Basic model theory

Model theory is used in every theoretical branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics. But these wide-ranging appeals to model theory have created a highly fragmented literature. On the one hand, many philosophically significant results are found only in mathematics textbooks: these are aimed squarely at mathematicians; they typically presuppose that the reader has a serious background in mathematics; and little clue is given as to their philosophical significance. On the other hand, the philosophical applications of these results are scattered across disconnected pockets of papers. The first aim of this book, then, is to consider the philosophical uses of model theory, focusing on the central topics of reference, realism, and doxology. Its second aim is to address important questions in the philosophy of model theory, such as: sameness of theories and structure, the boundaries of logic, and the classification of mathematical structures. Philosophy and Model Theory will be accessible to anyone who has completed an introductory logic course. It does not assume that readers have encountered model theory before, but starts right at the beginning, discussing philosophical issues that arise even with conceptually basic model theory. Moreover, the book is largely self-contained: model-theoretic notions are defined as and when they are needed for the philosophical discussion, and many of the most philosophically significant results are given accessible proofs.

Download Full-text

The model theory of differential fields with finitely many commuting derivations

Journal of Symbolic Logic ◽

10.2307/2586576 ◽

2000 ◽

Vol 65 (2) ◽

pp. 885-913 ◽

Cited By ~ 16

Author(s):

Tracey McGrail

Keyword(s):

Model Theory ◽

Stable Model ◽

Basic Model ◽

Differential Fields

AbstractIn this paper we set out the basic model theory of differential fields of characteristic 0, which have finitely many commuting derivations. We give axioms for the theory of differentially closed differential fields with m derivations and show that this theory is ω-stable, model complete, and quantifier-eliminable, and that it admits elimination of imaginaries. We give a characterization of forking and compute the rank of this theory to be ωm + 1.

Download Full-text

ON THE WAY TO A WIDER MODEL THEORY: COMPLETENESS THEOREMS FOR FIRST-ORDER LOGICS OF FORMAL INCONSISTENCY

The Review of Symbolic Logic ◽

10.1017/s1755020314000148 ◽

2014 ◽

Vol 7 (3) ◽

pp. 548-578 ◽

Cited By ~ 10

Author(s):

WALTER CARNIELLI ◽

MARCELO E. CONIGLIO ◽

RODRIGO PODIACKI ◽

TARCÍSIO RODRIGUES

Keyword(s):

Model Theory ◽

Classical Logic ◽

Large Family ◽

Paraconsistent Logics ◽

Traditional Logic ◽

First Order ◽

Basic Model ◽

Technical Details ◽

Completeness Theorems ◽

Refined Version

AbstractThis paper investigates the question of characterizing first-orderLFIs (logics of formal inconsistency) by means of two-valued semantics.LFIs are powerful paraconsistent logics that encode classical logic and permit a finer distinction between contradictions and inconsistencies, with a deep involvement in philosophical and foundational questions. Although focused on just one particular case, namely, the quantified logicQmbC, the method proposed here is completely general for this kind of logics, and can be easily extended to a large family of quantified paraconsistent logics, supplying a sound and complete semantical interpretation for such logics. However, certain subtleties involving term substitution and replacement, that are hidden in classical structures, have to be taken into account when one ventures into the realm of nonclassical reasoning. This paper shows how such difficulties can be overcome, and offers detailed proofs showing that a smooth treatment of semantical characterization can be given to all such logics. Although the paper is well-endowed in technical details and results, it has a significant philosophical aside: it shows how slight extensions of classical methods can be used to construct the basic model theory of logics that are weaker than traditional logic due to the absence of certain rules present in classical logic. Several such logics, however, as in the case of theLFIs treated here, are notorious for their wealth of models precisely because they do not make indiscriminate use of certain rules; these models thus require new methods. In the case of this paper, by just appealing to a refined version of the Principle of Explosion, or Pseudo-Scotus, some new constructions and crafty solutions to certain nonobvious subtleties are proposed. The result is that a richer extension of model theory can be inaugurated, with interest not only for paraconsistency, but hopefully to other enlargements of traditional logic.

Download Full-text