Abstract: A semiotic analysis of tic-tac-toe illustrates the use of multiple perspectives in human understanding of signs. Tagmemic theory is a semiotic framework that pays explicit attention to multiple perspectives. It can generate multiple complementary analyses of the same semiotic system. We illustrate using the game of tic-tac-toe. Our analysis illustrates how three distinct perspectives or views (particle, wave, and field) can be applied to the same semiotic system, resulting in radically different textures in analysis. At the same time, each analysis is in a sense complete, because all the information in the other analyses can be deduced in principle from the analysis using only one view. The result is suggestive for evaluating the strengths and limitations inherent in monoperspectival programs used for artificial intelligence.
Keywords: tagmemic theory, tic-tac-toe, artificial intelligence, multiperspectives
1. Why Analyze Tic-Tac-Toe?
We propose to analyze tic-tac-toe as a semiotic system, using tagmemic theory as our semiotic framework. Why do an extended analysis of something as simple as tic-tac-toe? And why use tagmemic theory to do it? A simple semiotic system like the game of tic-tac-toe (British, “noughts and crosses”) can serve as a convenient test example for examining larger issues. A simple beginning analysis of tic-tac-toe observes that there are two distinct signs, “noughts” (O) and “crosses” (X). These signs can be placed successively on a standard two-dimensional grid of nine boxes. Tic-tac-toe seems on the surface to receive a complete analysis quite easily using this analytical approach, which focuses on the two signs. Because of its simplicity however, tic-tac- toe is actually also a good candidate to illustrate the power of complementary analyses, using multiple observer perspectives. We can find a precedent for such an analysis in an earlier contribution (Poythress 2018): a multiperspectival semiotic analysis for traffic lights. Traffic lights constitute a simple semiotic system; however, this system shows surprising complexity when subjected to an analysis from multiple perspectives. The analysis of traffic lights in turn builds on a larger theoretical framework, influenced by information theory (Poythress 2013). In the present article we apply a similar strategy of analysis to the game of tic-tac-toe—a different kind of semiotic system—and for that very reason, it makes a useful second analysis from my previous example of the traffic lights.
Why, though, use tagmemic theory as our chosen semiotic framework? Tagmemic theory is best known as a framework for analyzing verbal language. But it offers a robust framework for wider-ranging semiotic analyses. The main architect of tagmemic theory, Kenneth L. Pike (1954–60: xv), deliberately and self-consciously designed tagmemic theory to be applicable to human action in general (Waterhouse 1974: 15). The theory treats verbal communication as a subset of human social action. Pike’s key work illustrates tagmemic theory not only through examples from verbal language but also through the analysis of other semiotic phenomena, such as an American football game (1954–60: 98–119), a church service (ibid.: 73–97), and behavior at a breakfast (using recorded data from breakfasts shared by Pike’s own family) (ibid.: 122–128). In addition, tagmemic theory explicitly affirms the value of multiple perspectives. It is useful in examining the complexity in meaning that belongs even to a simple semiotic system when we examine the system with an awareness of multiple perspectives. We may contrast this focus on complexity with the trend toward formalization. The twentieth century saw the growth of structural linguistics and with it, vigorous attempts to make linguistics rigorous by using quasi-mathematical formal models. Rigor and formalization can convey insights, but it is always possible to forget the messy complexities in human action with which one started. Semiotics, as a larger field encompassing linguistics, has similar challenges. A researcher can strive for rigor and formalization. He can also strive for maximal sensitivity to multiple associations and connections of meaning. The latter is one of our goals in analyzing tic-tac-toe.
2. Tic-tac-toe as a Semiotic System
Let us then consider the game of tic-tac-toe. Is it a semiotic system? Semiotics can be defined as the study of signs. The term “sign,” as formulated by Pike, is a form-meaning composite (Pike 1954–60: 9, 62–63, 1982: xiv, 16, 115–117). The meanings of that composite belong to a larger human community, and they may not be familiar outside the bounds of this community. This function of meaning holds true for tic-tac-toe. It has meaning only within a community that recognizes and understands tic-tac-toe as a game. What signs are found in tic-tac-toe? Tic-tac-toe has three signs: an O (called in the British version of the game a “nought”), an X (also called a “cross”), and a grid of nine boxes (#). Each of these has a distinct form, which we have just delineated by describing the geometrical shape. Each also has a meaning. Each appearance of an O indicates the play of one player, while each appearance of an X indicates a play by another player. The grid of nine boxes can be called a “playing field.” Each box represents a place where a player may place his mark. In addition to these elementary signs, tic-tac-toe has rules, as any game does. The rules, too, have their own signification. But in the usual course of human interaction, these rules remain in the background. They are tacit, rather than explicit—somewhat like the rules of grammar or spelling in a natural language. They become explicit primarily when a theoretical analyst makes them explicit. We take them up later in this article, as appropriate. Tic-tac-toe is a game. So, in most instances, it has significance only at the level of leisure activity. It is a game, not a serious work engagement. But even though games are less weighty, they do have meaning to those who engage in them, as well as to those who look on. The game of tic-tac-toe has a meaningful goal, namely, for a player to produce a line of three of his tokens in a row, or a column, or a diagonal. The first person to produce such a line of three tokens wins the game. So, yes, the game as a whole has human meaning, but it has meaning only within a culture that understands what the game is. Hence, the game as a whole is an instance of a semiotic system. Within the context of the game, the particular occurrences of Os and Xs have meaning. They are indeed form-meaning composites.
3. Particle, Wave, and Field Views
With these preliminaries behind us, we proceed to employ tagmemic theory as our semiotic framework for analyzing tic-tac-toe.1 One of the important features of tagmemic theory is its offer of multiple views of the same semiotic phenomena. Rather than offering a formalism that reduces the phenomena to one standard kind of analysis, there are three complementary views or perspectives. These three are customarily denominated the “particle view,” the “wave view,” and the “field view” (Pike 1959, 1982 3–4, 19–38; Waterhouse 1974: 6, 90; Poythress 2018: §3). Each of these views is capable on its own of taking in the phenomena. The distinct views, however, ask different questions concerning the phenomena. The choice of views is up to the observer/analyst.
3.1 A Particle View of Tic-tac-toe
The particle view of a semiotic system treats it as a collection of particle-like objects—each of which is a distinct unit (Pike 1982: 19–23). The particle view is usually the most intuitive, natural view for the naive participant in a semiotic system. In the case of tic-tac-toe, there are two main particle-like units. There is the “nought,” the O-shaped sign; and there is the “cross,” the X- shaped sign. These two units are clearly contrastive in relation to each other; that is, they are mutually exclusive. If one unit occurs at a particular location in the nine-box grid, the other does not occur there simultaneously.
We are speaking in a normative fashion. Tic-tac-toe, like any typical game, has a set of rules. Of course, we can contemplate situations in which the rules are violated. Such situations are analogous to situations in which a language user violates the rules of grammar or the rules of pronunciation, or spelling, or meaning (i.e., a person may use a word thinking it has one meaning when it actually has another). Similarly, in tic-tac-toe, we can imagine one child writing an X in the same box where the other player has already placed an O. He does so because he does not yet know the rules, or perhaps because he is frustrated and decides to violate the rules. The rules, however, still stand. Within the expectation for normal participants, it remains the case that Os exclude Xs and vice versa. The probabilities would be low for a simultaneous occurrence of the two. Moreover, the reactions of participants and bystanders show the unusual nature of such a joint occurrence. The other player tells the child, “Wait, you cannot do that. My O is already there.”
In addition, there is a third unit that can be treated as a particle-like unit, namely the framework consisting in the nine-box grid (#). If we consider the grid to occupy the whole space that it spans, it overlaps in time and space with any O or X that is placed within it. Rather than treating the grid as a distinct unit, we may choose to say that the grid is the framework for play. It functions like a game board—not like one of the pieces with which the game is played. In addition, if we think of a sports game, we may say that it functions like the tennis court, not like one of the players or the tennis balls. For a piece of written communication, it is analogous to the paper or the stationery—although perhaps the stationery has some design—but this design and the paper itself are the framework or context for the communication, not the communication itself.
What is the role of the nine-box grid in tic-tac-toe? There are games that are analogous to tic-tac-toe that use other grids with different numbers of boxes and different arrangements of boxes. We need to lay these analogues to one side in order to concentrate on tic-tac-toe in its traditional, well-known form. The game of tic-tac-toe necessarily has the nine-box grid. We cannot just dispense with the grid as if it did not matter for the game. It is a recognizable aspect of the game. It is what is called an “emic” unit, recognized as a meaningful unit by all who are familiar with the game. (The emic/etic distinction functions as an integral part of the framework of tagmemic theory and is one form of multiperspectival analysis; we may analyze the same phenomena either emically, as insider-participants, or etically, as outside analysts or uncomprehending observers (Pike 1954–60: 37–72; Waterhouse 1974: 6; Headland et al. 1990).
The grid, we say, is a necessity. Yet it does not function in a manner parallel to the two particle-like units, the O and the X. In a multidimensional analysis, we are free to observe the grid itself from more than one point-of-view. The most natural treatment, in my opinion, is to discuss it from a field view (see below). The grid presents us with the “field” of play. That is one point of view. If we like though, we can still treat the grid from a particle view. We do that most effectively if we treat the grid as a graphic sign that fills only the space actually occupied by the lines that belong to the grid. The empty spaces between the lines are not occupied by the grid.
More specifically, the grid can be broken up into two vertical lines and two horizontal lines. (This, by the way, is etic from the standpoint of game play; it may, however, be emic from the standpoint of the elements of artistic drawing and rendering.) Each of the two vertical lines crosses each of the horizontal lines. The joint effect is to create a central box surrounded on all sides by lines and then also eight other boxes bounded by the lines on two or three sides. All the boxes are regarded as empty. So the grid fills only the part of the area that is actually occupied by its lines. Hence, it does not “occupy” any of the space that is later going to be occupied by the Os and the Xs. The grid “contrasts” with the Os and the Xs as units. Where the lines are, the Os and the Xs must not be. Where an O or an X resides, there must be no grid lines. This kind of exclusion typifies the analytical treatment from the particle view.
We can clearly see that this feature of exclusion belongs to the rules of tic-tac-toe. We may illustrate it by imagining a violation of the rule. A child places his O overlapping parts of two or even four distinct boxes. His partner says, “No, you can’t do that. Your O must be completely inside just one of the boxes.”
3.2 A Wave View of Tic-tac-toe
Next, let us employ the wave view to analyze tic-tac-toe. In approaching a semiotic system, the wave view, also called the “dynamic view,” gives preference to dynamic developments of phenomena. It focuses on how one thing leads to another or naturally implies another (Pike 1982: 24–29).
The dynamics of tic-tac-toe become manifest in the progress of a game. The opening move is a response to the empty starting grid. After that, each move takes place in response to the previous move and to the overall configuration set up by the pattern of Os and Xs already on the board. This overall configuration—the pattern of Os and Xs at a particular moment within the game—is called a “game state.” As with many games, there is a “game tree” graph that shows all the possibilities for sequences of moves. It shows also how any particular move causes a transition from an earlier game state to a later game state (the one after the completion of the move in question) (Fudenberg and Tirole 1991; Griffin 2012; Ross 2021).
When we look at the game as a sequence of moves, the smallest units are the moves themselves. Each move consists in placing an O or an X in a single empty box within the nine- box grid. The moves are constrained, as we have observed, by the previous placement of markers in some of the boxes. So we can view a move not simply as the placement of a mark but also as the placement of a mark in the context of the previous state of markers already on the grid. The move presupposes this previous state. It in turn leads to the next state, which is the particular configuration once the new mark has been added. This sequence is a wave sequence. In short, the wave view of tic-tac-toe treats it as a series of configurations (or “states”) and moves. These states and moves succeed one another in time. They progress toward a conclusion (win or draw).
The wave perspective or wave view in tagmemics also pays attention to what is most central in a wave motion. Does the wave have what we would call a “peak” or “nucleus” that is the most prominent point of action—a defining center around which the rest of the action is organized? For a game like tic-tac-toe, the purpose is to win the game. From that point-of-view, the final winning state is the most prominent. It is the state that drives everything else that comes earlier. In the case of a game ending in a draw, the final state is still nuclear because it shows the final meaning of all the moves that led up to it. The meaning is that neither player won. The nine- box grid for tic-tac-toe holds a natural place here. It is the starting point for the development of the sequence of states and moves. It is the initial state, presupposed by all the succeeding states.
3.3 A Field View of Tic-tac-toe
In tagmemic theory, the field view focuses on features that may be simultaneously present at the same location and the same time (Pike 1982: 30–38; Poythress 2013: §9, 2018: §3.3). The field view organizes the emic units of a semiotic system into a grid with multiple dimensions. For example, a field view of phonology may organize the consonantal phonemes of one particular language in terms of three main dimensions: (1) point of articulation (for example, where is the air stream narrowed or cut off), (2) manner of articulation (stop or fricative), and (3) voicing (voiced, voiceless, and/or aspirated). Likewise, we may analyze the features of personal pronouns in English in four dimensions: (1) person (first-, second-, or third-person pronouns), (2) number (singular or plural), (3) grammatical function (subject, object, reflexive, possessive), and (4) gender (masculine, feminine, or neuter, applicable only for third-person singular pronouns). Each choice in one dimension is compatible with any of the choices in the other dimensions (except, as noted, that gender is a meaningfully marked choice only for the third-person singular case).
How might a field view apply to tic-tac-toe? The aspect most obviously inviting field analysis is the nine-box grid. The grid is a 3 x 3 grid with two dimensions. Moreover, vital to the concept of winning is the specification that one player wins when he succeeds in lining up three of his tokens in a single straight line. This specification imparts to the grid an extra structure. The nine boxes are linked in a specific way, so that each row, column or diagonal constitutes a single, larger whole that can lead to a win. Thus, each box has a specific relation to the other boxes. The field view is rightly also called the relational view. Each particle-like unit has multiple, simultaneous relations to neighboring particles. The multiple simultaneous features are what characterize the focus of the field view.
When a player places an individual token (either an O or an X), this placement can be viewed from any of the three views that belong to tagmemic theory as a semiotic theory. Using the particle view, we focus naturally on the token (or sign) itself as a sign with its own integrity that is in contrast with the other sign (an O is not an X). When we use the wave view, we focus on the dynamics of movement. The player makes a play by placing or inscribing his token. In the field view, we focus on the fact that the play is a play in relation to the grid itself and also in relation to all the other tokens that have already been played or may be played in subsequent moves. Since the aim is to achieve a lineup of three tokens, the player who is using some strategy begins to think ahead. He plans to block a lineup by his opponent and anticipates how to produce one or more opportunities to achieve a lineup himself by his future moves. He is thinking relationally. Or, to put it in other words, his thinking takes into account the field of relations between boxes and possible tokens in the boxes. As can be seen, since the field view focuses on relations, there are in fact many aspects on which to focus. The relation between the nine boxes in the grid is the most obvious, but there are others.
This threefold analysis in terms of three views or perspectives could be applied to the rules of tic-tac-toe, as well as to the first-order analysis of a particular game. Each rule can be considered from the particle view as a complete whole. From the wave view, it can be considered as having possibly developed from earlier, different rules of related games. From the field view, we can consider each rule in relation to all the other rules for the same game and related games.
4. The Game Tree of Tic-tac-toe
We earlier mentioned the game tree of tic-tac-toe. Any game with a prescribed series of discrete moves has an associated game tree (Ross 2021: §2.3). For those not familiar with the concept, some explanation is in order. A game tree represents, in diagrammatic form, the structure of the progress of the game. While a biological tree develops upward from the root, a game tree is frequently diagrammed in the opposite direction. The root is placed at the top and the branches that represent the moves in the game grow downward from the root.
The root of the tree is the start of the game. For tic-tac-toe, this root is represented by an empty nine-box grid. Just below the root are the possible choices for the first move. For tic-tac- toe there are nine of them, each corresponding to one of the boxes in the nine-box grid. These possibilities are represented by drawing nine branches or connecting lines, each of which connects downward from the root to one of the possible initial moves. The next layer down on the tree is generated by the second move. For any one of the nine branches generated by the first move, there are eight more branches reaching downward for the second move because the second move can place a token in any of the remaining eight boxes not occupied by the token placed during the first move. So the tree progresses downward for a maximum of nine moves. (See Figure 1.)
Fig. 1: The beginning of a game tree for tic-tac-toe
The wave view can treat the full tree as a development, or growth, or movement from the root downward. The wave view focuses on this movement. But the same diagram can also be viewed from the field view. The full tree is a system of relations. Each branch is a relation between two nodes. It represents a possible move, given the state at which the game has arrived through previous moves. The nodes to which the branches attach represent states of the game. The overall result is a plot of all the possibilities for playing the game. Any one possibility is singled out by traversing a route beginning from the root and traveling down through the branches until layer nine is reached after the completion of the ninth move. We may note that the game tree invites us also to notice some further possibilities for multidimensional classification. Each layer in the tree is distinct from all the other layers. Within each layer, there are closer so- called “genetic” relations to those nodes that are connected to the same node at the preceding layer.
We may use the field view to classify game states in still other ways. The entire set of nodes from a game tree can be organized into a gigantic array of nine dimensions. We have one distinct dimension for each box of the nine-box grid. Let us say that the first dimension in our new array is the dimension corresponding to the top-left box in the grid. Any game state can be classified in this dimension as belonging to one of three alternative categories: (a) the top-left box is empty, (b) the top-left box contains an O, or (c) the top-left box contains an X. The second dimension in our array is the dimension corresponding to the top-middle box in the tic-tac-toe nine-box grid. In this second dimension, a game state is classified according to what is in the top- middle box. Again there are three alternatives, and so we proceed for each of the nine dimensions. The result is a massive field representation of the game states of tic-tac-toe. Note that it is not possible, by following the rules of the game, to arrive at some of the positions in the nine-dimensional array, but it is only possible in those positions where the number of Os is either equal to the number of Xs or different by one.
5. Perspectives as Complementary Suites of Questions
What have we achieved through the use of the three views or perspectives: the particle, wave, and field views? We have confirmed that the particle, wave, and field views are complementary rather than mutually exclusive. We may also observe that an analysis from one perspective alone can furnish all the information needed to deduce answers to questions occurring in the other two perspectives.
To the particle view belong questions of the form: Does an O occur at location L at time T? or Does an X occur at location L at time T? Suppose that we obtain answers to all these questions, with sufficient precision for various Ls and Ts. Then we have all the information that we need about the sequence of play in one game or even a series of games. By comparing the answers about the presence of an X at the same location at two neighboring times, we can find out whether a player placed an X at that location at a moment in between the two times. We can also obtain information about what Os and Xs were already there just before he made the play. So in one sense, the particle view alone allows us to obtain a complete analysis of the game. That sense of completeness is what allows an analyst to think that his analytical point-of-view is definitive. The advocate for one point of view might thus think that his approach is in competition with other views. Yet, it is not the only possible point of view. A robust semiotic approach needs to take into account the choice of an analyst to use one particular view. Other views are possible, and these views highlight aspects of human action that may be left in the background or unacknowledged, if we are content with the analysis already presented from a single point of view.
Now consider the wave view. The wave view focuses on questions concerning moves. When used by itself, it also can furnish full information about what happened in any one game of tic-tac-toe. We have to have a starting point for the moves. The starting point is an empty nine- box grid. Then we ask questions as to what the first move is, what the second is, and so on, up to a maximum of nine moves. Suppose we collect the list of all the moves in succession. Then we have complete information about the whole game. Using this information, we can deduce all the information about when and where there are Os and Xs, over the whole course of the game. (There are technicalities, related to how much information we need to obtain regarding exact times for each move. Exactly how much time it takes for a player to make his next move would typically be regarded as incidental to the game. It is an etic detail, rather than an emic aspect, of the game. We can deal with the differences in this time information by supplying more exact information about the time associated with each move, or we may loosen up the requirements for the particle view by allowing a less precise measurement of the time during which we may perceive the presence of an O or an X at a particular location.)
As a result of our ability to make inferences, we conclude that the picture derived from the wave view is just as complete as the picture from the particle view. Yet we have two distinct views, not one. How can this be? The difference consists mainly in what is in focus. In a particle view, the tokens or marks (Os and Xs) are in focus. In the wave view, the moves are in focus. However, this focus is a choice made by the investigator. He is free to alter it. He is free to stand back and make deductions in order to answer questions that belong most appropriately to either of the two distinct views.
As we might anticipate, a similar principle holds for the field view. We can illustrate the working of complementarity using two instances of the field view. The first use of the field view is more elementary. It treats each of the nine positions on the tic-tac-toe grid as a position in two dimensions. In the particle view we specify the location directly by pointing to a spot in space. In addition, we can ask: “Is there an O in the top-left box within the nine-box grid?” In the field view, we specify the location using two independent axes, a vertical axis (distinguishing the rows) and a horizontal axis (distinguishing the columns). So the question would be: Is the O I see in the top row? Then, as a separate question, we would ask: Is the O I see in the left column? These questions have independent answers. Only if both answers are true do we deduce that the O is in the top-left box. But either of these strategies, either the particle view or the field view, can yield full information about the location of each O and X.
We can also illustrate the potential of the field view using a second approach that focuses on the game tree. The game tree, as we saw earlier, is a complex array of relations. It is innately field-like. We look at the whole array in order to understand the game of tic-tac-toe in a general way. Any specific game state can then be singled out for attention merely by focusing on one node. If we want to specify one complete game, we do it by focusing on a single node at the “leaf-end” of the tree. If the root of the tree is placed at the top, the leaves are at the bottom at level 9. (Keep in mind that some of the games lead to a win at an earlier stage prior to level 9.) Each leaf represents a game state either immediately after the game is won or after nine moves have been made in succession and the nine-box grid is completely full. Given the game tree, together with a focus on one node, we obtain complete information about any one game. We simply trace backwards in time or upwards in space, beginning with the leaf and ending with the root. From this information we can deduce all the answers to the questions that arise either in the particle view or in the wave view.
In sum, it only takes one view, out of the three views, to achieve an analysis that provides full information in principle. We may also observe that this kind of fullness is a characteristic feature of the use of perspectives in tagmemic theory (Pike 1982: 3–9, 11–13) and elsewhere (Frame 2008). A perspective has the property of being a perspective on the whole of a given subject-matter. It leaves nothing out. Yet it also has a special focus that is not the only possible choice of focus. The observer has a role in this decision.
6. Interlocking of Perspectives
We may add that the three views interlock. A person cannot actually employ one view without including the other views or at least fragments of them, lurking in the background. The tokens of Os and Xs make no sense apart from a game in which someone employs them as marks. Their having meaning presupposes that an analyst understands a certain employment that is dynamic. The purpose of the marks is to make moves using them. So aspects of a wave view or dynamic view are there, tacitly in the background, even when we are momentarily focusing on the Os or Xs as particle-like units.
Likewise, the nine-box grid is there in the background whenever people play tic-tac-toe. That grid is a meaningful whole, the meaning of which largely arises from relations. The game of tic-tac-toe fails to be defined if we do not take into account the grid. Moreover, the game fails to have a goal if we are not able to talk about winning. And winning means forming a straight line of three tokens of the same kind. Any straight line that represents a win is a meaningful whole that intrinsically involves relations. If we look at the field-like character of the game tree, we still have to remember that the branches that connect nodes represent moves. These moves are dynamic. They are tracked using the particle-like marks—the Os and the Xs.
The psychology of game play also shows the joint presence of particle-like, wave-like, and field-like realities to which players pay attention to from time to time. Let us illustrate. It is natural for a complete novice at tic-tac-toe to be aware primarily of his mark. “I am supposed to use an O as my mark,” he says to himself, and “Whenever it is my turn, I draw an O.” Yet he is soon involved in the game. He begins to develop a tacit sense of the dynamic movement of the game. He knows that the game is heading toward a win, a loss, or a draw. He soon takes the circular shape for granted. His focus is on decisions as to the next move given the preceding configuration or game state. So, he is closer to using a wave perspective when he is actually involved in playing a game. Then he begins to think forwards and backwards in time, asking himself what his opponent will do in response. He is not literally drawing a game tree in his head, but he is beginning to use the concepts that are involved in a game tree. He is seeing future possible moves. His immediate decision to place an O in the top-left box is taken in relation to other configurations that possibly lie ahead—that is, multiple nodes of the game tree. Unless he begins to diagram things, he probably does not literally imagine a tree in his mind, but he understands the choices of moves in relation to other moves. He is tacitly relying on a field perspective.
In sum, actual human involvement in a game like tic-tac-toe includes joint reliance on all three views—particle, wave, and field. Human understanding of what it means to play a game uses knowledge of pieces (particles), knowledge of moves (waves), and knowledge of relations among choices (fields). When human beings join in games, one or other of these elements may be in conscious focus. But all three have to be there, at least tacitly in the background, or the game itself does not make sense as a coherent whole in which people participate.2
7. Unit, Hierarchy, and Context
We can further illustrate the operation of multiple perspectives by focusing on three more aspects integral to tagmemic theory as a semiotic theory: unit, hierarchy, and context (Pike 1982: 13–18). These three come into view as a kind of reflection of the particle, wave, and field views, respectively, when we apply these views in a creative way to the study of the structure of a semiotic system.
A unit is a particle-like aspect of semiotic systems. Each unit has an inner unity and is also defined by its contrast with other, distinct units. What are the units in tic-tac-toe? There are two static units, namely an O and an X. These contrast with each other. There are also dynamic units, namely moves. Placing an O or an X is a move, and a move is an emic unit within the overall game of tic-tac-toe.
Hierarchy is the term designating a pattern of progressive inclusion of smaller units within larger units. The smaller units have functions within larger units in which they are embedded. The structure of hierarchy correlates naturally with the wave view because larger units imply the existence of smaller pieces within them. We move from large to small or small to large units. What is the hierarchy in tic-tac-toe? The hierarchy of static units has three main levels.
The lowest level is the level of atomic units. Each O or X is an atomic unit because as an emic unit it is not further divisible. An X can be broken up into two intersecting lines, and an O can be broken up into smaller arcs, but both these cases involve an etic analysis that is below the threshold of focus when we are focusing on tic-tac-toe playing. These atomic units can combine into larger units at the next level of hierarchy. The larger units are game states. A game state consists in a pattern of Os and Xs arranged within the nine-box grid. The game state specifies not only each location for an O or an X, but also which boxes are empty. This level of a game state is the next level of hierarchy, and each unit at this level is an emic unit. Next up is the level of a full game. A full game consists in a definite sequence of successive game states. This full game is itself an emic, semiotic unit. The participants tacitly recognize its unity as a single game, in contrast to other possible or actual games. There can also be higher levels of hierarchy. But these higher levels depend on the larger context of game play. We can imagine a tournament where many games are being played simultaneously, or where there is a ladder of elimination in successive games, or where two players pair off to play a series of five or seven or more successive games. Depending on the setup, various groupings of sets of games may function as emic wholes, that is, units at a higher level. At even higher levels, game play merges into life itself. A tournament of tic-tac-toe gets integrated into a number of meaningful human activities in the course of a single calendar day.
In addition, we can focus on the existence of a hierarchy not of marks but of moves— dynamic units. A single atomic move consists in a player placing his piece in a single box. That is the lower level in the hierarchy. The complete sequence of moves leading to a win or a draw is the next higher emic grouping. So, likewise, we may analyze tournaments as clusters of moves rather than primarily clusters of game states.
If we choose to focus on field-like units, we can also find a hierarchy there. Consider the game tree. The lowest level consists in individual nodes and individual branches. The next level is a path through the tree, proceeding from root to leaf. The next level is the full tree, including all its nodes and branches. As usual, we may extend the hierarchy upward to encompass tic-tac- toe tournaments.
Finally, tagmemic theory has within its framework attention to context, or rather, it pays attention to plural contexts, for there are several. Context is a fundamentally relational idea and thus correlates naturally with the field view. What is/are the context or contexts for tic-tac-toe? There are always larger real-world contexts in which the game players participate. These are multiple in nature. However, it is here appropriate to focus first on what might be called “internal contexts,” the contexts that belong intrinsically to the game.
When tagmemic theory is applied to natural languages, there are three fundamental contexts for the functioning of signs. These are the referential (or semantic) context, the grammatical context, and the phonological context. In the context of written communication, phonology is replaced by graphology. Each of these involves a linguistic subsystem. There is a semantic subsystem (semantic units, or units of meaning), a grammatical subsystem (morphology and syntax), and a phonological (or graphological) subsystem (phonological units). In semiotic systems outside natural language, these three can often be completely fused. They may be difficult to distinguish. The simplicity of tic-tac-toe would lead to an expectation that there may not be clear distinctions, but we can still see some degree of distinction in subsystems.
8. Graphical, Meaning, and Grammatical Subsystems
Since tic-tac-toe involves graphic rather than auditory symbols, is there a graphological subsystem? Yes, it consists in two graphic alternatives—the O and the X. The difficulty is that these two written forms cannot be strictly separated from their meanings. As we observed earlier, they are form-meaning composites. However, we can imagine a form of tic-tac-toe that operates according to exactly the same rules but with the signs being A and B instead of O and X. We can also imagine playing tic-tac-toe not with marks but with white and black chess pawns that are placed on any of the nine boxes constituting the grid, or we may use red and black checker pieces. Whatever pieces we use, the game as a whole is in almost every meaningful sense the same game. It does not matter what written forms or pieces we use, as long as one type is used exclusively by the first player and the other type exclusively by the second player. This observation enables us to see that the graphological subsystem is indeed a distinct subsystem. It can be swapped out for the (A/B) subsystem or the (white/black) subsystem but still leave the meanings of the game intact. As Ferdinand de Saussure stressed, the meaning is found in the contrast between the two elements, not in the etic details of one element.
We can see also, by the same reasoning, that there is a distinct meaning or semantic subsystem. We must not equate this idea with the narrow conception of verbal semantics. Individual marks do have meanings. An X means “the play of player 1,” while an O means “the play of player 2.” A game state has meaning, for example, threatening a win on the next move. The full panoply of such meanings can be distinguished in principle from the choice of particular graphic forms (X/O, rather than A/B, or white/black chess pieces) used to represent the meanings.
Finally, there is a kind of tic-tac-toe analogue to the grammatical subsystem of a language. There is the grammar of the structure of relations among pieces on the board. A straight line of three Os is what we would call the “grammar” of a win. As before, each of these subsystems can be treated as a perspective on the whole. In any actual game, all three are simultaneously present. And any one of the three can be used by itself to give a “complete” account of the game.
9. Contrast, Variation, and Distribution
Finally, we may observe, in a sketchy way, that we can apply still another triad of perspectives— namely contrast, variation, and distribution. Each O contrasts with an X. Each O that is actually drawn in the course of a game is a variant manifestation of the emic category called “O.” If an O is drawn by hand, there will be slight variations in its shape. One O might be more like an oval. Another might not fully close. Last of all, each O has a distribution. The distribution in a particular case could consist in its location in relation to other marks that are placed within the nine-box grid. The distribution of Os in general is the whole pattern of how they occur in their many instances in relation to many game states.
10. Suggestive Implications for Artificial Intelligence
Because tic-tac-toe is a game, it is natural to explore how it fits into the larger challenge of playing games in general. This exploration naturally brings up the question of computer capabilities. What does it mean for computer to play a game in the manner of a human opponent? Because tic-tac-toe is a comparatively simple game, it is comparatively easy for students in a computer programming class to learn to write a program that will play the game, either fallibly or perfectly. Can the lessons from a simple game be extrapolated to much more complex decisions? The issue of complex decisions leads us to ask about artificial intelligence. Work in artificial intelligence holds promise for enhancing decision-making, not only if one has a computer that can play tic-tac-toe but also for subjects like medical diagnosis or playing Jeopardy (Best 2013).
A common strategy in computer programming is to use the simplest and clearest route, within the constraints of what a linear sequence of computer commands can do. This strategy corresponds to taking a single perspective in analyzing a game. As we showed, even a single perspective of tic-tac-toe is able to provide all the information that we need to deduce everything we need to know about a particular game. Hence, it is also enough to supply a computer program with the input from a single perspective. The output from the program needs only to furnish its information from a single perspective—the output needs to be in only one form.
The strategy of using “neural networks” in computer learning is a partial exception to this simplicity. The networks have a flexibility. Based on the guidance from large numbers of solved cases, the network adjusts its strengths and learns how to deal with new cases more effectively. After a time, those who set up the network do not know completely how it is arriving at its answers. That is impressive, in a way. But the inputs and the outputs are still monoperspectival. The program’s interaction with the world is not a mirror of human interaction. One result is that simple computer programs can be very effective and efficient with some kinds of games that are themselves simple, but the programs do not operate in a manner that genuinely mirrors the power of human creativity, which includes the power of multiple perspectives. They leave out much of the richness of what human beings understand tacitly and are capable of attending to, if they think it will help.
We may contrast this simple kind of program with what is needed for more complex games. Consider the TV game of Jeopardy. Watson, the IBM supercomputer, impressively defeated the most skilled of human players of the game (Best 2013). It was a great technical triumph—but the underlying programming was far from simple. Watson was programmed to allow it to weigh multiple verbal associations in multiple dimensions. In this respect, it imitated aspects of human multiperspectival attentiveness. This multiperspectival ability was very much at work in the game of Jeopardy. By contrast, this type of ability is not needed for a computer playing tic-tac-toe, but our previous analysis has highlighted the tacit presence of multiple perspectives for tic-tac-toe, when considered not as a purely mechanical problem for programming but as a semiotic system, against the background of human understanding and human involvement in meaning. Straightforward computational problem solving can be a powerful tool, but it inevitably lacks the human ability to choose perspectives, to create new perspectives, and to rise above a single limited perspective in the effort to understand. For complex games, computer programming itself becomes complex. In many situations, it is most effective if it in some way imitates the multiperspectival capabilities of human observers.
We may still debate what might be the limits of artificial intelligence. With faster and faster computations and multiple searches and/or neural networks to imitate multiple perspectives, we can anticipate greater computational abilities from computers. How well though can programming simulate self-awareness and the ability to self-consciously use a new perspective at any time? One of the noteworthy things about human participation in tic-tac-toe is that there is a larger human context (including understanding the meaning of playing a game) and a broad human capability of using multiple perspectives in the same game (Dreyfus and Dreyfus 1986). Simulating this robust interaction with a real-world environment is one of the paths that is being explored within artificial intelligence (Tatum 2023; Serov 2013).
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- We refer readers to the earlier semiotic analysis of a traffic-light system (Poythress 2018) and to a book-length treatment by Pike (1982) for a more elaborate explanation of the individual elements in tagmemic theory. ↩︎
- These observations are in harmony with Michael Polanyi’s stress on tacit knowledge (Polanyi 1966, 1958). Polanyi makes the point that there is always a background of understanding—“tacit knowledge”—that guides the conscious focus of a participant in human interaction. ↩︎