Polish notation as devised by Jan Lukasiewicz is not only a technical device. There is a theory of syntax which underlies it, a theory which may shed light on more general problems of intelligent behaviour as investigated in cognitive science as well as praxeology. Moreover, logicians have their own motivation to trace connexions between syntax and general laws of thought and action.

To wit, Cantor's well-ordering theorem, to the effect that IT IS ALWAYS POSSIBLE TO ARRANGE EVERY WELL-DEFINED SET IN THE FORM OF WELL-ORDERED SET, was regarded by him as a fundamental law of thought, rich in consequences and particularily remarkable for its general validity. The same view was expressed by Zermelo and, in a sense, by Hilbert. There are some questions as to interpretation of this principle, e.g., what the phrase "well-defined" should mean, but there is a fairly common agreement as to its validity in the realm of mathematics.

What about its universal validity in the empirical world? It seems rather dubious unless one proves things like that: that even a most chaotic collection of gas particles has a hidden structure which could be rendered as a well-ordered set. In spite of such justified doubts, it seems reasonable to look for empirical domains in which the well-ordering principle could be applied. Among those worth considering there are collections of well-formed strings of a language.

Obviously, Cantor's principle holds for such collections, in particular that defined by the grammar governing Polih notation; its principles are developed in what nowadays is known as a version of categorial grammar, going back to Kazimierz Ajdukiewicz. As far as other grammars are concerned, the problem deserves a separate research. What is examined in this study, it is a praxeological generalization of the ordering principle which holds for Polish notation.

1. A programme for syntax of action

Polish notation has found fruitful applications in computer science, especially its versions termed as reverse Polish notation (used in machine codes and such programs as PosScript), and Cambridge Polish notation (used in programming languages of higher order as LISP); the latter reintroduces perentheses without losing, though, the syntactic principle characteristic of this notation. When addressed to computers, a string of symbols in Polish notation does not express any truth about the world. Instead, it forms a sequence of imperatives to control machine's work; this produces a link between Polish notation and a theory of action.

These applications are motivated by greater simplicity (from computer's point of view) of parentheses-free strings in machine codes as well as main syntactic idea of the notation. It is the idea of a hierachical sequence of operations, i.e. functions in the mathematical sense, being materialized in machine procedures. This hierarchy is what makes it possible to discern syntactic constituents without segmentation by parentheses, provided that the number and order of arguments of each functor (i.e. function sign) is previously given, e.g. in the vocabulary of the language in question. The behaviour which is controlled by instructions recorded in a machine code provides us with a fitting model to be applied in cognitive science and biology. In particular, human behaviour can be considered as realizing a program, in particular a program for planning a future action.

There is a widely known case of Koehler's chimpanzee Sultan who fitted a bamboo stick into another, after some futile attempts to solve the problem of grasping fruit that was out of his reach. In spite of the whole distance between a human and an ape, a human would react in a similar way, as in the case in question this is the only correct solution; therefore when reconstructing the structure of that action we are entitled to make use of our human thought-experiment.

The planning proceeds from more general to more concrete statements concerning what should be done; the latter express ideas which cannot arise unless the problem is stated in general terms. In the following list of planning steps, the longer the sequence of digits, the more concrete step is numbered by it.

• 1. to have a tool adjusted to the distance (between the agent and the fruit)
  • 1.1. to shorten the distance
    
  • 1.2. to lengthen the stick (serving as the tool)
    • 1.1.1. to come closer to the fruit
    • 1.2.1. to join two sticks
      • 1.2.1.1. to fasten stick no.1 to stick no.2
      • 1.2.1.2. to fasten stick no.2 to stick no.1.

Actions are arguments of the following functions:

*< assigns the desired result to planned means;
& combines two or more actions into a whole;
::: is the function which can be called sequential conjunction of three arguments; for instance, :::xyz means that x, y and z form a compound process, in which y follows x, and z follows y.

This is just an example of the syntax of planning actions. It represents only one class of functions whereas there may by much more of them. A systematic study of such grammars should lead to a theory of action, or praxeology (in the sense given this term by Ludwig von Mises and Tadeusz Kotarbinski). The constrains imposed by syntactical principles of Polish notation should grant planning a high level of preciseness because of the necessity of defining all the functions involved, and putting each of them into an exactly defined place in the string.

2. A test for Polish notation - parsing of reported speech

When recommending Polish notation for extra-linguistic applications, the author is bound to test it in linguistic area more difficult in this respect than are languages of logic and mathematics. There is in natural languages a syntactic construction which is headache for anybody who tried to handle it with the Polish notation grammar, namely reported speech. (The present author coped with it in the aricle 'How freely can categories be assigned to expressions of natural language? A case study' in: W.Buszkowski, W.Marciszewski and J.van Benthem (Eds), Categorial Grammar, Benjamins, Amsterdam 1988).

Let the asterisk represent the particle "that" as involved in the structure nv*s, where "x" represents a name, "v" a verb from the class containing "says", "believes", "knows", etc., and "s" hints at a sentence. There is a combinatory method to obtain various syntactic structures, and among them the following.

• [1] *(vn)s ... * links the sentences vn and s.
• [2] (*v)ns ... * makes a new functor out of v, and this functor, in turn, combines n and s into a sentence.
• [3] vn(*s) .... makes a name out of s, while v acts as functor to combine the names n and *s into a sentence.

Obviously, there is implicit * after "said". This may mean that the author reporting God's saying treats "said" as abbreviation of "says that", hence "that" is a part of "says that"; there are reasons to interpret "says" as having category s/n, while * transforms it into category s/ns. This exemplifies form [2].

The explicit * nicely fits into category n/s since the phrase "that the man should be alone" is the name of the situation of which it is predicated "it is not good", according to schema [3]. Parsing [1] may be less convincing, nevertheless it is preferred by some linguists.

3. Syntax as a constitutive factor of human intelligence

It was stated above that the grammar underlying Polish notation can improve planning of actions, and so make them more likely to succeed. Since successful (in a long run) activity is a proof of intelligence, the above statement accords with the tenets, appearing in cognitive science, that intelligence derives from human syntactic abilities.

This point is defended, e.g., in William H. Calvin's numerous books and, most concisely, in his article "The Emergence of Intelligence", Scientific American, 271(4):100-107, October 1994. Here is the statement in question.

Language is the most defining feature of human intelligence: without syntax--the orderly arrangement of verbal ideas--we would be little more clever than a chimpanzee. Something very close to verbal syntax also seems to contribute to another outstanding feature of human intelligence, the ability to plan ahead.

To understand why humans are so intelligent, we need to understand how our ancestors remodeled the ape symbolic repertoire and enhanced it by inventing syntax. [...] Human planning abilities may stem from our talent for building syntactical, string-based conceptual structures larger than sentences. [...] In this way, syntax raises intelligence to a new level. By borrowing the mental structures for syntax to judge other combinations of possible actions, we can extend our planning abilities and our intelligence.

Anyway, if there is grammar at all whose principles would be close to those of planning structures in intelligent actions, it should take advantage of the concepts of operation and its operands, or functor and its arguments. This condition is undoubtedly satfisfied by the grammar of Polish notation.