Semantics and Ontology:\ud On the Modal Structure of an Epistemic Theory of Meaning

In this paper I shall confront three basic questions.\ud First, the relevance of epistemic structures, as formalized\ud and dealt with by current epistemic logics, for a\ud general Theory of meaning. Here I acknowledge M. Dummett"s\ud idea that a systematic account of what is meaning of\ud an arbitrary language subsystem must especially take into\ud account the inferential components of meaning itself. That\ud is, an analysis of meaning comprehension processes,\ud given in terms of epistemic logics and semantics for epistemic\ud notions.\ud The second and third questions relate to the ontological\ud and epistemological framework for this approach.\ud Concerning the epistemological aspects of an epistemic\ud theory of meaning, the question is: how epistemic logics\ud can eventually account for the informative character of\ud meaning comprehension processes. "Information� seems\ud to be built in the very formal structure of epistemic processes,\ud and should be exhibited in modal and possibleworld\ud semantics for propositional knowledge and belief.\ud However, it is not yet clear what is e.g. a possible world.\ud That is: how it can be defined semantically, other than by\ud accessibility rules which merely define it by considering its\ud set-theoretic relations with other sets-possible worlds.\ud Therefore, it is not clear which is the epistemological status\ud of propositional information contained in the structural\ud aspects of possible world semantics. The problem here\ud seems to be what kind of meaning one attributes to the\ud modal notion of possibility, thus allowing semantical and\ud synctactical selectors for possibilities. This is a typically\ud Dummett-style problem.\ud The third question is linked with this epistemological\ud problem, since it is its ontological counterpart. It concerns\ud the limits of the logical space and of logical semantics for a\ud of meaning. That is, it is concerned with the kind of\ud structure described by inferential processes, thought, in a\ud fregean perspective, as pre-conditions of estentional\ud treatment of meaning itself. The second and third questions\ud relate to some observations in Wittgenstein"s Tractatus.\ud I shall also try to show how their behaviour limits the\ud explicative power of some semantics for epistemic logics\ud (Konolige"s and Levesque"s for knowledge and belief)

The Quantum Monadology

The modern theory of functional programming languages uses monads for encoding computational side-effects and side-contexts, beyond bare-bone program logic. Even though quantum computing is intrinsically side-effectful (as in quantum measurement) and context-dependent (as on mixed ancillary states), little of this monadic paradigm has previously been brought to bear on quantum programming languages. Here we systematically analyze the (co)monads on categories of parameterized module spectra which are induced by Grothendieck's "motivic yoga of operations" -- for the present purpose specialized to HC-modules and further to set-indexed complex vector spaces. Interpreting an indexed vector space as a collection of alternative possible quantum state spaces parameterized by quantum measurement results, as familiar from Proto-Quipper-semantics, we find that these (co)monads provide a comprehensive natural language for functional quantum programming with classical control and with "dynamic lifting" of quantum measurement results back into classical contexts. We close by indicating a domain-specific quantum programming language (QS) expressing these monadic quantum effects in transparent do-notation, embeddable into the recently constructed Linear Homotopy Type Theory (LHoTT) which interprets into parameterized module spectra. Once embedded into LHoTT, this should make for formally verifiable universal quantum programming with linear quantum types, classical control, dynamic lifting, and notably also with topological effects.Comment: 120 pages, various figure