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50 Cards in this Set
- Front
- Back
Not True
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False/Indeterminate
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Inductive Hypothesis
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Suppose that for every sentence P whose noc≤k for some arbitrary k in lN is such that P has property...
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P_k
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The sentence on the kth line of the derivation
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Gamma_k
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The set of open assumptions in whose scope P_k lies
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Universal Introduction
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P(a/x)
>(∀x)P a doesn't occur in an open assumption a doesn't occur in (∀x)P |
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Existential Introduction
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P(a/x)
>(∃x)P |
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Universal Elimination
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(∀x)P
P(a/x) |
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Existential Elimination
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(∃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 |
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P is t-f true iff
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P is true on every t-v assignment
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P and Q of SL are are t-f equivalent iff
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there is no t-v assignment on which P and Q have different truth values
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An argument of SL is t-f valid iff
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there is not t-v assignment on which all the premises are true and the conclusion is false
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A derivation is SD
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a series of sentences of SL, each which is an assumption or is obtained from previous sentences by 1 of the rules of SD
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Theorem in SD
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Any sentence in SL that can be derived from the empty set
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Inconsistent in SD
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There's a derivation of P and ~P from Γ
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If Γ⊨P and Γ⊆Γ', then
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Γ'⊨P
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If ΓU{P}⊨Q, then
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Γ⊨P⊃Q
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If Γ⊨Q and Γ⊨~Q, then
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Γ is t-f inconsistent
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If ΓU{Q} is t-f inconsistent, then
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Γ⊨~Q
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Outline Completeness
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Γ⊨P
=> ΓU{~P} is t-f inconsistent => ΓU{~P} is inconsistent in SD => Γ⊢P |
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Immediate Sentential Components
Sentential Components |
2 Sentences connected by main connective
Sentential Components are made up of -Sentence itself -Immediate Sentential Components -Sentential Components of Immediate Sentential Components |
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Name the 2 types of assumptions
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Primary & Auxiliary
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A sentence P is true iff
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what the sentence says or expresses is the case
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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
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An argument of SL is valid in SD iff
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the conclusion of the argument is derivable in SD from the set consisting of the premises
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Equivalence in SD
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Q is derivable in SD from {P} and P is derivable in SD from {Q}
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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. |
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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.
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Bound variable
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A variable x is bound iff it is in the scope of an x-quantifier.
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Free Variable
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An occurrence of a variable x in a formula P of PL that is not bound
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ΓU{P} is inconsistent in SD iff
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Γ⊢~P
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Γ⊢~P iff
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ΓU{P} is inconsistent in SD
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ΓU{~P} is inconsistent in SD iff
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Γ⊢P
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Γ⊢P iff
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ΓU{~P} is inconsistent in SD
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For any sentence P of SL, if P is not ϵΓ*, then
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Γ*U{P} is inconsistent in SD
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If Γ*U{P} is consistent in SD, then
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PϵΓ*
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For any sentence P of SL, if Γ*⊢P, then
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PϵΓ*
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6.4.11a
What's an easy way to prove stuff from 6.4.11? |
~PϵΓ* iff P is not ϵΓ*
Create a derivation |
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6.4.11b
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P&QϵΓ* iff PϵΓ* and QϵΓ*
Proven by &E |
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Chris supports neither union members nor administrators
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~(∃x)((Ux&Ax)&Scx)
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d[u/x]
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A variant of a variable assignment d that assigns the same value to each variable as d does except it assigns u to x.
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A sentence P of PL is derivable in PD from a set Γ of sentences of PL iff
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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
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An argument of PL is valid in PD iff
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the conclusion of the argument is derivable in PD from the set consisting of the premises
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A sentence P is a theorem in PD iff
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P is derivable in PD from {}.
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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
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Quantificationally True
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A sentence P of PL is quantificationally true iff P is true on every interpretation I
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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
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A set Γ of sentences of PL is quantificationally consistent iff
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there is at least 1 interpretation on which all the members are true.
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A set Γ of sentences of PL is quantificationally valid iff
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There is no interpretation in which all the premises are true and the conclusion is false
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A set Γ of sentences of PL quantificationally entails a sentence P of PL iff
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there is no interpretation which all the members of Γ are true and P is false
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A sentence is true on an interpretation I iff
false iff |
Every variable assignment for I satisfies P
No variable assignment for I satisfies P |