*Barry Taylor*

- Published in print:
- 2006
- Published Online:
- September 2006
- ISBN:
- 9780199286690
- eISBN:
- 9780191604065
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199286698.003.0003
- Subject:
- Philosophy, Metaphysics/Epistemology

This chapter sets out the relevant core of Putnam’s case. Section 3.1 extracts three arguments from Putnam’s writings: the Arguments from Cardinality, Completeness, and Permutation. Of these, section ...
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This chapter sets out the relevant core of Putnam’s case. Section 3.1 extracts three arguments from Putnam’s writings: the Arguments from Cardinality, Completeness, and Permutation. Of these, section 3.2 argues that only the second is of direct relevance. Section 3.3 examines attempts to frame constraints based on causal and psycho-behavioural reductions of reference. Section 3.4 investigates the Translational Reference Constraint (TRC), a constraint on reference which does not rely on a reduction of reference but makes essential use of translation (from object language to metalanguage) to sort out the models which get reference right. The claims made in this section, however, require foundation in a theory of translation, sufficient to sustain the assumptions it makes about that controversial and opaque notion. This foundation is supplied in section 3.5, whose general tenor is Davidsonian, its key notion being that of a ‘hermeneutic theory’, i.e., a Davidsonian theory of interpretation cast into model-theoretic terms. With Translational Truth Constraint (TTC) now identified as the most fundamental constraint on intendedness, it remains to see if it will suffice to rule out as unintended all the models of ideal theory whose existence the Completeness Theorem guarantees. The issue is examined in section 3.6.Less

This chapter sets out the relevant core of Putnam’s case. Section 3.1 extracts three arguments from Putnam’s writings: the Arguments from Cardinality, Completeness, and Permutation. Of these, section 3.2 argues that only the second is of direct relevance. Section 3.3 examines attempts to frame constraints based on causal and psycho-behavioural reductions of reference. Section 3.4 investigates the Translational Reference Constraint (TRC), a constraint on reference which does not rely on a reduction of reference but makes essential use of translation (from object language to metalanguage) to sort out the models which get reference right. The claims made in this section, however, require foundation in a theory of translation, sufficient to sustain the assumptions it makes about that controversial and opaque notion. This foundation is supplied in section 3.5, whose general tenor is Davidsonian, its key notion being that of a ‘hermeneutic theory’, i.e., a Davidsonian theory of interpretation cast into model-theoretic terms. With Translational Truth Constraint (TTC) now identified as the most fundamental constraint on intendedness, it remains to see if it will suffice to rule out as unintended all the models of ideal theory whose existence the Completeness Theorem guarantees. The issue is examined in section 3.6.

*Barry Taylor*

- Published in print:
- 2006
- Published Online:
- September 2006
- ISBN:
- 9780199286690
- eISBN:
- 9780191604065
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199286698.003.0004
- Subject:
- Philosophy, Metaphysics/Epistemology

This chapter examines other ways the Argument from Completeness might be attacked. It identifies two strategies that the realist might deploy in order to avoid the difficulties engendered by the ...
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This chapter examines other ways the Argument from Completeness might be attacked. It identifies two strategies that the realist might deploy in order to avoid the difficulties engendered by the applicability of the Completeness Theorem to the ideal theory. The first is to insist that the theory be cast in some non-first-order language which resists completeness. The second is to allow ideal theory to continue to be rendered in first-order form, but to argue for a semantics in which interpretations take such a new shape that there is no longer any guarantee that any consistent theory has a model in the new sense.Less

This chapter examines other ways the Argument from Completeness might be attacked. It identifies two strategies that the realist might deploy in order to avoid the difficulties engendered by the applicability of the Completeness Theorem to the ideal theory. The first is to insist that the theory be cast in some non-first-order language which resists completeness. The second is to allow ideal theory to continue to be rendered in first-order form, but to argue for a semantics in which interpretations take such a new shape that there is no longer any guarantee that any consistent theory has a model in the new sense.

*Tim Button and Sean Walsh*

- Published in print:
- 2018
- Published Online:
- May 2018
- ISBN:
- 9780198790396
- eISBN:
- 9780191863424
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198790396.003.0004
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

One of the most famous philosophical applications of model theory is Robinson’s attempt to salvage infinitesimals. Infinitesimals are quantities whose absolute value is smaller than that of any given ...
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One of the most famous philosophical applications of model theory is Robinson’s attempt to salvage infinitesimals. Infinitesimals are quantities whose absolute value is smaller than that of any given positive real number. Robinson used his non-standard analysis to formalize and vindicate the Leibnizian approach to the calculus. Against this, the historian Bos has questioned whether the infinitesimals of Robinson's non-standard analysis have the same structure as those of Leibniz. We offer a response to Bos, by building valuations into Robinson's non-standard analysis. This chapter also introduces some related discussions of independent interest (compactness, instrumentalism, and o-minimality) and contains a proof of The Compactness Theorem and Gödel’s Completeness Theorem.Less

One of the most famous philosophical applications of model theory is Robinson’s attempt to salvage infinitesimals. Infinitesimals are quantities whose absolute value is smaller than that of any given positive real number. Robinson used his non-standard analysis to formalize and vindicate the Leibnizian approach to the calculus. Against this, the historian Bos has questioned whether the infinitesimals of Robinson's non-standard analysis have the same structure as those of Leibniz. We offer a response to Bos, by building valuations into Robinson's non-standard analysis. This chapter also introduces some related discussions of independent interest (compactness, instrumentalism, and o-minimality) and contains a proof of The Compactness Theorem and Gödel’s Completeness Theorem.