Modalities as predicates. Modalities like necessity and possibility, may be analysed logically in essentially two ways: either as predicates, or as operators. In the first case they are applied to singular terms, whereas in the second case they are applied to formula, but in both cases the application gives us new formula. Thus the distinction between the operator and the predicate conception of necessity is made on the syntactical level at first. Both conceptions are tied to certain semantics respectively. If “necessary” and “possible” are regarded as predicates, they are interpreted as properties of objects and a decision has to be made concerning what precisely they should be predicates of: syntactical entities like sentences, or contents of syntactical entities like propositions (let us ignore further options like utterances or mental objects). In either case, necessity and possibility are properties of such entities, or, perhaps, relations between such entities and further objects. If “necessary” and “possible” are regarded as operators, they do not express properties or relations like predicate and relation expressions; necessity does not apply to anything— much like the logical connectives or the quantifiers. In this sense the operator conception of necessity is radically deflationary. Similar considerations apply not only to necessity but also to the notions of knowledge, belief, future and past truth, obligation and so on, which have been treated in analogous fashions as necessity.

Language and Logic

The simplest sentences of a natural language have two components: a subject and a predicate. ‘Africa is a continent’ and ‘One-half is a natural number’ are examples. In the first ‘Africa’ is the subject and ‘a continent’ is the predicate, while in the second ‘One-half’ is the subject and ‘a natural number’ is the predicate. The verb ‘is’ asserts that the subject has the property described by the predicate; briefly that the predicate is applied to the subject, or that the subject is an argument for the predicate. The resulting sentence may or may not be true.

Some sentences, although in the subject-predicate form, have a predicate with an embedded subject. For example the sentence ‘London is south of Paris’ has ‘London’ as subject and ‘south of Paris’ as predicate. The subject ‘Paris’ is embedded in the predicate. Indeed the sentence can be understood to have two subjects ‘London’ and ‘Paris’ and a predicate ‘south of’, often called a relation. The number of subjects to which a predicate may be applied is called the arity of the predicate. Thus ‘a continent’ and ‘a natural number’ have arity one, while ‘south of’ has arity two.

From an arity two predicate an arity one predicate can be formed by applying the first of its two subjects. Thus for example ‘south of Paris’ is an arity one predicate. Later, notation will be introduced for the predicate suggested by ‘London is south of’. In a slight generalization of the meaning of predicate, a sentence is understood to be a predicate of arity zero since a sentence results when it is applied to no subjects. Thus ‘Africa is a continent’ and ‘London is south of Paris’ are predicates of arity zero. Generalizing further, it will be assumed that there may be predicates of any given finite arity; that is, predicates which may be applied to any given finite number of subjects.

§1. Introduction. Generally speaking, the concepts of Hausdorff measure and dimension are a generalization of Lebesgue measure theory. In the early 20th century, Hausdorff [9] used Caratheodory's construction of measures to define a whole family of outer measures. For examining a set of a peculiar topological or geometrical nature Lebesgue measure often is too coarse to investigate the features of the set, so one may ‘pick’ a measure from this family of outer measures that is suited to study this particular set. This is one reason why Hausdorff measure and dimension became a prominent tool in fractal geometry.

Hausdorff dimension is extensively studied in the context of dynamical systems, too. On the Cantor space, the space of all infinite binary sequences, the interplay between dimension and concepts from dynamical systems such as entropy becomes really close. Results of Besicovitch [3] and Eggleston [7] early brought forth a correspondence between the Hausdorff dimension of frequency sets (i.e., sets of sequences in which every symbol occurs with a certain frequency) and the entropy of a process creating such sequences as typical outcomes. Besides, under certain conditions the Hausdorff dimension of a set in the Cantor space equals the topological entropy of this set, viewed as a shift space.

The Terms of ITT

Church described in [30] a logic of sense and denotation based on the simple theory of types [28]; that is, using the terminology of Carnap, a logic of intension and extension. ITT differs from that logic in two ways: First ITT is based on TT and not on the simple theory of types; but of greater importance, ITT identifies the intension of a predicate with its name while the logic of Church treats intensions as separate entities with their own types and notation. The justification for this identification is the belief that in a given context a user discovers the intension of a predicate from its name.

The types of ITT, an intensional type theory, are the types of TT, and the constants and variables of ITT are those of TT. The terms of ITT are an extension of those of TT. Definition 29 of term of TT in §2.1.1 is extended for ITT by a fourth clause that introduces a secondary typing for some of the terms of ITT. Since secondary typing is the main feature of ITT that distinguishes it from TT, it will be motivated in §3.1.1 before being formally expressed in clause (4) of Definition 43 in §3.1.2.

Motivation for secondary typing. The purpose of secondary typing in ITT is to provide a simple but unambiguous way of distinguishing between an occurrence of a predicate name where it is being used and an occurrence where it is being mentioned. The necessity for recognizing this distinction has been stressed many times and a systematic use of quotes is traditionally employed for expressing it; see for example [24] or [120]. But the systematic use of quotes is awkward in a formal logic and subject to abuses as described by Church in footnote 136 of [31]. Secondary typing exploits the typing notation of TT to distinguish between a used predicate name and a mentioned predicate name and is not subject to the abuses cited by Church.

§1. Introduction: Moschovakis's approach to intensionality. G. Frege introduced two concepts which are central tomodern formal approaches to natural language semantics; i.e., the notion of reference (denotation, extension, Bedeutung) and sense (intension, Sinn) of proper names2. The sense of a proper name is wherin the mode of presentation (of the denotation) is contained. For Frege proper names include not only expressions such as Peter, Shakespeare but also definite descriptions like the point of intersection of line l1 and l2 and furthermore sentences which are names for truth values. Sentences denote the True or the False. The sense of a sentence is the proposition (Gedanke) the sentence expresses. In the tradition of possible world semantics the proposition a sentence expresses ismodelled via the set ofworlds in which the sentence is true. This strategy leads to well known problems with propositional attitudes and other intensional constructions in natural languages since it predicts for example that the sentences in (1) are equivalent.

1. (1) a. Jacob knows that the square root of four equals two.

2. b. Jacob knows that any group G is isomorphic to a transformation group.

Even an example as simple as (1) shows that the standard concept of proposition in possible world semantics is not a faithful reconstruction of Frege's notion sense.

Frege developed his notion of sense for two related but conceptually different reasons. We already introduced the first one by considering propositional attitudes. The problem here is how to develop a general concept which can handle the semantics of Frege's ungerade Rede. The second problem is how to distinguish a statement like a = a which is rather uninformative from the informative statement a = b or phrased differently how to account for the semantic difference between (2-a) and (2-b).

1. (2) a. Scott is Scott.

2. b. Scott is the author ofWaverly.

Frege's intuitive concept of sense therefore was meant both to model information and provide denotations for intensional constructions.

[12] develops a formal analysis of sense and denotation which is certainly closer to Frege's intentions than is possible world semantics. Moschovakis's motivations are (at least) twofold. The first motivation is to give a rigorous definition of the concept algorithm [13] and thereby provide the basics for a mathematical theory of algorithms.

Informal background. Many logicians and philosophers, the present writer included, are familiar with the term “intension” (with an s) but rather less familiar with the term “intention” (with a t). Outside philosophy and logic the situation is reverse: “intention” is used by all, “intension” by few, if any. A miniature history of the development of the two terms was given by E. J. Lemmon [6]:

This elegant quotation is offered for what it is worth. It is of some help in explaining philosophers’ terminology, but unfortunately it leaves unexplained the connexion with everyday usage in the context of action. And it is the latter that is of concern in this paper.

We study a system in which one agent (“the agent”) operates in some environment (“the world”). The world may or may not be dynamic, in the sense that things can happen even if the agent does not do anything; when it is, it is convenient to postulate yet another agent, called “Nature”.