Is propositional calculus categorical?
| Abstract | According to the standard definition, a first-order theory is categorical if all its models are isomorphic. The idea behind this definition obviously is that of capturing semantic notions in axiomatic terms: to be categorical is to be, in this respect, successful. Thus, for example, we may want to axiomatically delimit the concept of natural number, as it is given by the pre-theoretic semantic intuitions and reconstructed by the standard model. The well-known results state that this cannot be done within first-order logic, but it can be done within second-order one. Now let us consider the following question: can we axiomatically capture the semantic concept of conjunction? Such question, to be sure, does not make sense within the standard framework: we cannot construe it as asking whether we can form a first-order (or, for that matter, whatever-order) theory with an extralogical binary propositional operator so that its only model (up to isomorphism) maps the operator on the intended binary truth-function. The obvious reason is that the framework of standard logic does not allow for extralogical constants of this type. But of course there is also a deeper reason: an existence of a constant with this semantics is presupposed by the very definition of the framework1. Hence the question about the axiomatic capturability of concunction, if we can make sense of it at all, cannot be asked within the framework of standard logic, we would have to go to a more abstract level. To be able to make sense of the question we would have to think about a propositional ‘proto-language’, with uninterpreted logical constants, and to try to search out axioms which would fix the denotations of the constants as the intended truth-functions. Can we do this? It might seem that the answer to this question is yielded by the completeness theorem for the standard propositional calculus: this theorem states that the axiomatic delimitation of the calculus and the semantic delimitation converge to the same result.. | |||||||||
| Keywords | No keywords specified (fix it) | |||||||||
| Categories | ||||||||||
| Options |
|
|||||||||
Download options| PhilPapers Archive |
Upload a copy of this paper Check publisher's policy on self-archival Papers currently archived: 5,701 |
| External links |
  Try with proxy. |
| Through your library | Only published papers are available at libraries |
Similar books and articlesAlexander Bochman & Dov M. Gabbay (2012). Sequential Dynamic Logic. Journal of Logic, Language and Information 21 (3):279-298.
Dexter Kozen (1988). A Finite Model Theorem for the Propositional Μ-Calculus. Studia Logica 47 (3):233 - 241.
Edward N. Zalta (1997). The Modal Object Calculus and its Interpretation. In M. de Rijke (ed.), Advances in Intensional Logic. Kluwer.
Tomasz Połacik (1994). Second Order Propositional Operators Over Cantor Space. Studia Logica 53 (1):93 - 105.
Don Pigozzi & Antonino Salibra (1995). The Abstract Variable-Binding Calculus. Studia Logica 55 (1):129 - 179.
Ronald Fagin, Joseph Y. Halpern & Moshe Y. Vardi (1992). What is an Inference Rule? Journal of Symbolic Logic 57 (3):1018-1045.
Henri Galinon (2009). A Note on Generalized Functional Completeness in the Realm of Elementrary Logic. Bulletin of the Section of Logic 38 (1):1-9.
Jaroslav Peregrin (2003). Meaning and Inference. In Timothy Childers & Ondrej Majer (eds.), Logica Yearbook 2002. Filosofia.
Tomasz Połacik (1998). Propositional Quantification in the Monadic Fragment of Intuitionistic Logic. Journal of Symbolic Logic 63 (1):269-300.
Analytics
Monthly downloads |
Added to index2009-01-28Total downloads18 ( #67,558 of 549,090 )Recent downloads (6 months)2 ( #37,333 of 549,090 )How can I increase my downloads? |
My notes
Discussion
