Phi 112 - Intermediate Symbolic Logic (The Logic Book)

Front 1

Quantificational Indeterminacy (set)

Back 1

iff neither {P} nor {~ P} has a closed truth-tree

Front 2

Closed Truth-Tree

Back 2

A truth-tree each of whose branches is closed

Front 3

Identity Elimination

Front 4

Conjunction Decomposition

Front 5

(∀y) (∀x) Lyx

Back 5

"Everyone likes everyone."

Front 6

Negated Negation Decomposition

Front 7

Existential Elimination

Front 8

(∀x) (∃y) Lxy

Back 8

"Everyone likes someone."

Front 9

State ‘(∀x)Lxm’ in terms of (∃x)

Back 9

∼ (∃x) ∼ Lxm

Front 10

Biconditional Introduction

Front 11

Closed Branch

Back 11

A branch containing both an atomic sentence and the negation of that sentence

Front 12

Equivalence in PD

Back 12

Sentences P and Q are equivalent in PD iff P is derivable in PD from {Q} and Q is derivable from {P}

Front 13

Sentence of PL

Back 13

Where all variables are metavariables:

A formula P of PL is a sentence of PL iff no occurrence of a variable in P is free (unquantified).

Front 14

Negated Conditional Decomposition

Front 15

Universal Decomposition

Front 16

Implication

Front 17

Is ‘(∀z)(Haz ⊃ (∀z)(Fz ⊃ Gza))’ a formula of PL?

Back 17

No: quantifier appears more than once within its own scope

Front 18

Universal Introduction in PDE

Front 19

(∃y) (∀x) Lxy

Back 19

"Someone is liked by everyone."

Front 20

Negation Introduction

Front 21

Atomic formula of PL

Back 21

Where all variables are metavariables:

Every expression of PL that is either a sentence letter of PL or an n-place predicate of PL followed by n-individual terms of PL is an atomic formula. No connectives or quantifiers.

Front 22

Quantificational Consistency (set)

Back 22

Set does not have a closed truth-tree

Front 23

Quantificational Validity (set)

Back 23

An argument of PL with finite premises is q-valid iff the set consisting of the premises and the negation of the conclusion is closed

Front 24

Hypothetical Syllogism

Front 25

Quantificational Entailment (set)

Back 25

A finite set Delta of PL q-entails P iff Delta U {~ P} has a closed truth-tree

Front 26

Completed Open Branch

Back 26

A finite branch on which each sentence is one of the following:
1. A literal (atomic or negated atomic sentence)
2. A decomposed compound sentence that is not a universally quantified sentence
3. A universally quantified sentence (\/x)P so that P(a/x) occurs on the branch for each constant a that occurs on the branch and at least one substitution instance P(a/x) occurs on the branch

Front 27

Existential Introduction in PDE

Front 28

Quantifier of PL

Back 28

Where all variables are metavariables:

An expression of PL of the form (∀x) or (∃x).
(∀x) = Universal quantifier.
(∃x) = Existential quantifier.

Front 29

Disjunction Decomposition

Front 30

(∃y) (∃x) Lyx

Back 30

"Someone likes someone."

Front 31

Exportation

Front 32

Universal Introduction

Front 33

Formula of PL

Back 33

Where all variables are metavariables:

1. Every atomic formula of PL is a formula of PL.
2. If P is a formula of PL, so is ∼ P.
3. If P and Q are formulas of PL, so are (P & Q), (P ∨ Q), (P ⊃ Q), and (P ≡ Q).
4. If P is a formula of PL that contains at least one occurrence of x and no x-quantifier, then (∀x)P and (∃x)P are both formulas of PL.
5. Nothing is a formula of PL unless it can be formed by repeated applications of clauses 1 to 4.

Front 34

Expression of PL

Back 34

Any sequence of the vocabulary of PL

Front 35

Theorem in PD

Back 35

A sentence P of PL is a theorem of PD iff it is derivable from the empty set

Front 36

True on an interpretation

Back 36

where all variables are metavariables:

a sentence P of PL is true on an interpretation I iff every variable assignment d (for I) satisfies P on I

Front 37

Negated Biconditional Decomposition

Front 38

Negation Elimination

Front 39

Form E

Back 39

Where all variables are metavariables:

(∀x) (P ⊃ ~Q)

Front 40

Negated Universal Decomposition

Front 41

Open sentence of PL

Back 41

A forumla of PL that is not a sentence of PL.

Front 42

Equivalence

Front 43

Is ‘(∀y)(Hay ⊃ (Fy ⊃ Gya))’ a forumula of PL?

Back 43

Yes

Front 44

Existential Elimination

Front 45

Symbolize in pseudo-English: 'The Roman general who defeated Pompey invaded both Gaul and Germany.'

Back 45

There is exactly one thing that is a Roman general and defeated Pompey and that thing invaded both Gaul and Germany.

Front 46

Form O

Back 46

Where all variables are metavariables:

(∃x) (P & ~Q)

Front 47

Expand for UD with constants a, b, and c: (∃x)(Fx ⊃ Gx)

Back 47

[(Fa ⊃ Ga) ∨ (Fb ⊃ Gb)] ∨ (Fc ⊃ Gc)

Front 48

Identity Decomposition

Front 49

Modus Tollens

Front 50

Open Branch

Back 50

A branch that is not closed

Front 51

Reiteration

Front 52

What is the contradictory of Form A (∀x)(P⊃Q)?

Back 52

Form O: (∃x)(P & ~Q)

Front 53

Negated Conjunction Decomposition

Front 54

Quantificational Falsity (truth-tree)

Back 54

iff set {P} has closed truth-tree

Front 55

Quantificational Equivalence

Back 55

Sentences meta-P and meta-Q of PL/E are quantificationally equivalent iff there is no interpretation where meta-P and meta-Q have different truth-values

Front 56

Quantifier Negation (QN)

Front 57

State ‘(∃x)Lxm’ in terms of (∀x)

Back 57

~ (∀x) ~ Lxm

Front 58

Identity Introduction

Front 59

Quantificational Entailment (|=)

Back 59

a set Γ of sentences of PL quantificationally entails a sentence meta-P of PL iff there is no interpretation where every member of Γ is true and meta-P is false

Front 60

Inconsistency in PD

Back 60

A set Gamma of sentences of PL is inconsistent in PD iff there is a sentence P where P and ~ P are derivable in PD from Gamma

Front 61

what are the only quantificational properties thatcan be proven directly?

Back 61

consistency and indeterminacy

Front 62

What is the contradictory of Form E (∀x)(P ⊃ ~Q)?

Back 62

Form I (∃x) (P&Q)

Front 63

Disjunction Elimination

Front 64

Quantificational Truth (truth-tree)

Back 64

iff the set {~ P} has a closed truth tree

Front 65

Quantificational Indeterminacy

Back 65

A sentence meta-P of PL/PLE is quantificationally indeterminate iff meta-P is neither quantificationally true nor quantificationally false

Front 66

Existential Decomposition

Front 67

Existential Elimination in PDE

Front 68

Quantificational consistency

Back 68

a set of sentences of PL is quantificationally consistent iff there is at least one interpretation on whch all members of the set are true

Front 69

Conditional Decomposition

Front 70

Where all variables are metavariables and P does not contain x,
(∀x)Ax ⊃ P is equivalent to:

Back 70

(∃x)(Ax ⊃ P)

Front 71

Quantificational validity

Back 71

An argument of PL/E with a finite number of premises is quantificationally valid iff there is no interpretation on which every premise is true and the conclusion is false

Front 72

Universal Decomposition in PLE

Front 73

Commutation

Front 74

Logical operator of PL

Back 74

An expression of PL that is either a quantifier or truth-functional connective

Front 75

Disjunctive Syllogism

Front 76

Quantificational Inconsistency

Back 76

a set of sentences of PL is quantificationally inconsistent iff there is no interpretation on which all members of the set are true

Front 77

Completed Truth-Tree

Back 77

A truth-tree each of whose branches either is closed or is a completed open branch

Front 78

Conjunction Elimination

Front 79

Open Truth-Tree

Back 79

A truth-tree that is not closed

Front 80

Form A

Back 80

Where all variables are metavariables:
(∀x) (P ⊃ Q)

Front 81

Expand for UD containing constants a and c:
(∀x)(Gac ∨ Fx)

Back 81

(Gac ∨ Fa) & (Gac ∨ Fc)

Front 82

Negated Disjunction Decomposition

Front 83

Quantificational Equivalence (set)

Back 83

iff the set {~ (P<>Q)} has a closed truth-tree

Front 84

"Everybody loves somebody sometime"

Back 84

U.D.: people and times
Px: x is a person
Tx: x is a time
Lxyz: x loves y at z

(\/x)(Px ⊃ (∃y)(∃z)[(Py & Tz) & Lxyz]

Front 85

Vocabulary of PL

Back 85

Sentence letters
Predicates
Individual terms (constants/variables)
Truth-functional connectives
Quantifier symbols
Punctuation marks

Front 86

Universal Elimination in PDE

Front 87

Quantificational Falsity

Back 87

A sentence meta-P of PL/PLE is quantificationally false iff meta-P is false on every interpretation

Front 88

False on an interpretation

Back 88

where all variables are metavariables:

a sentence P of PL is true on an interpretation I iff no variable assignment d (for I) satisfies P on I

Front 89

Existential Introduction

Front 90

Disjunction Introduction

Front 91

Where all variables are metavariables and P does not contain x,
(∃x)Ax ⊃ P is equivalent to:

Back 91

(∀x)(Ax ⊃ P)

Front 92

Validity in PD

Back 92

An argument of PL is valid in PD iff the conclusion of the argument is derivable from the set consisting of the premises

Front 93

Derivability in PD

Back 93

Sentence P is derivable in PD from set Delta iff there is a derivation in PD in which all of the primary assumptions are members of set Delta and P appears within the scope of only the primary assumptions

Front 94

Biconditional Decomposition

Front 95

Negated Existential Decomposition

Front 96

Quantificational truth

Back 96

A sentence meta-P of PL/PLE is quantificationally true iff meta-P is true on every interpretation

Front 97

Form I

Back 97

Where all variables are metavariables:

(∃x) (P & Q)

Front 98

Universal Elimination

Front 99

Distribution

Front 100

Is ‘(∀x)(Haz ⊃ (∀z)(Fz ⊃ Gza))’ a forumla of PL?

Back 100

No: variable x does not appear within scope of x-quantifier

Front 101

Conjunction Introduction

Front 102

Quantificational Inconsistency (set)

Back 102

Set has a closed truth-tree

Front 103

Expand for UD containing constants a and b:
(∀x)(Wx ⊃ (∃y)Cxy)

Back 103

(Wa ⊃ (Caa ∨ Cab)) & (Wb ⊃ (Cba ∨ Cbb))

Front 104

Biconditional Elimination