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One can also define a substitution-invariant consequence relation semantically using a class of mathematical objects (algebras, Kripke models, topological models, etc.) and a satisfaction relation.
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Notice that in the above quote we first get a substitution of ‘random’ for ‘chance’ in the phrases ‘random variation’ and ‘chance variation’, and then at least the suggestion that the concept of ‘random drift’ can be characterized as ‘changes in frequencies of variations due to chance’, where the contrast class consists of similar changes due to natural selection.
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So, substitution fails but not because the substitution affects the reference of the terms.
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In augite there can be extensive substitution of aluminum for tetrahedral silicon.
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(Throughout this discussion, by “substitution” we shall mean uniform substitution of one presumably univocal expression for another in all its occurrences in a sentence.)
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Another critical point of Bolzano's theory requires elucidation: the notion of “appropriate” substitution (substitution of appropriate ideas).
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Turing discussed a thesis that is closely related to Turing’s thesis, namely for every systematic method there is a corresponding substitution-puzzle (where ‘substitution-puzzle’, like ‘computable by Turing machine’, is a rigorously defined concept).
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Which notation to use for substitution seems to be a personal matter.
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Since “mice” is not synonymous with “ophthalmologist”, (16) is not a substitution of the required sort.
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The last phenomenon is best illustrated in the works of the German middle class meistersingers from the 15th and 16th centuries.
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These “substitution rules” were not the familiar rules for “substituting equals for equals”; rather they were closer to what are called instantiation rules today.
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Its officials worry that MIC2025 commits China to being world-class in all those sectors.
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The class of attitude verbs for which substitution problems arise — the “problematic” attitude verbs — can be divided into two subclasses: one consisting of verbs which do not grammatically tolerate substitutions (e.g., intransitive verbs such as ‘insist’, ‘complain’, ‘say’, and VPs of the form ‘Aux Adj’, such as ‘is pleased’, ‘was surprised’); the other consisting of verbs which grammatically tolerate substitutions but for which truth-value is not necessarily preserved (e.g., ‘expect’, ‘anticipate’, ‘bet’, ‘gather’, ‘judge’, ‘claim’, ‘maintain’, ‘hold’, ‘judge’, ‘feel’, ‘remember’, ‘know’, ‘recognize’, ‘find’).
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In “Do We Believe in Penal Substitution?”
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Extensive atomic substitution occurs between the ideal end-member compositions.
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The early American poet Anne Bradstreet used substitution to great effect in the following lines from “The Author to Her Book”:
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Dummett’s restriction, by comparison, bars not only the substitution of these propositions, but also the substitution of logically complex proposition that are clearly knowable.
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It is valid for any uniform substitution of its categorematic terms.
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When the method of variation is applied to a proposition, three different cases may arise: either the class of propositions obtained by substitution contains only true propositions, or it contains only false propositions, or it contains both true and false propositions.
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In this section we discuss the development of the epsilon-substitution method for obtaining consistency results for strong systems; these results are of a mathematical nature.
substitution class
noun communication
- the class of all items that can be substituted into the same position (or slot) in a grammatical sentence (are in paradigmatic relation with one another)
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