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Single Idea 10012
[filed under theme 5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic
]
Full Idea
Complete proof procedures are available not only for quantification theory, but for quantification theory and identity together. Gödel showed that the theory is still complete if we add self-identity and the indiscernability of identicals.
Gist of Idea
Quantification theory can still be proved complete if we add identity
Source
Willard Quine (Philosophy of Logic [1970], Ch.5)
Book Ref
Quine,Willard: 'Philosophy of Logic' [Prentice-Hall 1970], p.62
A Reaction
Hence one talks of first-order logic 'with identity', even though, as Quine observes, it is unclear whether identity is actually a logical or a mathematical notion.
The
25 ideas
from 'Philosophy of Logic'
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10014
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Quine rejects second-order logic, saying that predicates refer to multiple objects
[Quine, by Hodes]
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9012
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Talk of 'truth' when sentences are mentioned; it reminds us that reality is the point of sentences
[Quine]
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9011
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Truth is redundant for single sentences; we do better to simply speak the sentence
[Quine]
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9009
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Single words are strongly synonymous if their interchange preserves truth
[Quine]
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9007
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It makes no sense to say that two sentences express the same proposition
[Quine]
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9008
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There is no rule for separating the information from other features of sentences
[Quine]
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9010
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We can abandon propositions, and just talk of sentences and equivalence
[Quine]
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9013
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We can eliminate 'or' from our basic theory, by paraphrasing 'p or q' as 'not(not-p and not-q)'
[Quine]
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9015
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Universal quantification is widespread, but it is definable in terms of existential quantification
[Quine]
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9016
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Names are not essential, because naming can be turned into predication
[Quine]
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9014
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Some conditionals can be explained just by negation and conjunction: not(p and not-q)
[Quine]
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9017
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Predicates are not names; predicates are the other parties to predication
[Quine]
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9018
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A physical object is the four-dimensional material content of a portion of space-time
[Quine]
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9019
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Four-d objects helps predication of what no longer exists, and quantification over items from different times
[Quine]
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9020
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My logical grammar has sentences by predication, then negation, conjunction, and existential quantification
[Quine]
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9021
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A good way of explaining an expression is saying what conditions make its contexts true
[Quine]
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10705
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Putting a predicate letter in a quantifier is to make it the name of an entity
[Quine]
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10828
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Quantifying over predicates is treating them as names of entities
[Quine]
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10012
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Quantification theory can still be proved complete if we add identity
[Quine]
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9024
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Excluded middle has three different definitions
[Quine]
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9025
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You can't base quantification on substituting names for variables, if the irrationals cannot all be named
[Quine]
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9026
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Some quantifications could be false substitutionally and true objectually, because of nameless objects
[Quine]
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9023
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If you say that a contradiction is true, you change the meaning of 'not', and so change the subject
[Quine]
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9027
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A sentence is logically true if all sentences with that grammatical structure are true
[Quine]
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9028
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Maybe logical truth reflects reality, but in different ways in different languages
[Quine]
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