
Contents
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12.1 The promise of truth-internalism 12.1 The promise of truth-internalism
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12.2 Truth operators 12.2 Truth operators
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Defining new operators Defining new operators
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Truth with language–object relations Truth with language–object relations
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Expressive limitations Expressive limitations
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12.3 Internalism about model theory and internal realism 12.3 Internalism about model theory and internal realism
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Putnam’s internal realism Putnam’s internal realism
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From set theory to model theory From set theory to model theory
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Dealing with ‘interpretations’ Dealing with ‘interpretations’
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What to say about non-standard models What to say about non-standard models
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Arithmetical truth with language-object relations Arithmetical truth with language-object relations
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Expressive limitations Expressive limitations
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12.4 Truth in higher-order logic 12.4 Truth in higher-order logic
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12.5 Two general issues for truth-internalism 12.5 Two general issues for truth-internalism
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12.A Satisfaction in higher-order logic 12.A Satisfaction in higher-order logic
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Syntax and deduction for higher-order languages Syntax and deduction for higher-order languages
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Initial approach, and Tarski’s trick Initial approach, and Tarski’s trick
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Defining the entities for satisfaction Defining the entities for satisfaction
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The satisfaction clauses The satisfaction clauses
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Existence and uniqueness of this relation Existence and uniqueness of this relation
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Checking this works Checking this works
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Generalising to nth-order logics Generalising to nth-order logics
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Combining this with internal categoricity Combining this with internal categoricity
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12 Internal categoricity and truth
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Published:March 2018
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Abstract
This chapter considers whether internal categoricity can be used to leverage any claims about mathematical truth. We begin by noting that internal categoricity allows us to introduce a truth-operator which gives an object-language expression to the supervaluationist semantics. In this way, the univocity discussed in previous chapters might seem to secure an object-language expression of determinacy of truth-value; but this hope falls short, because such truth-operators must be carefully distinguished from truth-predicates. To introduce these truth-predicates, we outline an internalist attitude towards model theory itself. We then use this to illuminate the cryptic conclusions of Putnam's justly-famous paper ‘Models and Reality’. We close this chapter by presenting Tarski’s famous result that truth for lower-order languages can be defined in higher-order languages.
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