Log335

Front 1

Not True

Back 1

False/Indeterminate

Front 2

Inductive Hypothesis

Back 2

Suppose that for every sentence P whose noc≤k for some arbitrary k in lN is such that P has property...

Front 3

P_k

Back 3

The sentence on the kth line of the derivation

Front 4

Gamma_k

Back 4

The set of open assumptions in whose scope P_k lies

Front 5

Universal Introduction

Back 5

P(a/x)
>(∀x)P

a doesn't occur in an open assumption
a doesn't occur in (∀x)P

Front 6

Existential Introduction

Back 6

P(a/x)
>(∃x)P

Front 7

Universal Elimination

Back 7

(∀x)P
P(a/x)

Front 8

Existential Elimination

Back 8

(∃x)P
| P(a/x)
⊢-------
| Q
Q

a isn't in an open assumption
a doesn't occur in (∃x)P
a doesn't occur in Q

Front 9

P is t-f true iff

Back 9

P is true on every t-v assignment

Front 10

P and Q of SL are are t-f equivalent iff

Back 10

there is no t-v assignment on which P and Q have different truth values

Front 11

An argument of SL is t-f valid iff

Back 11

there is not t-v assignment on which all the premises are true and the conclusion is false

Front 12

A derivation is SD

Back 12

a series of sentences of SL, each which is an assumption or is obtained from previous sentences by 1 of the rules of SD

Front 13

Theorem in SD

Back 13

Any sentence in SL that can be derived from the empty set

Front 14

Inconsistent in SD

Back 14

There's a derivation of P and ~P from Γ

Front 15

If Γ⊨P and Γ⊆Γ', then

Back 15

Γ'⊨P

Front 16

If ΓU{P}⊨Q, then

Back 16

Γ⊨P⊃Q

Front 17

If Γ⊨Q and Γ⊨~Q, then

Back 17

Γ is t-f inconsistent

Front 18

If ΓU{Q} is t-f inconsistent, then

Back 18

Γ⊨~Q

Front 19

Outline Completeness

Back 19

Γ⊨P
=> ΓU{~P} is t-f inconsistent
=> ΓU{~P} is inconsistent in SD
=> Γ⊢P

Front 20

Immediate Sentential Components
Sentential Components

Back 20

2 Sentences connected by main connective
Sentential Components are made up of
-Sentence itself
-Immediate Sentential Components
-Sentential Components of Immediate Sentential Components

Front 21

Name the 2 types of assumptions

Back 21

Primary & Auxiliary

Front 22

A sentence P is true iff

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what the sentence says or expresses is the case

Front 23

A sentence P of SL is derivable in SD from a set Γ iff

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there is a derivation in SD in which all the primary assumptions are member of Γ and P occurs within the scope of only the primary assumptions

Front 24

An argument of SL is valid in SD iff

Back 24

the conclusion of the argument is derivable in SD from the set consisting of the premises

Front 25

Equivalence in SD

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Q is derivable in SD from {P} and P is derivable in SD from {Q}

Front 26

Define an interpretation

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-Define a UD
-To each 1 place predicate, assign a subset (possibly empty) of UD
-To each 2 place predicate, assign a set of ordered pairs, whose elements come from UD.
-To each 3-place predicate, assign a set of ordered triples, whose elements come from UD.
...
To each individual constant, assign 1 object from UD.

Front 27

The scope of a quantifier in a formula P of PL is

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the subformula Q of P of which that quantifier is the main logical operator.

Front 28

Bound variable

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A variable x is bound iff it is in the scope of an x-quantifier.

Front 29

Free Variable

Back 29

An occurrence of a variable x in a formula P of PL that is not bound

Front 30

ΓU{P} is inconsistent in SD iff

Back 30

Γ⊢~P

Front 31

Γ⊢~P iff

Back 31

ΓU{P} is inconsistent in SD

Front 32

ΓU{~P} is inconsistent in SD iff

Back 32

Γ⊢P

Front 33

Γ⊢P iff

Back 33

ΓU{~P} is inconsistent in SD

Front 34

For any sentence P of SL, if P is not ϵΓ*, then

Back 34

Γ*U{P} is inconsistent in SD

Front 35

If Γ*U{P} is consistent in SD, then

Back 35

PϵΓ*

Front 36

For any sentence P of SL, if Γ*⊢P, then

Back 36

PϵΓ*

Front 37

6.4.11a

What's an easy way to prove stuff from 6.4.11?

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~PϵΓ* iff P is not ϵΓ*

Create a derivation

Front 38

6.4.11b

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P&QϵΓ* iff PϵΓ* and QϵΓ*

Proven by &E

Front 39

Chris supports neither union members nor administrators

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~(∃x)((Ux&Ax)&Scx)

Front 40

d[u/x]

Back 40

A variant of a variable assignment d that assigns the same value to each variable as d does except it assigns u to x.

Front 41

A sentence P of PL is derivable in PD from a set Γ of sentences of PL iff

Back 41

there is a derivation in PD in which all the primary assumptions are members of Γ and P occurs in the scope of only those assumptions

Front 42

An argument of PL is valid in PD iff

Back 42

the conclusion of the argument is derivable in PD from the set consisting of the premises

Front 43

A sentence P is a theorem in PD iff

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P is derivable in PD from {}.

Front 44

d satisfies (∃y)Gy iff

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there's at least 1 member of the UD such that d[a/y] satisfies Gy

Front 45

Quantificationally True

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A sentence P of PL is quantificationally true iff P is true on every interpretation I

Front 46

Sentences P and Q of SL are Quantificationally Equivalent iff

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there's no interpretation on which P and Q have different truth-values

Front 47

A set Γ of sentences of PL is quantificationally consistent iff

Back 47

there is at least 1 interpretation on which all the members are true.

Front 48

A set Γ of sentences of PL is quantificationally valid iff

Back 48

There is no interpretation in which all the premises are true and the conclusion is false

Front 49

A set Γ of sentences of PL quantificationally entails a sentence P of PL iff

Back 49

there is no interpretation which all the members of Γ are true and P is false

Front 50

A sentence is true on an interpretation I iff

false iff

Back 50

Every variable assignment for I satisfies P

No variable assignment for I satisfies P