Typing logic contents using Lingua Cosmica

Acta Astronautica 68 (2011) 535–538

Contents lists available at ScienceDirect

Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro

Typing logic contents using Lingua Cosmica Alexander Ollongren a,n, Douglas A. Vakoch b a b

LIACS, Leiden University, The Netherlands SETI Institute, Mountain View, CA, USA

a r t i c l e in fo

abstract

Article history: Received 18 February 2009 Received in revised form 7 August 2010 Accepted 16 August 2010 Available online 22 September 2010

This paper informs how elements of constructive type theory can be used effectively for clarifying textual messages meant for communication with ETI. Within the setting of a suitable environment consisting of declared terms, it is shown how logical contents of texts can be modelled and codified. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Interstellar messages Lingua Cosmica

1. Considerations The present paper is concerned with the problem of explaining to intelligent beings in the universe the logic of interstellar messages formulated in a language they do not know and understand. In addition it is assumed that the recipients of the messages have no linguistic system in common with the species transmitting them. Therefore, ‘natural’ language only cannot be used for interstellar communication. On the other hand, it might be supposed that all sufficiently developed intelligent species know abstract logic and are familiar with the particular mode used in this paper, constructive logic based on type theory. There may be a multitude of conventions in notation and quite some variation in the basics for this theory used by intelligent species, but problems presented by that can be overcome. The reason is that the mentioned logic uses a sparse set of rules and admits its own interpretation. For the formation of (typed) expressions it is not assumed that concrete objects like truth values (Booleans) and sets exist. Expressions are abstractly typed and there are only three abstract basic types (constants free of

n

Corresponding author. Tel.: + 31 715173169. E-mail addresses: [email protected], [email protected] (A. Ollongren), [email protected] (D.A. Vakoch). 0094-5765/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2010.08.017

interpretation) together with the mapping concept available as basic instruments for constructing them. The expressive power of the logic, however, is large. In order to illustrate some aspects of these considerations an ancient text from Plato’s dialogues in The Republic is selected. The logic contents of it are described in the context of the mentioned abstract setting.

2. Introduction A language for cosmic message construction (a Lingua Cosmica or LINCOS) can be ‘pure’ in the sense that the possible sentences of the language are strictly defined syntactically by means of grammatical rules. In previous papers by the first author, the idea was introduced of formulating a message for ETI using a (terrestrial) natural language in juxtaposition with a set of pictures (coded as bitmaps) illustrating the ‘story’ told—allowing interaction between text and pictures. An additional idea is that messages might be augmented with descriptions of the semantics of certain parts in another formalism at a higher level. The selected parts can be sections, in some cases also individual declarative sentences, but the expressions at the higher level are supposed to describe relations ‘living’ in the text (in general a kind of story) transmitted. The resulting two-level approach to message construction can been called a LINCOS system.

536

A. Ollongren, D.A. Vakoch / Acta Astronautica 68 (2011) 535–538

There are several formalisms from mathematics and logic available for use at the higher level. In the present contribution we show how elements of constructive type theory can be used elegantly for partly clarifying messages for ETI. There are several reasons for using abstract type theory. Evidently a powerful system is needed, while the expressions at the second level should rely on a limited set of primitives only. Furthermore, preference is given to systems for which computer implementations and extensions to higher order logic are feasible, as in type theory. We base our discussion here on the Calculus of Constructions, more in particular elementary constructs in the French Coq proof assistant [1] which has the important advantage that each expression can be checked for correctness using available computer implementations. Since type theory belongs to the realm of formal logic, we systematically explain by examples how to construct expressions in terms of propositions and predicates, representing logical content of texts. Each example and introductory or concluding lemma has been verified in Coq. An advantage of this approach is that text augmented with formal descriptions is easily understandable for humans. For those ET’s, who have attained a level of technological and intellectual sophistication comparable or beyond ours, the underlying logic might hopefully be familiar and recognised in the end, even though the notation employed is not. The two levels (text—explanations and annotations in logic) in conjunction with one another should be self-explanatory. 3. A text In order to illustrate the idea of using a meta level for explaining the logic of message content, we use here an ancient text, in fact from the Great Dialogues of Plato (427– 347 BC). In the first book from The Republic, a dialogue takes place in which Socrates (469–399 BC) tries to define ‘justice’. He does not succeed in doing so. The summary of book I, in the words of Rouse [2], written in 1956, reads as follows: Polemarchos invites Socrates and Glaucon to visit his father Cephalos’ house. Various other friends are there as well. Cephalos talks about old age: eventually the conversation turns to the subject of justice. How do you define justice? asks Socrates. Polemarchos puts forward Simonides’ definition – to render what is due [obligation] – but this on examination proves unsatisfactory. Here Thrasymachos breaks in, maintaining that the whole conversation so far has consisted of nothing but pious platitudes. Justice, he says, is whatever suits the strongest best. Might is right. A ruler is always just. Socrates suggests that even a ruler sometimes makes a mistake, and orders his subjects to do something which is really not to his advantage at all. Thrasymachos answers that in so far as he is mistaken he is not a true ruler. Socrates then argues that a doctor is primarily concerned to heal the sick, and only incidentally to make money: similarly, medicine seeks not its own advantage but the advantage of the human body. By analogy a ruler seeks the advantage of his subjects, not of himself. Thrasymachos then rushes off on a new tack. Injustice, he says, is

virtuous, and justice is vicious. Justice is everywhere at the mercy of injustice, which is reviled not because men fear to do it but because they fear to suffer it. Socrates sets out to disprove this view, and establishes that justice is apparently wise and virtuous, and at the same time more profitable than injustice. But, he says, he is still without a definition of justice. 4. Message content All expressions in this section are written in the conventions of LINCOS, itself based on constructive type theory (CTT). The expressions are type expressions and are easily recognisable. Each expression ‘has a type’—or ‘is of a type’. The fact that an expression E is of type T is written E:T and it can be said that E lives in T, or that E is a resident of T. In CTT three distinct constants are predefined: Type—abstract types, itself typed by Type: Type Set—general entities, itself typed by Set: Type Prop—logical propositions, itself typed by Prop: Type Prop need not have residents a priori, not even true and false. We shall provide it now with one by using CONSTANT thus introducing a constant: CONSTANT unsatisfactory:Prop. The unsatisfactory is inspired by Plato’s text. Since we shall first consider Simonides’ definition in the text, we introduce a section, and in it then his definition. SECTION Simonides. CONSTANT obligation:Set. HYPOTHESIS render: obligation-Prop. DEFINE justice: obligation-Prop: = [x:obligation](render x). render is assumed to be a type (a map) which couples residents of obligation (the actual obligations) to residents of Prop (propositions). justice on the other hand is defined as a map, ranging over obligation, parametrized with the formal local variable x ranging over obligation. Formally the variable x is introduced by the mathematical technique known as lambda abstraction. Let A be some actual obligation, so A:obligation. Then (render A):Prop and [A:obligation](render A) of type obligation-Prop is a way of stating that any (given) obligation A must be rendered. Note that in view of the definition (justice A) =(render A):Prop

A. Ollongren, D.A. Vakoch / Acta Astronautica 68 (2011) 535–538

The definition of justice is indeed unsatisfactory because there are no restrictions over obligations (examples in Book I). So we close the section with END Simonides, and open a new section expressing Thrasy-machos’ views: SECTION Thrasymachos CONSTANT strongest:Set. CONSTANT ruler:Set. DEFINE might: =strongest Note: there is no difference in type between might and strongest. FACT suit: (strongest-(Prop-Prop)). This is an introduction lemma, which couples residents of strongest to the type Prop-Prop, not just to Prop, for technical reasons. The proof of the lemma is omitted. DEFINE justice: strongest-(Prop-Prop):= [x:strongest; best:Prop] (suit x best). For given parameters S: strongest and B: best, (suit S B), will be the case. This is because (justice strongest best): (suit S B), in view of the lambda form [x: strongest; best: Prop] Note (suit S B): Prop. Using above definition justice ranges over strongest. In this way the concept of justice is coupled to the concept of strongest, also an unsatisfactory situation. Unclear is what is meant by best. CONSTANT is-right: might-Prop. HYPOTHESIS Might-is-right: (x:might)(is-right x). Here (x: might) is to be read as ‘for all residents of might’, i.e. using the logician’s ‘all’, i.e. universal quantification. CONSTANT is-just: ruler-Prop. HYPOTHESIS ruler-is-just: (x:ruler) (is-just x). This expresses the assumption that all rulers are just. DEFINE mistaken-ruler: ruler-Prop: = [x : ruler]  (is-just x).

537

The  used here is the logical negation, or not. Since mistaken rulers are introduced by definition and not separately, it can be shown that they do not exist. In this case we do that by writing the provable fact: FACT notexist:  (EX x:ruler 9 (mistaken-ruler x)). Expressing that there is no x of type ruler, for which (mistaken-ruler x) is the case. This situation could be changed and improved, but is not interesting enough to justify the effort. So the section is closed. END Thrasymachos. The most interesting views in the text are those of Socrates. Here they are expressed in a new section SECTION Socrates. Socrates argues in the text that a doctor is primarily concerned to heal the sick, and only incidentally to make money, so we introduce a doctor with a definition DEFINE (INDUCTIVE) doctor:Set: = IDD: Primarily-(incidentally- doctor) WITH primarily: Set: = IDP: healing-sick-primarily WITH incidentally: Set: = IDI: making-money-incidentally. This expresses Socrates’ argument—in the form of a rather sophisticated (inductive) definition of the function of a doctor. We use the inductive form here because the text mentions primarily and incidentally. The recursive nature of doctor is, however, not essential in the present discussion. Also the selectors (sometimes called constructors) IDD, IDP and IDI, useful for identification, are not essential in the rest of the sequel. Separate definitions are needed for healing, sick and making-money. Each of these can be considered as abstract entities, for which no details need to be given, so they can be defined as members of Set. They must be defined before they are used in doctor. CONSTANTS healing,sick,making-money: Set. Next an introduction lemma is formulated and proved. Using the lemma we define the wise doctor, the one who heals the sick. FACT doctor-heal-th-sick: healing-(sick-(doctor-Prop)).

538

A. Ollongren, D.A. Vakoch / Acta Astronautica 68 (2011) 535–538

The (elementary) proof is omitted DEFINE wise-doctor:= [h:healing;s:sick; d:doctor] (doctor-heal-th-sick h s d). Socrates establishes that justice is both wise and virtuous. So it is useful to introduce now the virtuous doctor, for sake of simplicity the one who does not make a lot of money. The entity money can be taken to be the same as the natural numbers, so

Should it be required to explain the ingredients of this kind of proofs, more extensive examples need to be supplied in the text. Above example shows, however, also without explicit proof, how to express the conditions for doctors to be wise and virtuous. Such doctors are seen to be just. Socrates’ conclusion that justice is apparently wise and virtuous cannot logically be based on the assumptions formulated above. By hypothesis it could of course be introduced consistent with the proof of the lemma just-doctor. So, indeed, no satisfactory definition of justice has been obtained.

DEFINE money: = nat. where nat:Set will do. First the introduction lemma: FACT doctor-make-money : money- (doctor-Prop). The (simple standard) proof is omitted. After this, the virtuous doctor can be defined:

END Socrates. In much the same way as shown above, the views of Thrasymachos upon second thoughts, can be formalised in LINCOS, but that would not add to the insights provided with the above examples. 5. Discussion

DEFINE virtuous-doctor: = [m:money; d:doctor] (doctor-make-money m d). (note the important negation ) Of course there might exist wise and virtuous doctors as well. DEFINE wise-and-virtuous-doctor:= [h:healing; m:money; d:doctor; s:sick] (and (wise-doctor h s d) (virtuous-doctor m d)). Now the stage is set for proving a theorem which says that a wise and virtuous doctor heals the sick and does not earn money (!): FACT just-doctor: (h: healing; m: money; d:doctor; s:sick) (wise-and-virtuousdoctor h m d s) ((doctor-heal-th-sick h s d) ( (doctor-make-money m d))). The lemma is self-explanatory, the proof (not so elementary) is not given here.

In the foregoing typing exercise some selected words in an example text (a summary of one of the Great Dialogues of Plato) are literally quoted. That practice helps recipients of a message containing text and type expressions to recognise that logic is involved. At the same time the sequence of statements, as they are arranged in the text, has been kept more or less in the annotations, but this may not always be feasible. Some but not all statements in the text are formalised. No effort has been made to treat separate sentences an sich. This is because it is not our aim to provide semantics for sentences (or other composite utterances) in natural language. Instead we wish to show how to clarify the logic contents of texts by (partially) typing them. Note that completeness is not aimed at. In our view only those aspects that might help to clarify the logic of textual contents (following judgements by the ‘logician in charge’), should be taken into consideration for formalisation.

Acknowledgement The first author expresses his appreciation for the valuable comments of the reviewer. All of the suggestions for improvements have been taken into account. Hopefully this has led to a better presentation. References [1] Ge´rard Huet, et al., The Coq Proof Assistant, A Tutorial, INRIA, 1999. [2] Great Dialogues of Plato (translated by W.H.D. Rouse), Mentor Books, 1956.