Iterated team semantics for a hierarchy of informational types☆
Introduction
Semantic frameworks are commonly based on the notion of truth that is captured as a relation between possible worlds and formulas. In contrast, the team semantics for propositional dependence logic [20], [21] is based on the idea that propositional dependence should not be defined in terms of truth relative to individual possible worlds. Above the layer of possible worlds, one needs to add the extra layer of teams (sets of possible worlds) and define dependency relations among statements relative to these teams. We have here an example of a peculiar semantic relativity: While atomic statements are primarily evaluated with respect to possible worlds, dependence statements are primarily evaluated with respect to teams. This paper is motivated by the view that this kind of relativity is a more integral part of natural language than it might seem and in order to capture it in full generality one should go beyond the two-layered framework (involving just possible worlds and sets of possible worlds) and employ a whole hierarchy of other types of semantic objects. These new semantic objects stratify propositional language into a hierarchy of informational types and allow us to represent higher-order dependencies as well as some tricky interaction between the dependence operator and other logical operators.
In the next section, we will further elaborate on this idea and formulate two principles that will govern our approach. These principles are formulated rather vaguely but their meaning is illustrated with a detailed motivational example (Section 3) and they are embodied in a precisely defined formal semantics (Section 4). We will show that this semantics preserves some important features of classical logic (Section 5) but also deviates from it in several respects (Section 6). These deviations allow us not only to define the dependence operator in terms of the standard propositional connectives (negation, conjunction, disjunction and implication) but they also reflect some puzzling phenomena that are not taken into account in classical logic and that are related to the role modalities and conditionals play in arguments. Section 7 shows some connections between the proposed iterated team semantics and Kripke semantics for intuitionistic logic. Section 8 explores how the stratification of propositional language into a hierarchy of informational types shapes syntactic principles of deductive reasoning.
Section snippets
Semantic relativity and syntactic sensitivity
The common logical approach to information can be called eliminative. A piece of information is understood as a classifier of possibilities: some possibilities support the information and some do not. Growth of information is represented as elimination of those possibilities that do not support the information. In the classical framework “possibilities” are represented as possible worlds. A piece of information understood as a classifier of possible worlds can be encoded as a set of possible
A motivational example
To illustrate our two principles with a more concrete example and to motivate the semantics that we will define in the next section, let us consider the following simple game. We have a deck of cards each of which has one of the three values: 1, 2, or 3. Two cards will be drawn randomly. Thus, the result of the game is just a pair of numbers from the set . This simple game can determine various kinds of contexts with respect to which sentences can be evaluated. We have not fixed any
Formal semantics
In this section we will define a formal semantics motivated by our two principles and the example from the previous section. We can specify the semantics only for the most basic propositional language involving just negation, conjunction, disjunction and implication. It will turn out that the other connectives that we want to consider, namely possibility, necessity and the dependence operator, can be defined in this basic language in our semantics.
Let At be a set of atomic formulas . We
Some classical features of
The semantics that we formulated in the previous section, with its notion of -validity, is rather non-standard due to the restrictions concerning the degrees. Nevertheless, it preserves some important features of classical logic which will be shown in this section. First of all, we can observe that -validity of arguments satisfies the basic principles of Tarskian consequence relations. Proposition 11 For any finite sets of formulas and any formulas the following holds: If then is
Some non-classical features of
In this section, we point to several non-classical features -validity. Nevertheless, we argue that these features are well-motivated and reflect some puzzling aspects of the informal notion of validity of arguments in natural language. We present this as an evidence indicating that the logic of informational types captures a structure of real phenomena that manifest themselves in natural languages.
First, let us observe how the deduction theorem may fail. For example, the argument is
Projective contexts and Kripke semantics for intuitionistic logic
This more technical section will be mainly focused on a fragment of the language L that will be called the projective fragment and formulas of which will be called projective formulas. It will be shown that for this fragment our semantics is interestingly related to Kripke semantics for intuitionistic logic. Before we introduce the projective fragment, we will define another one that will be called the stable fragment and formulas of which will be denoted as stable formulas. This terminology is
Deduction in the hierarchy
The logic of informational types is very atypical and it might not be clear how a syntactic characterization of such logic could look like. In this final section, we describe a system of natural deduction that is sound and quite rich so that we conjecture its completeness with respect to (though completeness will not be proved in this paper).
Two characteristic features of the system are: 1. Every hypothetical assumption is indexed by a natural number, semantically corresponding to a level
Conclusion
Let us finish the paper with discussion of the plans for future work related to the approach that we have developed in this paper. There are technical as well as philosophical issues that still need to be addressed. As regards the technical aspects we will attempt to prove Conjecture 1 in a subsequent paper. A further technical task is to extend this approach to the first-order language. The philosophical questions would be related to a careful comparison of our approach with that of [4]. In
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This paper is an outcome of the project Logical Structure of Information Channels, no. 21-23610M, supported by the Czech Science Foundation and realized at the Institute of Philosophy of the Czech Academy of Sciences. I am grateful to Christopher Gauker, Ivano Ciardelli and Vladimír Svoboda for inspiring discussions on the issues related to the topic of this paper.
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A preliminary version of the semantic approach developed in this paper was presented at the online Workshop on Logics of Dependence and Independence.