STUDIES IN LOGIC, GRAMMAR AND RHETORIC 1 (14)

Andrzej Malec

NORMS AND PROGRAMS^{1}

1.

This paper is devoted to an explication of some concepts of the theory of law (first of all: concepts of inference) by means taken from logic of programs.

In logic of programs there are expressions of the form: A{N}B, where A and B are formulas and N is a program. There are also some connectives for programs which allow us to construct complex programs. Since norms and programs have the same nature (both are rules of behaviour), it should be possible to adopt the above concepts for a formal theory of norms. In this paper we want to show how some methods taken from logic of programs can contribute to the formal apparatus for studying norms. However, we are not going to construct a system of logic of norms.

2.

In the paper we will use expressions of the form: A{M}B, where A and B are the descriptions of a fragment of the world – respectively – before and after the transformation prescribed by the norm M. Also we will use normative connectives begin …, … end, … and …, and not . . .. The connectives are defined as follows: (A1) (A{M}B & B{N}C) → A{begin M, N end}C, (A2) A{M and N}B ≡ (A{begin M, N end}B & A{begin N, M end}B), (A3) (A{M}B ∨ A{not M}B) → (A{not M}B ≡ ¬A{M}B), for any A, B, C, M, N.^{2}

1 The paper is a revised version of my talk delivered during The Prague International Colloquium on The Nature of Argument (Prague, September 27-30, 1994).

2 So, we have two kinds of variables: for state descriptions and for norms. We use the quantifiers and the symbols: ¬, →, ≡, &, ∨ as they are used in the classical quantification theory.

Andrzej Malec

3.

A norm is a rule of behaviour accepted by a certain community. To obey legal norms one should recognize them. In continental Europe most of legal norms are derived by certain rules from legal texts (statutes, codes, etc.).

Let us reconstruct the process of derivation. Suppose that a judge wants to find a norm relevant to a certain situation. She or he starts with a set of inscriptions contained in legal texts (e.g. Civil Code of Poland).

Firstly, in the process of interpretation these inscriptions are transformed into a set of rules of behaviour. To construct this set one needs only some relatively simple methods which are usually called “the rules of interpretation”. The obtained rules of behaviour are called “primary norms” since they are explicitly contained in legal texts.

Secondly, the judge in question can infer some norms from the primary norms by methods which are usually called “the rules of inference”. The inferred norms are called “secondary norms” since they are not explicitly contained in legal texts. However, the inference in question is not the inference in the sense of classical logic: imperative clauses are inferred from imperative clauses.

How can we grasp the idea of such inference? To answer this question we will explicate several relations of this kind in terms of the proposed apparatus.

4.

According to the rules of inference accepted by lawyers, the following relations should hold:

(1) Inf.a. ({Help your parents!}, {Help your mother and father!})^{3} And vice versa, of course!

(2) Inf.b. ({If you are a man help Earth!}, {If you are a Pole help Earth!})

(3) Inf.c. ({If somebody is drowning, give her or him your hand}, {If somebody is drowning, pull her or him out of the water})

(4) Inf.d. ({Dress Adam and Eve!}, {Dress Adam!}).

3 We read: the norm “Help your mother and father!” is inferred from the norm “Help your parents!”.

Norms and Programs

These examples represent four of several kinds of basic relations of inference between norms. Moreover, various kinds of relations of inference can be combined. For example:

(5) Inf.a+d. ({Stay with your parents!}, {Don’t leave your mother!}) should hold, since:

(6) Inf.a. ({Stay with your parents!}, {Don’t leave your parents!}) and

(7) Inf.d. ({Don’t leave your parents!}, {Don’t leave your mother!}). We propose to explicate the relations of inference illustrated by the

above examples in the following way:

Inf.a. ({M}, {N}) iff

for any A, B: A{M}B ≡ A{N}B

Inf.b. ({M}, {N}) iff

for any A, B: A{N}B → A{M}B

Inf.c. ({M}, {N}) iff

for any A: A{begin M, not N end}A

Inf.d. ({M}, {N}) iff

there exists K such that for any A, B: A{M}B ≡ A{K and N}B

5.

In legal sciences there are so called “the collision rules” which help us to solve conflicts of norms. Suppose, a judge has derived from a legal text two norms A and B, both relevant to the same situation. If the norms are in conflict, she or he needs the collision rules to know which norm should be suppressed. Three types of conflicts of norms are discussed in legal theory: logical contradiction, logical opposition and praxiological contradiction of norms. Let us set forth some examples. According to lawyers, the following relations should hold:

(8) LC({“Help him!”}, {“Don’t help him!”})^{4}

(9) LO({“Turn left!”}, {“Turn right!”})^{5}

4 We read: there is a logical contradiction between the norms: “Help him!” and “Don’t help him!”.

5 We read: there is a logical opposition between the norms: “Turn left!” and “Turn right!”.

(10) PC({“Give him a lifebelt!”}, {“Put a shark into the water!”})^{6}

The above relations can be explicated in the following way: LC({M}, {N}) iff for any A, B: A{M}B ≡ A{not N}B LO({M}, {N}) iff for any A, B: A{M}B ≡ ¬A{N}B PC({M}, {N}) iff for any A: A{begin M, N end}A.

6.

Undoubtedly the presented explications are preliminary: we did not construct a system of logic of norms. However, the paper shows that some ideas of logic of programs may prove very useful for the formal theory of norms. In particular, the idea of considering programs in relation to certain input data and output data can be used in that theory: every norm can be treated as the prescription of transforming of a fragment of the world. From that point of view several legal concepts can be explicated. It becomes possible to define inference and contradiction between imperative clauses (compare section 3).

6 We read: there is a praxiological opposition between the norms: “Give him a lifebelt!” and “Put a shark into the water!”