\input stf
\def\NieparzystyHead{\line{\hfill\HeaderFont\vphantom{/S} Norms and Programs}}
\def\ParzystyHead{\line{\HeaderFont\vphantom{/S}Andrzej Malec\hfill}}

\startpageno=105
\pageno=\startpageno

\pu\pu\pu


\lasta Andrzej Malec\\

% Dept. of Logic, The University of Bialystok
% ul.Liniarskiego 4, 15\--420 Bialystok, Poland

\lastt NORMS AND PROGRAMS\f{The paper is a~revised version of my talk delivered during {\it The Prague
International Colloquium on The Nature of Argument}
(Prague, September 27-30, 1994).}\\

\pretolerance=10000

\def\Begin{\hbox{\it begin}}
\def\End{\hbox{\it end}}
\def\And{\hbox{\it and}}
\def\Not{\hbox{\it not}}
\def\tekst#1{\hbox{\rm #1}}
\def\sc#1{\scriptstyle{#1}}

\def\M{\hbox{\rm M}}
\def\N{\hbox{\rm N}}
\def\K{\hbox{\rm K}}

\def\ww #1 #2\\{\smallskip\line{\hbox to13mm{#1\hfill}#2\hfill}}
\def\wwDWA #1 #2\\#3\\{\smallskip
\line{\hbox to13mm{#1\hfill}#2\hfill}
\line{\hbox to13mm{\hfill}#3\hfill}
}

\rcenterbezpu 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.

\mpu

\rcenter 2.\\

\mppu


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 $\Begin\dots,\dots\End$,
$\dots\And\dots$, and $\Not\dots$.
The connectives are defined as follows:

\ww (A1) $(A\{\M\}B\ \&\ B\{\N\}C) \rightarrow A\{\Begin\ \M,\N\ \End\}C$,\\

\ww (A2) $A\{\M\ \And\ \N\}B \equiv (A\{\Begin\ \M,\N\ \End\}B\ \&\ A\{\Begin\ \N,\M\ \End\}B)$,\\

\ww (A3) $(A\{\M\}B\ \lor\ A\{\Not\ \M\}B) \rightarrow (A\{\Not\ \M\}B \equiv \lnot A\{\M\}B)$,\\

\smallskip

\noindent for any $A,B,C,\M,\N$.\f{So, we have two kinds of variables:
                            for state descriptions and for
                            norms. We use the quantifiers and the symbols:
                            $\sc\lnot,\ \sc\rightarrow,\ \sc\equiv,\ \sc\&,\ \sc\lor$
                            as they are used in the classical quantification theory.}


\rcenterbezpu 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.


\rcenter 4.\\

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

\ww (1) Inf.a. $(\{\tekst{Help your parents!}\},
             \{\tekst{Help your mother and father!}\})$\f{We read: the norm ``Help your mother and father!"
is inferred from the norm ``Help your parents!".}\\

\smallskip

\noindent And vice versa, of course!

\wwDWA (2) Inf.b. $(\{\tekst{If you are a~man help Earth!}\},\{\tekst{If you are a~Pole help}$\\$\tekst{Earth!}\})$\\

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

\ww (4) Inf.d. $(\{\tekst{Dress Adam and Eve!}\},\{\tekst{Dress Adam!}\})$.\\

\smallskip

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:

\ww (5) Inf.a+d. $(\{\tekst{Stay with your parents!}\},\{\tekst{Don't leave your mother}!\})$\\

\smallskip

\noindent should hold, since:

\ww (6) Inf.a. $(\{\tekst{Stay with your parents!}\},\{\tekst{Don't leave your parents!}\})$\\

\smallskip

\noindent and

\ww (7) Inf.d. $(\{\tekst{Don't leave your parents!}\},\{\tekst{Don't leave your mother!}\})$.\\


\smallskip

We propose to explicate the relations of
inference illustrated by the above examples in the following
way:

\wwDWA ~ Inf.a. $(\{\M\},\{\N\})$ iff\\for any $A,B$: $A\{\M\}B\ \equiv\ A\{\N\}B$\\

\wwDWA ~ Inf.b. $(\{\M\},\{\N\})$ iff\\for any $A,B$: $A\{\N\}B\ \rightarrow\ A\{\M\}B$\\

\wwDWA ~ Inf.c. $(\{\M\},\{\N\})$ iff\\for any $A$: $A\{\Begin\ \M, \Not\ \N\ \End\}A$\\

\wwDWA ~ Inf.d. $(\{\M\},\{\N\})$ iff\\there exists $\K$ such that for any $A,B$: $A\{\M\}B\ \equiv\ A\{\K\ \And\ \N\}B$\\


\rcenter 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:

\ww (8) LC$(\{\tekst{``Help him!"}\},
         \{\tekst{``Don't help him!"}\})$\f{We read: there is a~logical contradiction between the
norms: ``Help him!" and ``Don't help him!".}\\

\ww (9) LO$(\{\tekst{``Turn left!"}\},
         \{\tekst{``Turn right!"}\})$\f{We read: there is a~logical opposition between the norms:
``Turn left!" and ``Turn right!".}\\

\ww (10) PC$(\{\tekst{``Give him a~lifebelt!"}\},
         \{\tekst{``Put a~shark into the water!"}\})$\f{We read: there is a~praxiological opposition between the norms:
``Give him a~lifebelt!" and ``Put a~shark into the water!"}\\

\smallskip

The above relations can be explicated in the following way:

\ww ~ LC$(\{\M\},\{\N\})$ iff for any $A,B$: $A\{\M\}B\ \equiv\ A\{\Not\ \N\}B$\\

\ww ~ LO$(\{\M\},\{\N\})$ iff for any $A,B$: $A\{\M\}B\ \equiv\ \lnot A\{\N\}B$\\

\ww ~ PC$(\{\M\},\{\N\})$ iff for any $A$: $A\{\Begin\ \M,\N\ \End\}A$.\\


\rcenter 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).