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Kolmogorov and The General Theory of
Problems
Wagner de Campos Sanz
Abstract This essay is our modest contribution to a volume in honor of our dear friend
and fellow logician Peter Schröeder-Heister.1The article has a double objective. One
is to reexamine Kolmogorov’s problem interpretation for intuitionistic logic and the
second is to render it through a clausal semantics without mentioning the concepts
of hypothesis or assumption in a explicit way. Schroeder-Heister has always stressed
the role of assumptions in proof-theoretic semantics. The clausal semantics to be
presented will be used for making remarks about assumptions and hypotheses, about
logical constant rules and harmony.
1 Introduction
The epistemology of mathematics has gone through an important change in the
last centuries: from focus in problems to focus in theorems. One landmark in
this transformation is Hilbert’s Foundations of Geometry. Compared to Euclid’s
Elements, all its propositiones are theorems while this is not true for the ancient
books. It is time for the pendulum to move back.
Kolmogorov (1932; p.151) had advanced an interpretation of intuitionistic logic
based on the concept of problem:2
Wagner de Campos Sanz
Universidade Federal de Goiás, Faculdade de Filosofia, Campus Samambaia, Goiânia–GO, 74690-
900, Brazil, e-mail: wsanz@ufg.br
1I’m indebted to Peter in many in different ways. More than once I was received by the whole
Tübingen group in a cordial and intellectually inspiring environment which has been possible
through the generosity and leading role of Peter in Logic and Proof-Theoretic Semantics.
2Kolmogorov’s interpretation is the third element in what has been conventionally called BHK
interpretation of intuitionistic logical constants. The abbreviation corresponds to the initials of
Brouwer, Heyting and Kolmogorov.
1
2 Wagner de Campos Sanz
Along with the development of theoretical logic, which systematizes the schemes of proofs
of theoretical truths, it is also possible to systematize the schemes of solution of problems, for
example, geometrical construction problems [. . . ] The second section in which the general
intuitionistic presuppositions are accepted, presents a critical analysis of intuitionistic logic.
It is shown that this logic should be replaced by the calculus of problems, since the objects
under consideration are in fact problems, rather than theoretical propositions.
Years later, in a comment about his paper, the author evaluated it as follows
(Tikhomirov 1991; p. 452):
On the interpretation of intuitionistic logic (IIL) was written with the hope that the logic of
solutions of problems would later become a regular part of courses on logic. It was intended
to construct a unified logical apparatus dealing with objects of two types—propositions and
problems.
The objective here is to reformulate Kolmogorov’s semantical theory of problems
in the path already taken in Sanz (2012). The semantics to be offered turns to be
identical to Hypo Semantics (Sanz 2019) with a new dressing.
2 Kolmogorov on problems
Kolmogorov’s article is schematic and it does not contain an explanation of what is
a problem, only a few carefully chosen examples (Kolmogorov 1932; pp. 151–152):
1. Find four integers x,y,zand nsuch that xn+yn=zn,n>2.
2. Prove that Fermat’s theorem is false.
3. Construct a circle passing through three given points (x,y,z).
4. Given one root of the equation ax2+bx +c=0, find the other root.
5. Assuming that the number πhas a rational expression, π=m/n, find a similar
expression for the number e.
According to Coquand (2007; §,2.5), the perspective of interpreting intuitionistic
logic as a calculus of problems is an important antecedent of a later distinction: that
between formulae-as-types and λ-terms which would then correspond to a separation
of problems and solutions, respectively. Although this can be a productive way of
interpreting Kolmogorov about the semantics of problems, it is not clear that he
would subscribe a sharp separation between the formal apparatuses for representing
problems and solutions.
Kolmogorov (1932; p. 151) characterized his semantics of problems for logical
molecular operators as follows:
If aand bare problems, then a∧bdenotes the problem “solve both problems aand
b”, while a∨bstands for “solve at least one of the problems aand b”. Further, a⊃b
is the problem [(FIRST)] “given a solution to problem a, solve problem b” or, which is
the same, [(SECOND)] “reduce the solution of problem bto the solution of problem a.
[...] ¬adenotes the problem “assuming that there is a solution to problem a, derive a
contradiction”.3
3Square brackets were added by us.
Kolmogorov and The General Theory of Problems 3
In the five problems quoted above, we notice that the principal verb is imperative.
They are to be considered as the command of an action. The semantic explanations,
in their turn, notoriously contain the command: “solve”. Implication is the only
exception. It has two explanations: (FIRST) and (SECOND). And the author regards
them as equivalent. The expression “solution” is used in both explanations for naming
and the expression “reduce” in the (SECOND) explanation is used for commanding.
It is not clear how the semantics of implication could be formulated by using the
command “solve” in the antecedent.4In the reverse direction, an attempt to employ
the expression “solution” for explaining conjunction and disjunction would require
the use of another command verb in place of “solve”, as in: “produce the solution of
. . . ”. But this is no real progress. It seems that a strange heterogeneity has infiltrated
in the above semantical explanations.
The (SECOND) explanation is partially misleading since the expression “reduce
the solution of b. . . ” seems to make reference to a specific solution. But a problem
could possess distinct solutions. A similar issue also affects the second part of the
(SECOND) explanation. In the so called BHK interpretation5, for any solution of a
corresponds a solution of b, and this is necessary in order to have a true reduction.
3 A semantics of problems
The whole list of explanations can be made homogeneous by adopting a new
organization of the background concepts such that the concept of reduction becomes
the principal relation in the semantical explanation of logical constants. The
expression Γbwill mean from now on “problem bhas been reduced to the
set of problems Γ” but only if it is guaranteed to obtain a solution of bin case of
having a solution, whatever be it, for each problem in Γ. This idea is stressed by
Veloso (1984; p. 26) when he considers the concept of reduction.6
Every intuitionistic molecular logical constant can be defined by means of the
following semantical clauses:7
Clauses for using a logical constant in the repertoire:
(∧l):Γ,c∧deΓ,c,de
(∨l):Γ,c∨deΓ,ceand Γ,de
(⊃l):Γ,c⊃degiven any problem a:(a,cd⇒Γ,ae)
(⊥l):Γ,⊥aalways (where ais a basic problem)
Clauses for using a logical constant in the focus:
4By saying “first solve a, then solve b” another connective distinct of implication is being used.
5See the second note.
6He suggests that we should not consider arbitrary links between problems, but only those
guaranteeing that solvability of bis obtained from solvability of a, as described previously.
7Kolmogorov didn’t use the absurd as a logical constant in his paper, we here follow the late
tradition on the subject.
4 Wagner de Campos Sanz
(∧r):Γc∧dΓcand Γe
(∨r):Γc∨dgiven any problem a((Γ,caand Γ,da)⇒Γa)
(⊃r):Γc⊃dΓ,cd8
(⊥r):Γ⊥given any basic problem a:Γa
The repertoire of problems is the set in left side of the semantical symbol, and
the focal problem (of which there is only one) is in the right side of the semantical
symbol. Clauses are of two kinds: (i) those explaining the use of logical constants
in repertoires (finite sets, for the time being); (ii) those explaining the use of logical
constants as a focal problem. The symbol “” expresses a necessary and sufficient
condition,9that is an explanation of meaning which has explicandum in the left side
and explicans in the right side.10 Negation is explicitly defined as: ¬a≡a⊃ ⊥.
Given any problem, we can fairly say that either it has been solved or it has not.
If it was solved, then either it has been positively solved - when a correct solution
is given to the problem - or it has been negatively solved - when a positive solution
has shown to be impossible. The expression "to be solvable" is ambiguous and it can
mean to be positively solvable. But it can also mean to be either positively solvable or
negatively solvable. These distinctions are relevant for the discussion coming next.
The logical constants usual in intuitionistic sentential logic have now an
homogeneous explanation in terms of reduction between problems. For example,
a conjunction in the focus is to be read as: c∧dhas been reduced to repertoire Γ
when, and only when, chas been reduced to Γand dhas been reduced to Γ. Although
the command “solve” could mean either positively solve or negatively solve11 , the
intended reading is probably that of positively solve. Since the notion of reduction is
at hand, the command “solve” can be dismissed. That is, categorically and positively
solve c∧dis the same as to show that it has been reduced to the empty repertoire of
problems. And, to negatively solve it means to show that the reduction is impossible.
Implication a⊃bis now positively solved categorically when bhas been reduced
to aaccording to clause (⊃r). Observe that by saying “has been reduced” it should
be understood that an action took place.
The fictious absurd (⊥) problem is considered as negatively solved. Thus, since
negation is defined via implication, the negation of a(¬a)means that the absurd
(impossible) problem has been reduced to a. At the same time that ais negatively
solved, ¬ais also positively solved. The absurd problem is taken as a basic problem
itself and it is semantically characterized as the problem to which all basic problems
reduce. That is, its solution is the [or a] panacea. Hence, the concept of contradiction
originally used in Kolmogorov’s paper can also be dismissed. As such, when the
author says that a conditional problem is meaningless and then consider it as solved,
this is equivalent to say that the antecedent problem is impossible and the whole
8As a semantical rule, we can read it as follows: in order for the reducibility problem of dto cto be
reduced to the set of problems Γit is necessary and sufficient that dbe reduced to the set Γ∪ { c}.
9Harmony is an inherent property of explanations stated in terms of necessary and sufficient
conditions.
10 Thus, it is not an explicit definition.
11 Which then means that one of two possible conclusions is expected.
Kolmogorov and The General Theory of Problems 5
conditional problem is reduced to the empty repertoire, since every basic problem
reduces to the absurd problem by definition.
The above clauses must be complemented by the following semantical principles
making explicit the properties of reduction:
cc(Identity)
Γd⇒Γ,cd(Load problem)
(Γ,cd&Γc) ⇒ Γd(Drop problem)
The above semantics of problems is called Reduction Semantics.12 13
Going back to Kolmogorov’s paper in order to determine if the above picture fits
in, notice that the author intends with his calculus “. . . to systematize the schemes of
solution of problems”, as those in geometrical constructions. In order to judge how
good is the above approach in this respect, we’ll consider a few elements in Book I
of Euclid’s Elements.
4 What is a problem?
Before examining geometry, it is important to analyze what is a problem, how they
can be conceived, question that was not explicitly answered by the author.
Problems involve actions. It asks or commands an action: to find something,
to draw something, etc. Solutions too involve actions, and they can be divided
in two kinds, because problems are themselves of two kinds: token-problems and
type-problems. For example, a problem like (1) in the above quotation enumerating
problems, the one asking to find four numbers, is a token-problem. Its solution, when
there is one, involves an act of exhibition of a result or state-of-affairs. A problem
like (3) is a type-problem, since its solution must be a recipe/algorithm showing how
to fulfill what is being commanded or asked for given x,yand z.
Actually, all solutions can be characterized as recipes. Any token-problems can
be understood as solved by them: recipes containing a final act of exhibition. In other
terms, here the recipe is to be thought as being recoverable from the succession of
actions that produced the result.
Problems and solutions are basically actions. A solution is a recipe describing
actions which when realized will solve a problem, while a problem is equivalent to
ask or command an action. Because of this same inner nature, we think it unnecessary
to consider two different representational structures for problems and solutions as
Coquand seemed to imply.
12 A simpler version of (Id) with basic cis enough for obtaining the full version. In that case,
all repertoire and focus clauses are needed. The reverse implications of (Load) and (Drop) do not
hold. A simpler version of (Drop) with basic cis enough for obtaining full (Drop). In that case, all
repertoire and focus clauses for logical constants are needed. Notice however that having the three
full principles in hand the repertoire clauses are derivable from the focus clauses and vice-versa.
13 When the variables a,b,c, etc., are interpreted as sentences we call it Hypo[thesis] Semantics
(Sanz 2019).
6 Wagner de Campos Sanz
5 Solutions and problems
Kolmogorov intended the calculus of problems to deal with objects of two types
in a unified fashion.14 And since the expression of problems and propositions
corresponds to different kinds of utterances, imperative and assertoric (or indicative)
moods, it might seem necessary to dispose of two different structures for representing
propositions and problems, like typed λ-calculus.
Notwithstanding, the author stated, although only in an implicit fashion, what
was his unifying solution to the above question. It is not that of having two distinct
apparatuses for representing problems and propositions. His unifying view can be
extracted from the discussion about the principle of excluded middle, when he says
that the expression `a∨ ¬ashould be read as asking to (p. 156, our emphasis):
[. . . ] find a general method which for any problem aallows one either to find its solution or
to derive a contradiction from the existence of such a solution! In particular, if the problem
aconsists of proving a proposition, one must have a general method which allows one either
to prove each proposition or to reduce it to a contradiction.
Very clear! Proving a proposition is just one kind of problems.
The concept of a problem is apparently more general than the concept of a
theorem.15 Actually, Kolmogorov’s calculus has room for propositions and their
proofs, since one kind of basic problems that must be considered are problems of
the kind (prove p), for a basic proposition p. In consequence, (prove p∧prove q),
(prove p∨prove q),(a⊃prove q), etc., are problems too.
Deductions and connectives for propositions are given as follows:16
(deduce p∧qfrom Γ) ≡ (deduce pfrom Γ)∧(deduce qfrom Γ)
(deduce p∨qfrom Γ) ≡ (r)(((deduce rfrom Γ,p)∧(deduce rfrom Γ,q))
⊃ (deduce rfrom Γ))
(deduce p⊃qfrom Γ)≡(deduce qfrom Γ,p)
(deduce ⊥from Γ) ≡ (atomic p)(deduce pfrom Γ)
The special case where the Γis empty in the definiendum corresponds to a definition
of closed proofs, i.e.: (prove p)≡(deduce p f r om ∅). It can easily be realized
that (prove(p⊃q)) cannot be obtained from (prove p) ⊃ (prove q), whose
equivalent is (prove p)(prove q).
A positive solution to the problem (prove that the number πhas a rational
expression) is impossible (i.e., it has only a negative solution), therefore: (*) (prove
that the number πhas a rational expression) ⊥.17 As a consequence ¬(prove
14 Problem (5) in the above quotation illustrates this need, since this conditional problem - asking to
find a way to express number eas a rational expression - has in the antecedent position a proposition:
“that the number πhas a rational expression”.
15 Hence the change in focusfrom problems to theorems in the history of epistemology is regrettable.
16 The proposition ⊥is atomic.
17 Kolmogorov (1932; p. 151) says: “Here and elsewhere the proof of the fact that a problem is
meaningless is considered as a solution of this problem”. An example of meaningless or empty
problem is the conditional problem (5) in the quotation. It has a false antecedent.
Kolmogorov and The General Theory of Problems 7
that the number πhas a rational expression). According to clause (⊥r), (*) means:
for any basic problem p, (prove that the number πhas a rational expression) p.18
6 Solutions and problems in ancient geometry
Now turning to Book I of Euclids Elements, can we use the semantics of problems
for analyzing it?
Well, there is a precedent attempt on this subject by VonPlato and Mäenpää
(1990). The authors develop an interpretation based on Martin-Löf´s Type Theory
where each of the first three postulates is taken as a constructive function. This
interpretation is discussed and criticized in Naibo (2018). VonPlaton and Mäenpää´s
article is also one of the basis over which an interesting exegesis of Euclidean
Geometry has been proposed by Sidoli (2018). Another important source is Beeson
(2010) on what concerns a constructivist interpretation of ancient geometry.
According to VonPlato and Mäenpää, since Martin-Löf´s Type Theory is a
development of Kolmogorov´s problem interpretation, their examination of geometry
is a problem interpretation of constructions. However, some doubts concerning this
problem interpretation must be considered. Although postulate I.1 can be interpreted
as a constructive function producing a straight-line, given two points, it is doubtful
that straight-lines depend on points as parameters in order to be produced. Also, there
seems to be no canonical ways of introducing points in ancient geometry. Finally,
it is not clear that constructive functions are a fair way of rendering solutions for
problems.
The three beginning propositiones of Book I are naturally read as problems not
as theorems. Their proofs show how to fulfill what is being asked or commanded:
a construction. Actually, the proofs are divided into two main parts: kastaskeue and
apodeixis. The kastaskeue contains a recipe or solution for the construction problem.
The apodeixis contains a demonstration that this solution really produces what is
demanded. Once propositio I.1 – “... to construct an equilateral triangle on a given
finite straight-line”19 – has been resolved and the solution has been proven correct,
it can be used for obtaining a solution of propositio I.2 – “... to place a straight-line
equal to a given straight-line at a given point (as an extremity)”. Therefore, the
propositio that was formerly read as a problem is now used as part of a solution for
a new problem. Thus, the distinction between problems and solutions is superficial,
contrary to what some might believe. Besides, the proof of propositio I.1 contains
the resolution of two kinds of problems: a construction problem (dealt with in the
kastakeue) and the correctness-proof problem (dealt with in the apodeixis).
Other propositiones, like I.47 – i. e., Pythagoras’ – are theorems. Their proofs also
contain kataskeue and apodeixis. When proving I.47, similar kinds of problems are
to be solved again: first, a construction and, second, a property-proof problem. In fact
18 Actually a different problem can also be solved: (prove ¬(that the number πhas a rational
expression)). It is equivalent to: (deduce ⊥f rom that the number πhas a rational expression).
19 All quotations of euclidean geometry are taken from Fitzpatrick’s translation of Heiberg 1885.
8 Wagner de Campos Sanz
both are intertwined and, although proofs in modern reconstructions are completely
discursive, the original geometrical proofs are never exclusively discursive since they
involve [practical] problems.
The semantics of problems delineated above fits quite well the propositiones of
Book I. We think they also fit fair enough other books, but we don’t examine it here.
Now, it can be asked, how adequately does the semantics of problems explain the
relation between postulates and common notions to propositiones? Here the answer
depends on how to interpret the postulates and the common notions.
Common notions are generic in the sense that they apply to different objects:
lines, angles, triangles, etc. They can be interpreted as schematic inference rules
involving relations between the respective objects – for example, common notion
I.2: “if equal things are added to equal things then the wholes are equal”. “Things”
is a parametric word, and if common notions are to be interpreted as inference rules,
they are schematic inference rules.
Postulates, in turn, are the fundamental principles governing geometry. The first
three postulates of Book I are of practical nature since they are about actions: “... to
draw a straight-line from any point to any point”; “... to produce a finite straight-line
continuously in a straight-line”; “... to draw a circle with any center and radius”.
They can easily and fairly be read as problems. Problems of a special kind: either
these problems are considered immediately solved, not requiring that a solution be
presented, or they are just supposed to be positively solvable.
The fourth postulate in Book I – “that all right angles are equal to one another” –
looks like on the surface as a statement, not a practical principle. But it can be related
to a problem: that of knowing when two angles classified as right-angles are indeed
equal. Remember that right-angles are defined as the angles formed by the incidence
of two straight lines when all four angles era equal between themselves. How could
one then guarantee that two distinct incidences will have their eight right-angles all
equal?20 In a sense, the solution seems to be immediately postulated in I.4: all them
are to be taken as equal. But, of course, this is not true of any kind of surface. Only
surfaces having a certain homogeneity in curvature. Thus, it seems fair to say that
the fourth postulate imposes a restriction on which kind of surfaces that are to be
considered in Euclidean geometry: those presenting homogeneous curvature.
The fifth postulate is the problem of determining when two straight lines are
parallel. And the condition is that a straight line crossing both lines form internal
angles equal to two right angles. For homogeneous curved spaces the fifth postulate
imposes a further restriction requiring them to be flat, i.e, of zero curvature.21
Whatever the choice for interpreting postulates, in all propositiones the proofs
achieve a problem reduction: from more complex problem to less complex until
20 That the postulates hold for plane surfaces is a well known fact. An example of a surface where
two distinct perpendicular incidences determining four equal angles each have different angles inter
them is the surface of a cone, just consider the point in the vertex in relation to any other point.
21 The surface of a sphere is homogeneous and curve, some parallel lines meet each other on it.
It is a mind boggling fact that during more than two millennia many have attempted to prove the
parallel postulate from the other four. This is intuitively impossible considering the examples just
given.
Kolmogorov and The General Theory of Problems 9
the bottom ones, i.e., the postulated problems and the common notions. It does not
make sense to ask a correctness-proof for both postulates and common notions. It is
enough with supposing them as solvable problems to which all other propositiones-
problems will be reduced. Rephrasing Kolmogorov (1932; p. 151), but avoiding
any commitment with the existence of solutions, and remembering that problem
reduction involves a specific relation between solutions already pointed before:
If we can reduce the solution of problem bto the solution of problem aand the solution of
problem cto the solution of problem b, then the solution of ccan also be reduced to the
solution of a.
7 Problems and practical principles
The treatment of postulates as practical principles is not new. There is a modern
tradition culminating in Kant which also does so (Critique of Pure Reason,
A234/B287, our emphasis): 22
Now in mathematics a postulate is the practical proposition that contains nothing except the
synthesis through which we first give ourselves an object and generate its concept, e.g., to
describe a circle with a given line from a given point on a plane; and a proposition of this
sort cannot be proved, since the procedure that it demands is precisely that through which
we first generate the concept of such a figure.23
More than that, problemata were seen as practical propositions (Hechsel Logic,
p. 88):
. . . , and problemata, practical propositions which require a solution.
As also (Jäsche Logic, § 38, our emphasis):
A postulate is a practical, immediately certain proposition, or a principle that determines
a possible action, in the case of which it is presupposed that the way of executing it is
immediately certain. Problems (problemata) are demonstrable propositions that require a
directive, or ones that express an action, the manner of whose execution is not immediately
certain. [. . . ] Note 2: A problem involves (1.) the question, which contains what is to
be accomplished, (2.) the resolution, which contains the way in which what is to be
accomplished can be executed, and (3.) the demonstration that when I have proceeded
thus, what is required will occur.
8 Adequacy of Kolmogov’s problem semantics interpretation
The next task is to examine the second part of Kolmogorov’s paper and to averiguate
how the semantical elucidation advanced above clarifies the calculus of problems.
22 I’m grateful to A. Lassalle-Casanave who pointed me this passage and the next two.
23 Kant points to a very important fact from our perspective. On his interpretation, the same
procedure that is going to generate a circle under the postulate I.3 is also the procedure behind the
definition of what is a circle.
10 Wagner de Campos Sanz
Let’s consider Kolmogorov’s validation for Heyting’s axiom 2.12, i.e., ` (a⊃b) ⊃
((a∧c)⊃(b∧c)):
For example, in problem 2.12, assuming that the solution of bhas already been reduced to
the solution of a, one should reduce the solution of b∧cto that of a∧c. Let a solution of
a∧cbe given. This means that we are given both a solution of aand a solution of c. By the
hypothesis, we can derive a solution of bfrom that of a, and, since a solution of cis known,
we obtain solutions of both problems band cand hence a solution of problem b∧c.
The expression “the solution of”, we remember, has been dismissed in our
semantics. The above argumentation can be now formulated in very detailed steps
following the semantical clauses given above: (1) a∧ca∧cby (Id); (2)
a⊃b,a∧ca∧cfrom 1 by (Load); (3) a⊃b,a∧cafrom (2) by the
necessary condition of (∧r); (4) a⊃b,a∧ccfrom (2) by the necessary condition
(∧r); (5) a⊃ba⊃bby (Id); (6) a,a⊃bbfrom (5) by the necessary condition
of (⊃r); (7) a,a⊃b,a∧cbby (Load) from (6); (8) a⊃b,a∧cbfrom (3) and
(7) by (Drop); (9) a⊃b,a∧cb∧cfrom (4) and (8) by the sufficient condition
of (∧r); (10) a⊃b(a∧c) ⊃ (b∧c)from (9) by the sufficient condition of (⊃r);
and finally (11) (a⊃b) ⊃ ((a∧c) ⊃ (b∧c)) by the sufficient condition of (⊃r).
That is, axiom 2.12 is semantically valid.24
Kolmogorov’s proof makes sense only if we understand it as a semantical
validation of 2.12, otherwise it would not be an axiom.25
Lemma 1 All axioms in the calculus of problems are valid.
Proof Straightforward.
Next, Kolmogorov presents three rules for extending the set of solved problems.
All three rules are validated by the following semantical implications: I. p∧q⇒
p; II. (pand p⊃q) ⇒q; III. p⇒p[a,b,c,.. .
q,r,s,.. . ].26 All three are provable in
the semantics of problems. Then:
Theorem 1 Kolmogorov’s calculus of problems is sound with respect to the above
semantics of problems (Reduction Semantics).
24 This same proof could be given in a shorter version if we had used natural deduction rules with
suppositions: Suppose that a⊃bis positively solvable (i.e., suppose that bis reducible to a), one
should next reduce b∧cto a∧c. Suppose a∧cto be positively solvable. This means that both a
is positively solvable and cis positively solvable (by conjunction elimination). By the hypothesis,
bis also positively solvable (by implication elimination) and, since cis positively solvable, b∧c
is positively solvable (by conjunction introduction). Intuitionists tend to equate the supposition of
being positively solvable wit the supposition of having a proof. But this is misleading.
25 If we don’t, then 2.12 can be proved as a theorem of a natural deduction calculus as above. Natural
deduction rules are valid because of the meaning they have, meaning made explicit by the principles
and clauses in the semantics of problems. The supposition of a problem being positively solvable
is not to be identified with the supposition that I (or that someone) has the solution guaranteeing
the problem to be positively solvable. Although the second implies the first, the reverse is not the
case. From the supposition that I have a solution to the NP=P problem, it follows that I should win
a prize. But from the supposition of NP=P being positively solvable, it clearly does not follow that
I should win a prize.
26 p[a,b,c,. . .
q,r,s,. . . ]is the result of substituting a,b,c, .. . by q,r,s, ... inside p, respectively.
Kolmogorov and The General Theory of Problems 11
Proof By Lemma 1and the validity of the rules I–III.
Completeness is more laborious.
Double-line inferences by (Došen 1989) can be seen as an inferential explanation
of logical constants. Unfortunately, there are three cases in which it is not possible to
give such explanations: (∨r),(⊥r)and (⊃l). For them, the necessary and sufficient
condition contains an universal quantification over all sentences. But this goes beyond
the structural resources that Došen defended as enough for the formulation of double-
line inferences. In complement, (Došen 2016) defends the idea that for explaining
logical constants we must rely on consequence as our basic semantical relation. We
agree with the author, and we believe that Reduction Semantics corresponds to a
full development of this idea. Their meaning is explained in two distinct uses: in
repertoires and in the focus.27
It happens that each repertoire clause can be proven to be equivalent to the
respective focus clause.28 Hence all logical constants are sufficiently explained by
one of the clauses. Let’s say, those not containing quantification, i.e., those that
can be read as double-line inferences. This observation provides the essentials for a
completness proof.
Completeness is obtained by the following actions. First, define a sequent calculus
with rules for the semantic principles of reduction – (Id), (Load) and (Drop) – and
double-line inferences corresponding to the following clauses: (∨l),(∧r),(⊃r)and
(⊥r).29 Also, as a calculus, the repertoire has to be treated like a list, a multiset indeed.
Then, one extra rule is needed: contraction in the left side. Second, prove that each
of the semantical clauses and principles are valid metatheoretical properties of this
calculus.30 This is almost immediate. Third, prove that the calculus just described is
theoremhood-equivalent to Kolmogorov’s calculus of problems.31
27 Došen (2016; § 6.4) makes some conjectures about who would be the original author of double-
line inference rules, believing Gentzen to be the first. Each logical constant clause allows a
reading from left to right and vice-versa. Each of such readings corresponds to an inference rule
interpretation, or double-line reading, when the condition does not contain universal quantification.
(Došen 1989; p. 75) does not present intuitionistic double-line rules for implication, in the repertoire,
and for disjunction, in the focus. Those cases involve quantification over all formulas in the formal
language, as in our clauses above. Also, his examination starts with classical logic, and there we
find a double-line presentation conflating thinning and cut into one syntactical rule, which can be
horizontally rendered as: Γ`∆⇔Γ`∆∪ {a}and Γ∪ { a} ` ∆. The (Drop) principle in Reduction
Semantics cannot be presented like that. Usually, proof-theoretic logicians seem to favour a more
syntaxist approach to consequence, thus avoiding semantical clauses. We do the opposite.
28 For a proof of this property see Sanz (2019)
29 Not all clauses are needed in virtue of the fact that left and right clauses are interderivable by
using the semantic principles concerning reduction.
30 Just consider that (⊃ r)is the semantical version of deduction theorem.
31 More details of the completness proof are described in (Sanz 2019).
12 Wagner de Campos Sanz
9 Why problem semantics and intuitionistic logic?
Problem semantics was the conceptual way in which Kolmogorov interpreted
intuitionistic logic. We think that there are two reasons why Kolmogorov adopted
such approach.
The first is the fact that tertium non datur (`a∨ ¬a) is not a valid principle in the
calculus of problems and in intuitionistic logic. He interprets the sign “`”, differently
from Heyting, as meaning quantification (Kolmogorov 1932; p. 156):32
For a function p(a,b,c, .. .)of undefined problems a,b,c, . .. we simply write `
p(a,b,c, ...)instead of (a)(b)(c)...p(a,b,c, .. .). Hence, p(a,b,c, .. .)denotes the
problem “find a general method for solving the problem p(a,b,c, .. .)for each individual
choice of the problems a,b,c, ...”.
His interpretation is constructive in the measure it considers the problem of
proving logical formulas as requiring a general method. In our opinion, Kolmogorov
might be implying that [intuitionistic] logic is a part of a theory of problems. In
particular, `a∨ ¬awould be valid if (a) (a∨ ¬a)were valid, which means that we
should have a general method for solving problems of form a∨¬afor each individual
choice of a.33
The observation concerning the requirement of a method of solution brings us
to the second reason. It concerns the interpretation of existential propositions, when
the proof of existence does not exhibit the object:
Brouwer does not, however, intend to exclude existential propositions from mathematics
completely. He only explains that an existential proposition should not be stated without
presenting the corresponding construction. At the same time, according to Brouwer, an
existential proposition is not a mere indication of the fact that we have already found the
desired element of K. In this case the existential proposition would be false prior to the
invention of the construction and true after that. Thus, propositions of a completely new
type arise, which, although their content does not change in time, can nevertheless be stated
only under certain conditions.
The natural question which can arise is whether this specific type of proposition is a
mere fiction. Indeed, the problem “find an element of a set Kpossessing a property
A” is posed. This problem actually has a certain sense independent of the state of our
knowledge. If this problem has been solved, that is, if the corresponding element xis
found, we obtain the empirical proposition “our problem is now solved”. Thus, Brouwer’s
existential proposition is partitioned into two elements: an objective component (problem)
and a subjective component (its solution). (Kolmogorov 1932; p. 157)
Intuitionists do not accept [classical] negation of an universal to be a basis for the
inference of an existential judgement involving an infinite collection. According to
32 Our emphasis.
33 Kolmogorov (1932; p. 156): “formula [`a∨ ¬a] reads as follows: find a general method which
for any problem aallows one either to find its solution or to derive a contradiction from the existence
of such a solution.” Since a formalized language has to be previously presented before defining
its semantics, all basic problems in a calculus of problems would also be exhaustively enumerated
beforehand. The only hope of validating tertium non datur principle occurs when language is such
that we possess a general method for solving all basic problems enumerated on it. For a non-specific
language such a general method is impossible.
Kolmogorov and The General Theory of Problems 13
the conceptual background set by our author, the meaning of an existential problem
cannot depend on the possession of a construction exhibiting the required element
because a previous understanding of the problem is needed in order to even look for
that construction. Problems have an objectivity that solutions might lack.
10 Towards a general theory of problems
Kolmogorov’s point of view immersed intuitionistic logic into a general theory of
problems. He saw the schemes for the solutions of geometrical problems as a model
of how to approach logic in terms of problems and their solutions.
Veloso (1984; p. 29) sees in problem decomposition one of the two main strategies
for solving problems, together with reduction, i.e., transformation of problems. He
illustrates his perspective on decomposition with a quotation that we reproduce
here:34
A common approach to solving a problem is to partition the problem into smaller parts, find
the solutions for the parts, and then combine the solutions for the parts into a solution for
the whole.
As we see it, decomposition of problems is one of the principal subjects to be
approached in a general theory of problems. And, we believe, logical constants are the
natural patterns according to which problems are composed and decomposed. From
our perspective, it is Kolmogorov who deserves to be acknowledged for noticing
that.
Let’s illustrate decomposition by considering the whole proof of propositio I.1.
This proof has a construction solution part – to build the equilateral triangle over
straight line AB – which is solved first; and it has a correctness-proof solution part –
to prove that AB=BC=AC – which has to be done after the first part. Now, focusing
the construction part, it is also done in two steps. First, the two circles of radius
AB and BA, in any order. Second, and only after the first part is done, the straight
lines AC and BC, in any order. Drawing a circle and drawing a straight line are two
problems that we either suppose or assume35 to be solvable, according to postulates
I.1 and I.3. The usual conjunction (∧) is a natural way of composing problems when
the order is irrelevant: (draw circle of radius AB with center A) ∧(draw circle of
radius BA with center B). It is used again in the composition of: (draw straight line
AC) ∧(draw straight line BC). But, the next step requires the distinction of a before
and an after, which we are going to represent by “{”36, a before-after conjunction:
34 The original is in A.V. Aho et alli., The Design and Analysis of Computer Algorithms, Addison-
Wesley, Reading, 1975, p. 60.
35 The way we are understanding these two concepts makes them different. For more details see the
concluding section.
36 In programming languages this before-after conjunction is usually represented by the semicolon
“;”. But, inside a text, this symbol might produce confusion. And this is the reason why we didn’t
use it.
14 Wagner de Campos Sanz
{(draw circle of radius AB with center A) ∧(draw circle of radius BA with center
B)} {{(draw straight line AC) ∧(draw straight line BC)}.37 Finally, adding the
proof of correctness for the construction: {{(draw circle of radius AB with center
A) ∧(draw circle of radius BA with center B)} {{(draw straight line AC) ∧(draw
straight line BC)}} {(prove that AB=BC=AC). That is, the solution of a problem
can be obtained by decomposition and composition of problems already considered
solved.38
11 Problems and logical constants
The semantical clauses for the before-after conjunction are as follows, where l,m
and nare ordered indexes indicating the accomplishment of a task:
In the repertoire:
({l):Γ,c{dnefor any given problem p((plcand pmd)for l<
m⇒p,Γnesuch that m≤n)
In the focus:
({r):Γnc{dfor l<m≤n:(Γlcand Γmd)
Clearly, before-after conjunction is not among the usual intuitionistic logical
constants, neither among the classical. Also, this constant cannot be defined through
other usual logical constants. As a result, accepting Kolmogorov’s point of view that
intuitionistic logic is a calculus of problems, intuitionistic logic has to be viewed as
incomplete in terms of its expressivity.39
Doubts concerning the logicality of the before-after conjunction are understandable.
This doubt brings to the forefront another question. A decision of how to interpret
logic involves already a decision of what is a logical constant. And, apparently, there
are competing different initial decisions: (i) logic is the science of logical truths; (ii)
logic is the science of formal deductions (Došen 1989; p. 364)40; (iii) logic is part
of a general theory of problems or a general theory on problems resolution.41
37 It can be read as follow:{(draw circle of radius AB with center A) and (draw circle of radius BA
with center B)} then {(draw straight line AC) and (draw straight line BC)}
38 One item seems to be missing in the given solution: the action of determining the intersection
point C. We notice that the Euclidean text contains a reference to point C in both circles drawn:
circle BCD and circle ACE, thus apparently bypassing the issue.
39 In intuitionistic type theory, the before-after conjunction is expressed by means of function
composition. If f(x)and g(x)are two constructive functions, then f og(x)is the ordered application
of gto xand fto the result of that. But, it is doubtful that to draw a straight-line should depend on
parameters. The act of drawing a straight-line creates so to say the starting and the stopping points.
40 This corresponds to the oldest and more traditional conception of logic, but it is not at all the
most accepted way of conceiving the subject matter nowadays among mathematicians and computer
scientists, for example.
41 We gave a hint above of how this theory applies to ancient geometry.
Kolmogorov and The General Theory of Problems 15
12 Sketches of a first-order problem semantics
Although Kolmogorov didn’t examine first-order quantificational problems, we can
advance one small step on this subject.
The semantical clauses presented above have in their definiens some metalogical
constants. For their correct understanding, we must read those constants in accordance
with the constructivist explanation. Quantification over sentences were used in (⊃l),
(∨r)and (⊥r). This quantification is predicative since the set where the elements are
taken has been defined previously: the formal language.
The clauses for first order logic are the following, where xis an object language
individual variable, iis an object language individual parameter and ta metalanguage
variable for language terms:42
Repertoire:
((x)l):Γ,(x)c[i
x]efor any given problem p((for any term t,pc[i
t]) ⇒
Γ,pe)
((∃x)l):Γ,(∃x)c[i
x]efor any given term t(Γ,c[i
t]e)
Focus:
((x)r):Γ(x)c[i
x]for any given term t,Γc[i
t]
((∃x)l):Γ(∃ x)c[i
x]for any given problem e((for any term t,Γ,c[i
t]e) ⇒
Γe)
After stating them, the problem of how to understand open formulas presents
itself. Closed formulas represent problems, this is clear. Open formulas should
represent problem forms where certain specific expressions may vary. The question
is: they represent what? Natural candidates for being object of quantification in
first order logic are the initial parameters of an action – lines, points, circles, etc.,
thinking in terms of geometry. Nonetheless, as lines, circles, etc., may very well
be seen as the result of an action, actions might also be considered as elements of
quantification. Whatever we decide, the objects must also include propositions, at
least in the measure we want to deal with proving problems.
13 Conclusion
The above semantical considerations offer a perspective from which we can say a
few words about proof-theoretic semantics. Schroeder-Heister (2016; § 4) obser ves
that:
Proof-theoretic semantics has been occupied almost exclusively with logical reasoning, and,
in particular, with the meaning of logical constants. Even though the way we can acquire
knowledge logically is extremely interesting, this is not and should not form the central
42 c[i
x]represents the result of substituting xin place of every occurrence of iin c, etc.
16 Wagner de Campos Sanz
pre-occupation of proof-theoretic semantics. The methods used in proof-theoretic semantics
extend beyond logic, often so that their application in logic is nothing but a special case of
these more general methods.
We could hardly be more eloquent promoting the idea that logic should not be
the central pre-occupation of proof-theoretic semantics. Indeed, the examination
of ancient geometry and its proofs when looked under the light of a semantics of
problems not only reinstate a traditional way of considering knowledge as also inserts
logic into a wider scope, that of problem resolution.
Schroeder-Heister (2016) drives attention to definitional systems as an example of
the wider scope he has in mind when mentioning extensions beyond logic. Another
way of widening the picture consists in viewing logic as part of a general theory of
problems. It is Kolmogorov’s merit to have realized that in the XXth century.
In the history of mathematics, problems were by large the central theoretical
concern, being later displaced by the assertion-theoremhood perspective. However,
a great deal of contemporary science, from Medicine to Social Sciences, has always
been directed at problems resolution. Our knowledge seems to be mainly of that.
Instead of maintaining a sharp division between theoretical knowledge and
practical knowledge, other attitude might prove to be more profitable as the example
of ancient geometry above illustrates. The benefits for logic are considerable. If logic
is seen just as part of a general theory of problems, teaching logic would be a kind of
propaedeutical step into the broadest investigation on problems and solutions. From
this perspective, the study of algorithms and their complexity is an adjacent area of
concern in a general theory of problems.43
From a closer perspective, the above developments present some implications for
two items that Schroeder-Heister (2016) considered: the nature of hypotheses and
assumptions, in one side, and the old question concerning harmony among inference
rules. We start with second.
The above semantics of problems roughly shares the same perspective of double-
line inference rules by Došen (1980). For double-line rules, it is immediate that
harmony holds as, for example, in: Γc∧d⇐⇒ Γcand Γe. Here
the logical constant occurs in the focus. Direction ⇐=always corresponds to a
sufficient condition reading (in natural deduction, an introduction rule); direction
=⇒always corresponds to a necessary condition reading (in natural deduction, an
elimination rule). However, logical constants can also be inserted in repertoires, like
in: Γ,c∧de⇐⇒ Γ,c,de. Nonetheless, the sufficient condition direction now
corresponds to a natural deduction elimination rule applied on top of hypotheses,
and the necessary condition direction to a natural deduction introduction rule on
top of hypotheses. A reasonable definition for harmony is then: when necessary and
sufficient conditions coincide.
Dosen’s approach to double-line rules is reductive in the sense that he claims
logical constants to be characterizable through structural resources. It occurs that all
double-line rules correspond to a semantical clause, but not all semantical clauses
can be stated by means of simples structural double-line rules. Five cases cannot
43 For a discussion concerning the relation between logic and problems see Cellucci (2013).
Kolmogorov and The General Theory of Problems 17
be rendered by simple structural resources: disjunction in the focus, implication
in the repertoire, absurd in the focus, existential in the focus and universal in the
repertoire.44
Disjunction in the focus is a particularly interesting case since it implies that
the traditional natural deduction introduction rule is not in harmony with the
traditional natural deduction elimination rule. It expresses, nonetheless, a correct
sufficient condition, althought not one that coincide with the necessary condition
expressed by the elimination rule. However, if disjunction natural deduction rules
lack local harmony, they have another kind of harmonious relation. Completeness
for intuitionistic propositional logic with respect to Reduction Semantics guarantees
their correct complementarity.
From our perspective, the above concept of harmony does not necessarily imply
the normalization properties for natural deduction calculae. In order to perform
normalization one action is essential: that of plugging derivations. This basic
semantical property is expressed by the following transitivity rule: (Γcand
∆,cd) ⇒ Γ,∆d. This principle is normally derivable from: Drop and Load.
For a natural deduction calculus, a failure in normalization might occur even with
an harmonious constant. For example, if the given rules violate Load principle.45
Concerning assumptions and hypotheses, a distinguished subject in Schroeder-
Heister’s paper, we propose a distinction. We propose to reserve the concept of
hypothesis for the objects under supposition, primarily to those occurrences in
repertoires like cin cd. The concept assumption is reserved for those cases where
we have cbeing supposed as in the metaimplication c⇒d. Assumptions
would then be hypotheses of a specific nature: those where we suppose the validity
or the possession of a proof/solution for a proposition/problem. This is equivalent
to suppose the assertion of the proposition when assertibility is understood as the
possession of a proof, like intuitionists do. Supposing is by itself an act and it is
entirely different from asserting. When talking about problems these two concepts
were used only in a unessential way.
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