Types
By Richard Weyhrauch

This note is called: [TYPES]
The notation used in this note is described in [NOTATION]
Under development

Preface

Our goal is

GOAL: to build an artifact that can think

The purpose of this note is to introduce the idea of a mental image as created by a mental act and to give a description of how we will represent these ideas as data structures in a computer.

This note together with the series [NOTES] providse both a phyosophical framework and an operational biueprint for our goal.

One source of ideas is the literature on logic before the discovery of set theory and formalism. We only peripherially address the recently developed theoriesof reasoning implicit in this work but rather we are interedted in what is in a head/mind. The notions introduced here make use of the ideas of Boole, De Morgan, C.S. Pierce, Whatley and Lewis Carrol. This note owes a lot to Carroll as he synthesizes the others in a way that is clear (at least to me) and was clearly recognized by Russell[note 1]. I start with the words of Carroll but very soon interpret the ideas he presents in a modern symbolic framework that acknowledges the techniques since Frege even as it diverges on substance. Some of the notions introduced in this note are now commonplace in computer science but largely unused by logicians. We also acknowledge the direct importance of both ancient phylosophy&emdash;in particular Aristotle&emdash;and medieval logicians like William of Ockham.

Things

1) The universe contains things.

[For example. "I", "London", "roses", "old English books", "the letter which I recieved yesterday"]

2) Things have attributes.

[For example. "large", "red", "old", "which I recieved yesterday"]

3) One thing may have many attributes and one attribute may belong to many things.

[Thus, the thing "a rose" may have the atteibutes "red", "scented", "full-blown", &c. and the attribute "red" may belong to the Thing "a rose", "a brick", "a ribbon", &c.]

These (with the addition of numbering by us) are the words of Lewis Carrol. symbolic logic, pp xx-xx, but now we start to paraphrase in order to coordinate with our vocabvulary and to allow for the influence of later work. Before we go on, however, we will takea closer look at what he said.

What I like about these 'principles' is that they are so straight forward. There is no wordy philosophical decoration but rather simple statements. This allows for a clean view of what he is after. (Of cource Carroll was interested in giving people tools for solving 'logic puzzles' but as usual with Carroll, you should not underestimate the sophisticated knowledge implicit in his work and his amazing writing talent.)

The first dictum is not a comprehension principle in the modern sense. It is a simple observation. Its statement predates Russell paradox to whuch we will return below.

The second dictum is typical of pre-set-theoretic writing. He does not say that attributes are properties (or determine 'sets') but just that they 'belong' to Things.

Types

Here we start paraphrasing in earnest. Carroll defines 'class'. We prefer the word 'type'. It answers the question 'what type of thing we are thinking of?' It is the term we use in our formal defINITION of IBML (the IBUKI modeling language) and it more closely resembles its usage in modern computer science. Further, we do not want this notion confused with the idea of 'class' as used either in set theory of computer terminology. That being said:

4) The formation of a typoe is a mental process in which we imagine that we put together, in a group, certain ihings. Such a group is called a type.

This idea introduces the idea of mental process.

5) A type is determined by its collection of attributes

This mental process may be performed in many ways but (as Carroll did) we start with three

5a) We may imagine that we have put together all Things. The Type so formed (i.e., the Type of "Things") contains the whole Universe. IBUKI calls this type '[top]'.

5b) We may think of the type [top], and may imagine that we have picked out from it all the things which possess certain attributes notpossessed by the whole type. This collection of attributes is said to particular to the type so formed. In this case, the type [top] is called a Genus with regard to the type so formed: the type, so formed, is called species of the type [top]: and its peculiar attributes are called differentia

As this process is entirely Mental, we can perform it whether there is, or is not, an existing thing which has those properties. If the is the type is said to be real, if not, it is said to be unreal or imaginary.

5c) We may think of a certain type, not [top] and may imagine that we have picked out from it all the Members of it which possess a certain attributes not possessed by the whole type. This list of attributes is said to be 'percular' to the smaller type so formed. In this case, the type thought of is called a 'Genus' with regard to the smaller type picked out from it: the smaller type is called a 'species' of the larger: and its peculiar set of attributers called its 'differentia'.

L6) A type containing only one Member is called an 'individual'.

This definition is labeled 'L6' because it appears in Carroll, but is not valid for our notion of type. We use the idea of 'example' and for us every type has an unlimited collection of examples.

Division

'Division' is a Mental Process, in which we think of a certain type of things, and imagine that we have divided it into two or more smaller types.

A type that has been obtained by a certain division, is said to be codivisional with every type obtained by that division.

Hence a type, obtained by division, is codivisional with itself.

Dichotomy

If we think of a certain type, and imagine that we have picked out from it a certain smaller type, it is evident that the remainder of the large type not possess the differentia of that smaller type. Hence it may be regarded as another smaller type whose differentia may be formed, from that of the type first picked out, by prefixing the word "not"; and we may imagine that we have divided the type first thought of into two smaller types, whose differentia are contradictory. This kind of division is called Dichotomy.

In performing this process, we may sometimes find that the attributes we have chosen are used so loosely, in ordinary conversation, that it is not easy to decide which of the things belong to the one type and which to the other. In such a case, it would be necessary to lay down some arbitrary rule, as to where the one type should end and the other begin.

Henceforwards let it be understood that, if a type of things be divided into two types, whose differentia have contrary meanings, each differentia is to be regarded as equivalent to the other with the word "not" prefixed.

After dividing a type, by the process of dichotomy, into two smaller types, we may sub-divide each of these into two still smaller types; and this process may be repeated over and over again, the number of types being doubled at each repetition.

Names

The word 'thing', which conveys the idea of a thing, without any idea of its attributes, represents any single thing. Any other word (or phrase), which conveys the idea of a thing, with the idea of its attributes represents any thing which possesses those attributes; i.e., it represents any Member of the type to which those attributes are peculiar.

Such a word (or phrase) is called a Name; and, if there be an existing thing which it represents, it is said to be a name of that thing.

Just as a type is said to be real, or unreal, according as there is, or is not, an existing thing in it, so also a name is said to be real, or unreal, according as there is, or is not, an existing thing represented by it.

Every name is either a substantive only, or else a phrase consisting of a substantive and one or more adjectives (or phrases used as adjectives).

Every Name, except 'thing', may usually be expressed in three different forms&emdash;

7a) The substantive 'thing', and one or more adjectives (or phrases used as adjectives) conveying the ideas of the attributes;

7b) A substantive, conveying the idea of a thing with the ideas of some of the attributes, and one or more adjectives (or phrases used as adjectives) conveying the ideas of the other attributes;

7c) A substantive conveying the idea of a thing with the ideas of all the attributes,

A name, whose substantive is in the plural number, may be used to represent either

8a) Members of a type, regarded as separate things;

or

8b) a whole type, regarded as one single thing.

Definitions

 Carroll, Lewis, The Game of Logic, London: Macmillan, 1896 [new edition, 1887]
 Carroll, A logical Paradox, Mind, 3(11), July 1894, p. 436-438.
 Carroll, What the Tortoise said to Achilles,
   Mind, 4(14), April 1895, p. 278-280.
 Carroll, Lewis, Symbolic Logic. Part I: Elementary,
   London: Macmillan, 1896 [4th edition, 1897]

 Abeles, Francine F. (ed.),
   The Logic Pamphlets of Charles Lutwidge Dodgson and Related Pieces,
   Charlottesville-London:    Lewis Carroll Society of North America
   University Press of Virginia 2010.
 Bartley III, William Warren (ed.), Lewis Carroll's Symbolic Logic.
   Part I: elementary (1896. Fifth Edition),
   Part II: Advanced (Never Previously Published),
   New York: Clarkson N. Potter, 1977 [new edition, 1968]
 Moktefi, aMIROUCHE Lewis Carroll's Logic.
   in Dov M. Gabbay & John Woods (eds.),
   British Logic in the Nineteenth-Century,
   series: Handbook of the History of logic, vol. 4,
   Amsterdam: North-Holland, 2008, pp. 457-505.
 Venn, John, Symbolic Logic, London: Macmillan, 1881 [2nd edition, 1894]
 Wakeling, Edward (ed.),
   Lewis Carroll's Diaries: the Private Journals of Charles Lutwidge Dodgson (Lewis Carroll),
   10 volumes, The Lewis Carroll Society, 1993-2007.