Theo M. V. Janssen

Frege, Contextuality and Compositionality

There are two principles which bear the name “Fregesprinciple:” the principle of compositionality, and the contextprinciple. The aim of this contribution is to investigate whether thisis justified: did Frege accept both principles at the same time, did hehold the one principle but not the other, or did he, at some moment,change his opinion? The conclusion is as follows. There is a developmentin Freges position. In the period of Grundlagen he followed to a strict form of contextuality. He repeatedcontextuality in later writings, but became less strict. From 1914 on,pushed by the needs of research, he comes close to compositionality. Buthe could never make the final step toward compositionality forprincipled reasons, therefore he always would reject compositionality.

Equivalence and quantifier rules for logic with imperfect information

In this paper, we present a prenex form theorem for a version of Independence Friendly logic, a logic with imperfect information. Lifting classical results to such logics turns out not to be straightforward, because independence conditions make the formulas sensitive to signalling phenomena. In particular, nested quantification over the same variable is shown to cause problems. For instance, renaming of bound variables may change the interpretations of a formula, there are only restricted quantifier extraction theorems, and slashed connectives cannot be so easily removed. Thus we correct some claims from Hintikka [8], Caicedo & Krynicki [3] and Hodges [11]. We refine definitions, in particular the notion of equivalence, and sharpen preconditions, allowing us to restore (restricted versions of) those claims, including the prenex form theorem of Caicedo & Krynicki [3], and, as a side result, we obtain an application to Skolem forms of classical formulas. It is a known fact that a complete calculus for IF-logic is impossible, but with our results we establish several quantifier rules that form a partial calculus of equivalence for a general version of IF-logic reflecting general properties of information flow in games.

Chapter 7 – Compositionality

Publisher Summary This chapter provides a precise interpretation of the principle of compositionality and a mathematical model for the principle. The principle of compositionality is a restriction that rules out several proposals in the literature, and is certainly not vacuous. On the other hand, it shows that there are several methods to obtain a compositional meaning assignment; so it is not an impossible task. Compositionality is not a formal restriction on what can be achieved, but a methodology on how to proceed. The discussions in this chapter have pointed to several advantages of this methodology, in particular its heuristic value. It suggests solutions to semantic problems. It helps to find weak spots in non-compositional proposals; such proposals have a risk of being defective. Cases where an initially noncompositional proposal was turned into a compositional one, the analysis is improved considerably. Compositionality requires a decision on what in a given approach the basic semantic units are: if one has to build meanings from them, it has to be decided what these units are. Compositionality also requires a decision on what the basic units in syntax are, and how they are combined.

Independent Choices and the Interpretation of IF Logic

In this paper it is argued that Hintikkas game theoreticalsemantics for Independence Friendly logic does not formalize theintuitions about independent choices; it rather is aformalization of imperfect information. Furthermore it is shownthat the logic has several remarkable properties (e.g.,renaming of bound variables is not allowed). An alternativesemantics is proposed which formalizes intuitions aboutindependence.

Algebraic translations, correctness and algebraic compiler construction

Abstract Algebraic translation methods are argued for in many fields of science. Several examples will be considered: from the field of compiler construction, database updates, concurrent programming languages, logic, natural language translation, and natural language semantics. Special attention will be given to the notion ‘correctness of translation’. In all fields this notion can be defined as the commutativity of some diagram which connects languages, translation and meanings. For algebraically defined compilers, five different definitions are found in the literature. We argue which of these should be considered the ‘right’ one (it is not the standard choice). We conclude with a first step towards a general algebraic theory of translation.

On the Proper Treatment or Referencing, Dereferencing and Assignment

A Floyd-like semantics is presented for the assignment statement in a fragment of ALGOL 68. The fragment considered contains array identifiers, referencing, dereferencing and conditionals. The semantics is based upon an interpretation in a model of intensional logic, without use of addresses or stores. In doing so, several ideas developed by R. Montague concerning the treatment of semantics for natural languages are applied for the first time in the area of semantics of programming languages. We also consider an operational semantics, based on the same model and prove that the Floyd-like semantics is valid with respect to the operational one and always yields the strongest postcondition.

Compositional Natural Language Semantics using Independence Friendly Logic or Dependence Logic

Independence Friendly Logic, introduced by Hintikka, is a logic in which a quantifier can be marked for being independent of other quantifiers. Dependence logic, introduced by Väänänen, is a logic with the complementary approach: for a quantifier it can be indicated on which quantifiers it depends. These logics are claimed to be useful for many phenomena, for instance natural language semantics. In this contribution we will compare these two logics by investigating their application in a compositional analysis of the de dicto - de re ambiguity in natural language. It will be argued that Independence Friendly logic is suitable, whereas Dependence Logic is not.

A Mathematical Model for the CAT Framwork of EUROTRA

Eurotra is the machine translation project of the EEC. The basic ideas for the design of the system are given by the CAT framework, which is, together with various relaxations, presented in several publications (e.g. Arnold 1985, 1986 e.a., des Tombes e.a. 1985, Arnold & des Tombes 1987). In the present paper a mathematical model for the CAT framework will be developed. This will be a model of the structural aspects of the framework, such as the structure of the grammars and of the translation steps. The model uses notions and results from universal algebra; a branch of mathematics which deals with structures and their relations. The model is in a certain sense the same as the CAT framework, but it is build with different tools. Eurotra is a project of ongoing research with continuous practical experience, and this might cause changes in the original framework. Since the present paper is mainly based upon the publications mentioned above, it does not necessarily describe the present situation correctly (for your information, the author is not personally involved in the project). The aim of this paper is, however, not to present some version of Eurotra, but to argue for a more abstract and more mathematically based approach to Eurotra (and other machine translation systems). It will be shown that there are several advantages of such a mathematical approach. It brings new insights in the framework (see sections 4 and 5), and gives us a new appreciation of certain Eurotra proposals (see sections 6 and 7). Furthermore, the mathematical model for Eurotra will, I expect, be a good starting point for investigating later stages of the Eurotra system.

On Intensionality in Programming Languages

Programs are pieces of text, written in some programming language. These languages were designed for the special purpose of instructing computers. They also are used in communication among human beings for telling them how to instruct computers or for communicating algorithms which are not intended for computer execution. So programs are used for certain kinds of communication, hence they have some meaning. The branch of computer science called ‘semantics of programming languages’ deals with the relation between programs and their meanings.