=Paper=
{{Paper
|id=Vol-1283/paper2
|storemode=property
|title=Old and New Riddles on Concept Sharing
|pdfUrl=https://ceur-ws.org/Vol-1283/paper_3.pdf
|volume=Vol-1283
|dblpUrl=https://dblp.org/rec/conf/ecsi/BottazziMP14
}}
==Old and New Riddles on Concept Sharing==
https://ceur-ws.org/Vol-1283/paper_3.pdf
Old and New Riddles on Concept Sharing
Emanuele Bottazzi, Claudio Masolo, and Daniele Porello
Laboratory for Applied Ontology, ISTC-CNR, Trento, Italy
Abstract. We ask whether social interaction demands sharing social concepts.
We illustrate our point by depicting possible situations that emerge when two in-
dividuals play chess. We formalize our hypothesis in First Order Logic and we
show that the very idea of sharing social concepts poses an interesting challenge
both from the standpoint of knowledge representation and of philosophical con-
ceptual analysis. By endorsing a minimal notion of interaction, we conclude that
sharing social concepts is not necessary for social interaction. Then, we relate our
view to Wittgenstein’s and Kripke’s “Rule-following Considerations”.
1 Introduction
Many authors views social concepts as depending on human beliefs and intentions
[10,12] and, on this basis, some have argued that we have a certain form of “epistemic
privilege” with regard to them, protecting us from certain possibilities of ignorance and
error. We could be mistaken about the real nature of gold, as we were for centuries, but
we could not be mistaken about the nature of marriages, parties or chess pieces, since
they are human products, defined by us via collectively accepted rules [11]. Social con-
cepts are strictly tied to rules [3], that is, rules “create the very possibility” ([10]: 27) for
social, or more specifically, institutional activities—such as marriages, parties or chess
games—to take place. This view promotes a tight connection between social interac-
tion, concept sharing and rule following. For instance, in order to play chess, we need
to grasp the concepts involved in a chess game (moves, pawns, capture, etc), that is,
we need to accept the rules that define chess concepts. In this paper, we want to take a
closer look at such an entanglement. In particular, we want to understand what does it
mean to share social concepts and to what extent sharing is the grounding of interaction.
We illustrate our questions by means of the following scenarios. Two persons start
playing chess. Black moves after White and so on. They both know the rules of chess
and this is reflected by the fact that each agent acknowledges the move of the other as
a correct move. That means that the moves are done according to the rules of chess.
However, such an example represents only a very specific type of social interaction.
Consider now two persons that plays chess possibly for the first time. They are not sure
about the rules, for instance they may have a partial grasping of the rules that define
the pawns, they may make mistakes, they may change their minds during the game.
For example, White moves the rook as a queen, Black acknowledges this move because
she is not certain of what the rules are, and they may keep on playing by adjusting
their beliefs concerning the rules. We may question whether the two persons are still
playing chess, however it seems hard to question that they are interacting. In this case,
players may share very little of the rules of chess, hence they may share very little of
2 Bottazzi et al.
the concepts involved in the chess game. We believe that this situation covers a broader
spectrum of possible social interactions then any by the book chess game.
The question we want to approach is: What is it that two persons share whenever they
interact? We will assume that social concepts are at least partially defined by rules, that
is, the concept of a rook is at least partially defined by the admissible moves that a rook
can make. In order to simplify our analysis, we will not discuss issues related to com-
munication between interactants. Moreover, we concede that individuals have the same
access to the physical or non-social reality, in order to exclude forms of disagreement
that are not relevant to our point. For instance, players can recognize unambiguously
the positions and the shapes of the different pieces of wood, they know the size of the
board, and they understand that those pieces of wood stand for chess-pieces. We will
discuss what does it mean to share social concepts by representing the players’ views
of the chess game in First Order Logic and by formalizing a number of hypothesis on
social concepts. We will see that unpacking the very notion of sharing social concepts
is quite involving and that a number of delicate ontological choices have to be made.
We shall conclude that in many chess-like interactions, it appears that the two players
may endorse very different chess concepts. That is, sharing social concepts appears not
to be necessary for interaction.
We are assuming a minimal definition of interaction that is at least capable of ac-
counting for cases such as our weird game of chess. This minimal form of interaction is
intuitively a form of commitment to respond to the other’s actions. For instance, White
is interacting with Black because, whenever White moves, Black is committed to re-
spond. Even in case White’s move violates the rules that Black believes are holding,
Black can in principle still respond by revising her rules. This notion of interaction is
related to the notions of commitment and entitlement proposed by Robert Brandom [2].
The remainder of this paper is organized as follows. In Section 2, we present our for-
mal treatment and a number of examples of chess interactions. Section 3 draws some
connections between our points and Kripke’s interpretation of Wittgenstein’s ‘Rule-
following Considerations’ in his Philosophical Investigations.
2 The illustrative example of chess formally illustrated
We start from the descriptions given by the two players of what happened during the
game, i.e., from two subjective reports of the course of the game. We represent these
reports by means of two FOL-theories—ΠW for White and ΠB for Black—describing
how (and according to which rules) the chess moved and the way these moves are
grounded on the physical-level, i.e., the level of bits of wood, stones, etc. as opposed to
the game-level, i.e., the levels of rooks, bishops, etc. The physical-level is assumed to be
shared by the players, i.e., while the players can have different viewpoints on the game,
they agree on what happened in the physical world. This is a simplification hypothesis,
one could think that also the physical-level is prone to disagreement.
The shared physical-level includes times (TM), physical objects (OB), e.g., bits of
wood, stones, etc., positions on the board (PS), an existence predicate defined on both
physical objects and positions—E(x, t) stands for “x exists at time t”, and a location
predicate—L(x, p, t) stands for “at time t, the physical object x is located in position
Concept Sharing 3
a3 b3 c3
a2 b2 c2
a1 b1 c1
t0 t1 t2
Fig. 1. Position labeling and three snapshots of the physical world.
p", where L, as all the predicates with a temporal argument, satisfies (a1) (where R is
a generic relation that needs to be substituted by L in this case) and is functional, i.e.,
at a given time, the position of an object is unique (a2). The physicality of objects is
further characterized by (a3), i.e., existent objects are always located. Furthermore, the
‘board’ exists during the whole game without changing, i.e., the positions are eternal
(a4). For our aims an extremely simple model of the time of the game, the time of the
moves, is enough: time is discrete, finite, linear, and left-bounded. We indicate by t+1
the (immediate) successor of time t and by ti+1 the (immediate) successor of ti .
a1 R(x1 ,. . ., xn , t) → E(x1 , t) ∧ · · · ∧ E(xn , t)
a2 L(x, p, t) ∧ L(x, p0 , t) → p = p0
a3 OB (x) ∧ E(x, t) → ∃p(L(x, p, t))
a4 PS(x) → E(x, t)
The set of sentences that describe the existence and the position of the physical objects
during the game—the report about what happened in the physical world—is included in
both ΠW and ΠB . Focusing on a 3 × 3 fragment of a chessboard and labeling the posi-
tions as in Figure 1, the three snapshots of the physical world in Figure 1 are represented
by (f1), where and are constants for physical-objects: OB( ) and OB( ).
f1 L( ,a3,t0 ) ∧ L( ,b1,t0 ) ∧ L( ,a3,t1 ) ∧ L( ,b2,t1 ) ∧ L( ,a2,t2 ) ∧ L( ,b2,t2 )
Another shared aspect regards who is moving. We do not explicitly represent neither
the moves nor the player that is responsible for the transition between two configura-
tions of the world. We just assume that White starts the game, she is responsible for
the transaction from t0 to t1 , and that the players correctly alternate. For instance, in
the example in Figure 1, starting from an initial configuration at t0 , White does the first
move, she moves in b2, while Black does the second move, she moves in a2.
The game-level includes chess pieces (CH), an extension of the existence predicate
to chess-pieces, and a manifestation relation—M(x, c, t) stands for “at time t, the chess
piece c is manifested, is realized, by the physical object x”—that satisfies (a1), (a5), i.e.,
M is complete manifestation, (a6), i.e., chess-pieces always have a manifestation, and,
therefore, a location (through the location of its manifestation, see (d1)), and (a7), i.e.,
our model focuses only on the physical world involved in the game, objects are always
manifestations of chess-pieces (not necessarily of the same piece). The object x is the
physical substratum of the chess-piece c that has a more ‘abstract’ level of existence: c
can survive a change of its substratum, i.e., its re-identification through time is not based
solely on its physical manifestation. The idea is that the destruction or the damage of
4 Bottazzi et al.
the manifestation of a chess-piece does not necessarily compromise the continuation of
game, a replacement could be accepted by the players (see Example 2).
a5 M(x, c, t) ∧ M(x0 , c, t) → x = x0
a6 CH (c) ∧ E(c, t) → ∃x(M(x, c, t))
a7 OB (x) ∧ E(x, t) → ∃c(M(x, c, t))
d1 L(c, p, t) , ∃x(L(x, p, t) ∧ M(x, c, t))
On the one hand, the manifestation relation could be seen as a sort of existential depen-
dence or, more specifically, a material constitution. On the other hand, one could think
to a sort of denotation or reference, i.e., one could understand a chess-piece as a name, a
definite description, an individual concept (a ‘variable individual concept’ [1]), or, more
formally, as a (partial) function from times (possible worlds) to physical objects (indi-
viduals). In both cases, chess-pieces, e.g. “the white queen”, are distinct from chess-
types, e.g., “queen”: the white queen is a queen. We represent the pieces as individuals
(not as predicates subsumed by chess-types) for two reasons: (i) to be neutral with re-
spect to the two different views on pieces and (ii) to simplify the model: by representing
pieces as predicates, to understand whether or what piece moved one needs to explicitly
provide a re-identification criterion that can be leaved implicit in our proposal.1 Chess-
types are represented by means of predicates like Queen: e.g., Queen(wq)∧Queen(bq),
where wq and bq are individual constants naming chess-pieces, stands for “the white
queen wq and the black queen bq are both queens”.
As said, the theories of players ΠW and ΠB agree on the physical-level, i.e., they
share both the terminological and factual (assertional) knowledge about the physical
world, i.e., physical objects exist and are located objectively. The subjective percep-
tual capabilities of the players do not play any role at this level. Furthermore, ΠW
and ΠB both adopt the previous axioms about M, the players agree on the mean-
ing of M. However their factual knowledge about what (types of) pieces exist and
what are their manifestations can differ, i.e., the game-level is subjective. For instance,
players could disagree on the ‘interpretation’ of in the first snapshot in Figure 1,
i.e., M( , bq, t0 ) ∈ ΠW whereas M( , bk, t0 ) ∈ ΠB , where bq stands for ‘the black
queen’ and bk for ‘the black king’. In the following we provide some arguments for the
view that White and Black can play chess without ‘sharing’ any chess-entity. For this
reason we prefer to assume individual constants for chess-pieces to be local for each
report, e.g., bkW represents “the black king" according to White while bkB represents
“the black king" according to Black. Note that this is a formal move, it does not imply
an ontological commitment on the privateness of chess-pieces, the individual constants
must be seen here just as ‘names’.2
The previous framework allows to represent the time-sequences of configurations of
the objective physical-world and of the subjective game-worlds of the two players. Still,
1
Conceiving pieces as individual concepts is a classical reification move, see for instance [9].
2
The ontological dichotomy between a private, mental, or psychological nature vs. an abstract or
social one concerns the theories of concepts in general. Note that privateness “doesn’t preclude
the sharing of a mental representation, since two people can have the same type of mental
representation (...) When someone says that two people have the same concept, there is no
need to suppose that she is saying that they both possess the same token concept.” ([7], p.7)
Concept Sharing 5
the rules used by the players to accept these configurations as ‘valid’ (that allow the
players to continue the game) are not in the model. Rules are often seen as what enable
us to interpret what is going on in the brute (physical) reality as a game (institution).3
Different kinds of rules can be distinguished, we will limit ourselves to rules that define
the way chess-pieces move. In particular, we represent rules as check constraints that
apply to chess-types, i.e., all the pieces of the same type are submitted to the same
contratins, they move in the same way.4 Chess-types can be used to formalize rules by
means of sets of constraints on successive configurations of the game-world, i.e., for
each chess-type T there is a set of necessary conditions with form (f2). The meaning
of (f2) is: if at t a piece c of type T is located in the position p1 , in the presence of one
piece c0 in location p2 , then, at t+1, c cannot be located in (cannot move to) the position
p3 (p1 , p2 , and p3 are individual constants).5
f2 T (c) → (L(c, p1 , t) ∧ ∃c0 (L(c0 , p2 , t)) → ¬L(c, p3 , t+1))
We assume a sort of ‘rationality’ of players. Players stop to play if they are not able
to ‘solve a rule contradiction’ manifested in a physical move. This does not mean that
players cannot be ‘flexible’. This simply means that when a player detects a contradic-
tion, either she stops to play or she ‘accepts’ this contradiction by revising her rules or
her factual game-knowledge to accommodate the contradicting move. This is motivated
by our notion of interaction that is limited to the acceptance of moves, the others’ re-
sponses must be taken into account to continue to interact. Flexibility could then mean
(i) a revision of rules, i.e., in our framework, the substitution of a set of (f2)-constraints
for T with a non logically equivalent new set for a different type T 0 ;6 or (ii) a revision
of the factual game-knowledge, e.g., the ‘death’ of some pieces and the ‘birth’ of new
pieces, or a change in the manifestation of a given piece. One has then the problem of
understanding if a chess-piece can survive a revision of rules, i.e., if migrations from T
to its revision T 0 are allowed. Because chess-types are non temporally qualified, chess-
types are implicitly essential for pieces, i.e., the identity (and re-identification) of pieces
is based on how they move. To analyze type-migration, a temporal qualification of types
is then necessary: T (x, t) ∧ T 0 (y, t) could or could not be constrained to imply x=y.
The possibility of type-migration requires to modify (f2) as in (f3).
f3 T (c, t) ∧ T (c, t+1) → (L(c, p1 , t) ∧ ∃c0 (L(c0 , p2 , t)) → ¬L(c, p3 , t+1))
The fact that chess-pieces can survive a migration from T to T 0 highlights a con-
ceptual change of the player, e.g., the player changes her mind regarding what queens
are, she revises the rules for queens, i.e., intuitively, both T and T 0 individuate rules for
queens. A conceptual change that shifts T to T 0 from t to t+1 impacts all the instances
3
On that the literature is vast, starting from [13]. Often such rules are called constitutive [10].
4
This does not hold for captures. For instance, a black rook in a1 can capture a white queen in
b1 (and then move to b1). Vice versa a white rook in a1 cannot move to a position occupied
by a white queen. Because of the illustrative role of our model we do not consider captures.
5
The condition ∃c0 (L(c0 , p2 , t)) could be ‘empty’ (it is enough to instantiate it as a redundancy
∃c0 (L(c0 , p1 , t))), i.e., there could be positions that the T s cannot reach (in one move) inde-
pendently of the positions of the other pieces.
6
Alternatively one could relativize (f2) to times as (where t1 , . . . , tm are individual constants)
T(c) ∧ (t = t1 ∨ ... ∨ t = tm ) → (L(c, p1 , t) ∧ ∃c0 (L(c0 , p2 , t)) → ¬L(c, p, t+1))
6 Bottazzi et al.
of the concepts involved, all the instance, at t, of T migrate, at t0 , to T 0 . Vice versa,
chess-pieces cannot survive a misinterpretation—e.g., the player confuses a T 0 -piece
with a T -piece. To account for the distinction between conceptual change and misinter-
pretation, we group chess-types concepts into syntactically distinguishable classes of
concepts. This is done by grouping the predicates of our language. Let C be a set of
indexes and c ∈ C, we assume that the set of chess-types T1c , . . . , Tm
c
lists the possible
c
conceptual changes that a chess piece of type Ti may endure. A conceptual change can
then been represented as a re-classification, e.g., Queeni (wq, t) ∧ Queenj (wq, t+1).7
By contrast, the misinterpretation ‘destroys’ the previously existing piece and creates a
new one, i.e., no migrations from Tic1 to Tjc2 (with c1 6=c2 ) are possible.
Our view is formalized by means of the axioms (a8)-(a12).8 At any time, a chess-
piece is conceptualized in some way (a8) and this conceptualization is unique (a9),
(a10). Chess-pieces cannot migrate from a class of concepts to a different one (a10),
they cannot survive a change of class, e.g., Queeni (c, t) ∧ Rookj (c, t+1) is inconsistent.
We illustrate (a11) and (a12) by means of an example. Whenever a player realizes that
the rules for queen have to be changed from the ones that characterize Queeni to the
ones that characterize Queenj , she applies the new view to every queen (a11). Moreover
(a12) assures that a substitution of Queeni with Queenj at t+1 implies that Queenj has
no instances at t and that Queeni has no instances at t+1.
a8 CH(x) ∧ E(x, t) → i,c Tic (x, t)
W
a9 Tic (x, t) → j6=i ¬Tjc (x, t)
V
a10 Tic1 (x, t) → ∀t0 ( j,c2 6=c1 ¬Tjc2 (x, t0 ))
V
a11 Tic (x, t) ∧ Tic (y, t) ∧ Tjc (x, t+1) ∧ E(y, t+1) → Tjc (y, t+1)
a12 Tic (x, t) ∧ Tjc (x, t+1) → ¬∃y(Tic (y, t+1)) ∧ ¬∃y(Tjc (y, t)) (for i 6= j)
Note that we approached the distinction between conceptual changes and misinter-
pretation by syntactically fixing what migrations in type a piece can survive.9 We did
not provide a complete characterization of the range of possible changes of a chess-
piece, because it seems to require modal reasoning and it would commit us to define
what are the essential properties of a chess-piece. The only constraint we put is of a
semantical nature: we exclude that an agent who knows that queens and rooks are dif-
ferent things may consistently believe that something that is a queen follows the rule of
a rook. That is the meaning of our constraints on classes of chess-types.
As in the case of individual constants, to be as general as possible, we assume that
all the types are local to players. We note TiW (TiB ) the ones in the report ΠW (ΠB ).
In addition, we assume that all the predicates in the same class have non-equivalent
necessary conditions. This constraint does not hold for predicates in different classes.
For instance, at a given time White can adopt as rules for queens the rules he adopted
for rooks in the past.
7
To simplify the notation we write Queeni instead of Tqueen
i .
8
Note that our definition does not exclude that two different chess-pieces are associated with
the same set of predicates: for instance, we can assume that every rook is associated with the
same list of possible types.
9
This allows to group predicates a posteriori by looking at the migrations in a player’s report.
Concept Sharing 7
2.1 Examples
We discuss a number of simple examples of a somehow queer chess game, where play-
ers may change their mind about chess-pieces and about the rules that regulate chess-
types. These examples allow us to clarify what does it means to ‘share’ the game level.
Example 1. (Rook or Queen) Consider the three snapshots in Figure 1 represented by
(f1), i.e., first, White moves the to b2, then Black moves in a2. Because the factual
physical knowledge is shared among the players (f1) ∈ ΠW and (f1) ∈ ΠB .
This sequence of moves is compatible with a number of scenarios in each player’s
perspective on the game-level. Firstly, we may assume a standard plain case: players co-
ordinate and keep on playing because they share the same rules and they agree on what
chess pieces exist and on their manifestations. For instance, there are no revisions, we
have RookW (brW ,[t0 ,t2 ]), RookB (brB ,[t0 ,t2 ]), M( , brW , [t0 ,t2 ]), M( , brB , [t0 ,t2 ]),10
and the necessary conditions of RookW and RookB are equivalent (they are the standard
one for rooks).11 In this case we say that the players agree on interpreting as rook.
Similar for , the players agree on interpreting as queen.
However, that is not the only possible scenario that allows the game to go on. For
instance, suppose that, from t0 to t2 , White is interpreting as rook instead of queen
while Black is still interpreting as queen. Despite the disalignement, the two players
still go on playing because the discrepancies in what pieces exist have not manifested
in this phase of the game and could never manifest during a whole game, i.e., their
conceptual alignment is indeterminate.
Example 2. (A stone for a bit of wood) Example 1 shows that the agreement on the in-
terpretation of the physical level (and on the rules) is not necessary for keep on playing
the game. We consider now an example where there are new physical entities that appear
during the game: Black moves in b3, then White substitutes (originally in b1) with
putting it c2 (see Figure 2). Suppose White is interpreting both and as queen,
she changes the manifestation of the queen by using a stone, i.e., M( , wqW , [t0 , t1 ]),
M( , wqW , t2 ), and QueenW (wqW , [t0 , t2 ]). At this point Black may wonder what have
happened and may stop playing as she cannot understand which piece is the mani-
festation of (see (a7)). Or Black can assume a substitution, for instance agreeing with
White on interpreting both and as queen.12 Clearly, other scenarios are possible.
If it is true that rules create the very possibility of playing, as Searle states, it is also
true that there is a non trivial sense according to which they are not enough in order
to have the game. Whether or not is established by the rules that we can do something
like changing a manifestation for another, it seems that we can anyway do the change
without affecting the game-level. One could still follow the other rules that regulate
10
We write R(x1 , . . . , xn , [ti , ti+n ]) instead of R(x1 , . . . , xn , ti ) ∧ · · · ∧ R(x1 , . . . , xn , ti+n ).
11
Note that the necessary conditions are expressed only in term of the location L, a shared prim-
itive. The equivalence of these conditions is then purely objective. The proof of the ‘equiva-
lence’ of necessary conditions that involve private predicates is much more complex because
it requires a link between all the involved private predicates.
12
In the case of blindfold chess we could to say that a piece manifests itself through words. This
is debated in the literature, for a discussion see [4].
8 Bottazzi et al.
t0 t1 t2
Fig. 2. Substituting a piece of wood with a stone.
t0 t1 t2
Fig. 3. Other three snapshots of the physical world.
the chess moves without any problem, even by maneuvering a stone instead of a bit
of wood. This seems to reflect what one could call the twofold nature of institutional
reality, in the sense that it seems to depend on the one hand on what is accepted by those
who are dealing with it, but on the other also on what it is actually going on according
to the rules, a reality emerging from both rules and practices.
Example 3. (Rule Changing) Example 1 is a trivial source of possible disagreement, as
the standard moves of a queen include the moves of a rook. Consider now the scenario
where White moves in c1, then Black moves in b2 (see Figure 3).
Again different game-reports are compatible with this physical sequence. Suppose
that, at t0 , both the players agree on interpreting as rook. At t2 , Black moves by
violating the standard rook-rules. Remember that we are assuming that the players are
consistent with their rules and facts. This could mean that Black revised the rook-rules,
i.e., brB persists but RookB (brB , t0 ) and RookBi (brB , t1 ).
Different scenarios are possible. White can refuse to keep on playing, because she
maintains the standard rook-rule also at t1 , i.e., RookW (brW , [t0 , t1 ]) and this fact to-
gether with the physical-facts are inconsistent with M( , brW , [t0 ,t2 ]). In this case the
game ends at t2 because White refuses the Black’s move. Alternatively, White may ac-
cept the Black’s move by revising her rook-rules to be consistent with this move. These
new rules are compatible, but not necessarily identical, with Black’s rules. The physical
traces are not enough to completely determine the rules in ΠB and ΠW . One could also
think that White changed her opinion about the existence of pieces: M( , brW , t0 ) ∧
M( , bkW , [t1 , t2 ])—instead of M( , brW , [t0 , t2 ])—and KingW (bkW , [t1 , t2 ]).
This example shows that rules and pieces may keep on changing from time to time,
as in the case of players learning how to play chess. Moreover, the change tells us that
an actual game depends on moves and their acceptance, and this could be seen also as
undermining the idea that rules are necessarily stable, they are given once and for all, as
if, once the rule is in place, “all the steps are already taken”, as Wittgenstein says ([14]:
§219). Players may adapt, they may be open to change their rules to achieve a goal. It
seems then that there should be an “agreement not only in definitions but also (...) in
judgments” ([14]: §243). The actions and reactions of players are not simply important,
they are, so to speak, the only thing that somebody has, for understanding what is going
Concept Sharing 9
on when interacting with others. This is connected with the wittgensteinian idea of
meaning as use: “the meaning of a word is its use in the language” ([14]: §43). At this
point, one could say that the two players agree on the chess-pieces if they use them in
the same way. But the previous examples show that we cannot be sure that we are using
them in the same way by accessing only a limited sequence of configurations of the
physical world. Wittgenstein says, “this was our paradox: no course of action could be
determined by a rule, because every course of action can be made out to accord with
the rule.” ([14]: §201). Different rules can be followed to achieve the same moves.
The acceptance of changes of rules during the game is debatable. According to many
philosophers working on constitutive rules, if Black moves her rook diagonally, she is
maybe playing some game, but not chess. Nonetheless, in our daily life, it becomes
difficult to assess the exact boundaries of our institutional activities: changes and irreg-
ularities are quite common13 and, as seen, to continue to play, White may accept the
move of Black, revising her rules. Wittgenstein seems to be in this line of thought too:
We can easily imagine people amusing themselves in a field by playing with
a ball like this: starting various existing games, but playing several without
finishing them, and in between throwing the ball aimlessly into the air, chasing
one another with the ball, throwing it at one another for a joke, and so on. And
now someone says: The whole time they are playing a ball-game and therefore
are following definite rules at every throw. And is there not also the case where
we play, and make up the rules as we go along? And even where we alter them
— as we go along. ([14]: §83)
3 A potential illusion before the rules
Kripke in [5] interprets the aforementioned passage ([14]: §201) as the starting point to
pose a paradox that undermines the very possibility of rule following. If we consider
an individual in isolation the paradox in unsolvable, if we instead take into account
agreement amongst members of a community, only a ‘sceptical’ solution is possible.
Kripke’s example focuses on rules governing the word ‘plus’. The symbol ‘+’ de-
notes the mathematical function of addition. Suppose that I have to make a computation,
take ‘68+57’, I never performed before. In addition, suppose that my past computations
never involved numbers greater than 56. I answer ‘125’, but a very bizarre sceptic asks
me how I did it. The sceptic asks me to provide some proof that I did not change the
interpretation of ‘plus’ and ‘+’, i.e., that in the past I did use ‘plus’ and ‘+’ to denote
the addition and not, for example, to denote a function called ‘quus’ (symbolized by
‘⊕’) defined by: x⊕y = x+y, if x, y < 57; x⊕y = 5, otherwise ([5]: 9). Clearly,
following the rules for ‘plus’ is different from following the rules for ‘quus’, but my
past answers (my ‘external behaviors’) are compatible with the possibility I was not
adding but instead quadding. But the threat of the sceptic is even deeper, since it goes
in arguing that even when mental facts are accepted as proof still the problem exists.
Kripke argues that my mental states, my past intentions, my dispositions in my calcu-
lations, even an omniscient God that knows which continuation I was thinking of, do
13
Ethnomethodological studies on board games seem to go also in this direction, see e.g. [8].
10 Bottazzi et al.
not help in founding a proof. If no adequate explanation is forthcoming—i.e., a rule is
always compatible with different interpretations that lead to different results—then we
must give up the idea that in the past I was following the rules for addition instead of
the rules for quaddition. So there is no such thing as rule-following, no agreement or
disagreement in accord with the rule. This goes against the inexorability of rules we
mentioned in Example 3 and after Kripke has been called ‘meaning determinism’ [6],
the idea that when one possesses a concept, “all future applications of it are determined
(in the sense of being uniquely justified by the concept grasped)” ([5]: 107).14
In our framework, chess-pieces seem to play a role similar to symbols. Symbols
may have multiple physical realizations: for instance the same word can be written
or read aloud. Similarly, chess-pieces may have multiple manifestations, i.e., they ab-
stract from the physical level to enter the game level.15 In the Kripke example, symbols
stand for functions defined by rules. Our chess-pieces are classified by chess-types that
are given, but not completely defined, in terms of rules. Thus, we can simulate the
change in the interpretation of a symbol as a re-classification of a chess-piece x, i.e., as
Tic (x, t) ∧ Tjc (x, t0 ) (with i 6= j). This re-classification implies a change in the rules the
piece is submitted to, i.e., a change in the accepted moves. The computation required
in the example of Kripke corresponds then to an acceptance of a move, i.e., instead of
calculating the addition (quaddition) of two numbers, we check if the sequence of two
configurations satisfy our constraints. To check the correctness of a move at a given
time, to compute the result, one needs to follow some constraints or procedures. For
instance, from an extensional perspective, to check if 125 is the result of 68 + 57, it is
enough to verify if the triple h68, 57, 125i belongs to the function that is the interpreta-
tion of the symbol ‘+’. The set associated to ‘+’ could vary in time (if we change the
interpretation) but this does not mean that it is not possible, at every instant, to provide
a justification of the answer (given at that time). Our reports go exactly in this direction,
the classification Tic (x, t) tells us what are the constraints x is submitted to, and these
constraints can be used as justification. Clearly one observer could be in trouble in re-
constructing the history of (the meaning of) ‘+’ or finding the interpretation of ‘+’ that
is compatible with all the answers. But this is another problem.16
One objection to this dynamic view on rules is that for something to count as a rule,
it has to norm homogeneously the behavior of a player throughout the game (this seems
the position of Kripke). If this is so, the skeptical argument applies directly to our model
as well. However, if we ask a player what is the rule for the queen that she was follow-
ing during the game, a player may reply by saying that at any time she was changing the
rules for the queen to cope with chess board situation. For instance, this shift appears
explicitly in White’s report as a switch of concepts, say from QueenWi to QueenWj . That
is, White was not following a rule for moving the queen, for example she was just trying
to guess the right rule for the queen. But she may have no idea of what rule accounts
14
Adding the deontic dimension complicates the picture without solving the problem.
15
Note however that, at a given time, chess-pieces have a single manifestation, while symbols
may have multiple realizations. In addition, we allow for ambiguous manifestations, manifes-
tations that can be understood as different chess-pieces by different players.
16
Similarly, arguments about the memory of what a player has done have only an epistemic
impact.
Concept Sharing 11
for the full history of her moves. Kripke’s sceptical argument is convincing if rules are
statical and are perfectly accessible to players, on that, his examples are illuminating
of this view: rules are exemplified by mathematical functions, that is something unam-
biguously defined and hardly subject to change. In case of social norms, such as those
of chess game, the interpretation of a rule is challenging. Rules may be adapted to cope
with new situations, take again Wittgenstein’s example in §83: agents start various ex-
isting ball games never finishing them. According to Kripke, we would conclude that
they are not following a rule. However, their behavior is not hectic and they are still
coordinating and going along. Another way of viewing Wittgenstein’s example is by
saying that players agree on a rule that regulates how to change rules from time to time.
If this is so, our chess player may reply to the question on what rule she was follow-
ing by saying that she was following the rule that compel to change the rules for the
queen in a given manner. Alternatively, one can look at the migrations of chess-pieces
accepted by a player during the game. These migrations highlight (i) the acceptance of
an originally inconsistent move, and (ii) how the player solved the inconsistency. Thus
the ‘rule’ the player is following depends not only on the constraints provided by the
chess-types, but also on her acknowledgment of the other’s move.
The point is that skepticism about rule following strictly depends on what we view
as a rule. Our model allows for both readings, as we can interpret the players’ reports
in two ways. Firstly, we can view a players’ report as the history of his guessing of
what is a chess rule: the player keeps on adapting her hypothesis to the new chess board
situations. In this case, if the player is successful, at the end of the game, she may end
up figuring out what is ‘the’ (or ‘a’) rule of the queen. On the other hand, at the end
of the game, a player may have no idea of what is the rule of the queen. In any case,
we cannot know whether the player was following any rule throughout the game, and
scepticism applies. There is however another interpretation of the reports, that accounts
for Wittgenstein’s example of the ball game. Players can always play a queer game of
chess that demands them to change the rules of the queen from time to time. The rule
that the players are following is just the rule that compel them to change the rule of the
queen, and that would be their answer to the question what rule were you following. Of
course, analogous treatment applies to adding and quadding. In any case, according to
both interpretations rules are dynamic. This is due to the fact that we are trying to take
seriously, even if minimally, interaction into our model. Moreover, this is also in line
with the skeptical, ‘communitarian’ solution posed by Kripke, according to which to
follow a rule is to be under the scrutiny of others in a certain community: “if everyone
agrees upon a certain answer, then no one will feel justified in calling the answer wrong”
[5]. In our examples each player is ready to change her rules to continue the game, to
align, since she knows that these changes are often necessary to interact.
What is then a rule? Far from providing a definition, we can say that, at every phase
of the interaction, even though there is no shared rules or concepts—and we have to
give up the epistemic privilege about social reality—still the interactants often behave
as if there are. When we interact with others we could have no basis to know what rule
they are following, or if they are following any rule at all. We try to find a way to fit
a situation with others by reading off their behaviors, their actions and their reactions
according to our behaviors. This means that if we read off the behavior and we pre-
12 Bottazzi et al.
dict a move that the player actually makes and the player has somehow accepted it, it
could be possible that this reinforces in us a sense of following the same rule that al-
lows us to keep on playing. We can have different rules—that objectively can be just
personal constraints—and concepts, and pieces, and attributions but anyway keep going
on playing. So it is not that the rule “traces the lines along which it is to be followed
through the whole of space”, but it is true that it is in this way, as Wittgenstein con-
clude ([14]: §219), that the rule “strikes me”. Sameness of concepts is also potentially
illusory. Nonetheless, the interactants can believe that the concepts are shared, since
they are based on ‘correct’ (but just “up until now”, before the other breaches them)
predictions, and nothing prevents that we are just going on in the interaction by trials
and errors, or, better, by acceptances and refusals.
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