inwards using DeMorgans, MI, etc.

66
Q

True or False: Every argument form whose conclusion is a tautology is valid.

A

True.

67
Q

True or False: “A unless B” is equivalent to: “Either B or not A”.

A

False.

68
Q

True or False: It is possible for a compound sentence to contain only one component sentence.

A

True.

69
Q

What is Double Negation?

A

[DN]. Like [~E] except it goes both ways.
p :: ~~p

70
Q

What is DeMorgans?

A

[DM].
~(p&q) :: ~p v ~q
~(pvq) :: ~p & ~q

71
Q

What is Transportation?

A

[TRANS]. A rewriting of Modus Tollens.
p -> q :: ~q -> ~p
**keep in mind order and negations

72
Q

What is Material Implication?

A

[MI]. Meant to switch arrows to wedges and vice versa.
p -> q :: ~p v q

73
Q

What is Exportation?

A

[EXP].
(p&q) -> r :: p -> (q -> r)

74
Q

What is this form called and what kind of rule is it:
p -> q :: ~q -> ~p

A

Transportation, equivalence rule.

75
Q

What form is this and what kind of rule is it:
p -> q :: ~p v q

A

Material Implication, equivalence rule.

Philosophy Ch. 1, 2, & 3 (4 decks)

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