This paper aims to provide a new criterion of formal equivalence of the- ories that is suitable f... more This paper aims to provide a new criterion of formal equivalence of the- ories that is suitable for being supplemented with preservation conditions concerning interpretation. I argue that both the internal structure of mod- els and choices of morphisms between models are aspects of formalisms that are relevant when it comes to their interpretations. So an adequate crite- rion should take these aspects into account. The two most important cri- teria presently discussed in philosophy of science—generalised definitional equivalence (Morita equivalence) and categorical equivalence—are not opti- mal in this respect. Generalised definitional equivalence neglects choices of morphisms, whereas categorical equivalence neglects the internal structure of models. I put forward the new criterion of definable categorical equivalence which takes into account both aspects and connects the category-theoretic approach and the definability-theoretic approach to formal equivalence.
Several philosophers of science construe models of scientic theories as set-theoretic structures.... more Several philosophers of science construe models of scientic theories as set-theoretic structures. Some of them moreover claim that models should not be construed as structures in the sense of model theory because the latter are languagedependent. I argue that if we are ready to construe models as set-theoretic structures (strict semantic view), we could equally well construe them as model-theoretic structures of higher-order logic (liberal semantic view). I show that every family of set-theoretic structures has an associated language of higher-order logic and an up to signature isomorphism unique model-theoretic counterpart, which is able to serve the same purposes. This allows to carry over every syntactic criterion of equivalence for theories in the sense of the liberal semantic view to theories in the sense of the strict semantic view. Taken together, these results suggest that the recent dispute about the semantic view and its relation to the syntactic view can be resolved.
Causal set theory and the theory of linear structures (which has recently been developed by Tim M... more Causal set theory and the theory of linear structures (which has recently been developed by Tim Maudlin as an alternative to standard topology) share some of their main motivations. In view of that, I raise and answer the question how these two theories are related to each other and to standard topology. I show that causal set theory can be embedded into Maudlin's more general framework and I characterise what Maudlin's topological concepts boil down to when applied to discrete linear structures that correspond to causal sets. Moreover, I show that all topological aspects of causal sets that can be described in Maudlin's theory can also be described in the framework of standard topology. Finally, I discuss why these results are relevant for evaluating Maudlin's theory. The value of this theory depends crucially on whether it is true that (a) its conceptual framework is as expressive as that of standard topology when it comes to describing well-known continuous as well as discrete models of spacetime and (b) it is even more expressive or fruitful when it comes to analysing topological aspects of discrete structures that are intended as models of spacetime. On one hand, my theorems support (a). The theory is rich enough to incorporate causal set theory and its definitions of topological notions yield a plausible outcome in the case of causal sets. On the other hand, the results undermine (b). Standard topology, too, has the conceptual resources to capture those topological aspects of causal sets that are analysable within Maudlin's framework. This fact poses a challenge for the proponents of Maudlin's theory to prove it fruitful.
The system of CMS satisfying CAP permits unbounded divisibility and, hence, is only weakly atomis... more The system of CMS satisfying CAP permits unbounded divisibility and, hence, is only weakly atomistic.
This paper aims to provide a new criterion of formal equivalence of the- ories that is suitable f... more This paper aims to provide a new criterion of formal equivalence of the- ories that is suitable for being supplemented with preservation conditions concerning interpretation. I argue that both the internal structure of mod- els and choices of morphisms between models are aspects of formalisms that are relevant when it comes to their interpretations. So an adequate crite- rion should take these aspects into account. The two most important cri- teria presently discussed in philosophy of science—generalised definitional equivalence (Morita equivalence) and categorical equivalence—are not opti- mal in this respect. Generalised definitional equivalence neglects choices of morphisms, whereas categorical equivalence neglects the internal structure of models. I put forward the new criterion of definable categorical equivalence which takes into account both aspects and connects the category-theoretic approach and the definability-theoretic approach to formal equivalence.
Several philosophers of science construe models of scientic theories as set-theoretic structures.... more Several philosophers of science construe models of scientic theories as set-theoretic structures. Some of them moreover claim that models should not be construed as structures in the sense of model theory because the latter are languagedependent. I argue that if we are ready to construe models as set-theoretic structures (strict semantic view), we could equally well construe them as model-theoretic structures of higher-order logic (liberal semantic view). I show that every family of set-theoretic structures has an associated language of higher-order logic and an up to signature isomorphism unique model-theoretic counterpart, which is able to serve the same purposes. This allows to carry over every syntactic criterion of equivalence for theories in the sense of the liberal semantic view to theories in the sense of the strict semantic view. Taken together, these results suggest that the recent dispute about the semantic view and its relation to the syntactic view can be resolved.
Causal set theory and the theory of linear structures (which has recently been developed by Tim M... more Causal set theory and the theory of linear structures (which has recently been developed by Tim Maudlin as an alternative to standard topology) share some of their main motivations. In view of that, I raise and answer the question how these two theories are related to each other and to standard topology. I show that causal set theory can be embedded into Maudlin's more general framework and I characterise what Maudlin's topological concepts boil down to when applied to discrete linear structures that correspond to causal sets. Moreover, I show that all topological aspects of causal sets that can be described in Maudlin's theory can also be described in the framework of standard topology. Finally, I discuss why these results are relevant for evaluating Maudlin's theory. The value of this theory depends crucially on whether it is true that (a) its conceptual framework is as expressive as that of standard topology when it comes to describing well-known continuous as well as discrete models of spacetime and (b) it is even more expressive or fruitful when it comes to analysing topological aspects of discrete structures that are intended as models of spacetime. On one hand, my theorems support (a). The theory is rich enough to incorporate causal set theory and its definitions of topological notions yield a plausible outcome in the case of causal sets. On the other hand, the results undermine (b). Standard topology, too, has the conceptual resources to capture those topological aspects of causal sets that are analysable within Maudlin's framework. This fact poses a challenge for the proponents of Maudlin's theory to prove it fruitful.
The system of CMS satisfying CAP permits unbounded divisibility and, hence, is only weakly atomis... more The system of CMS satisfying CAP permits unbounded divisibility and, hence, is only weakly atomistic.
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