Tractatus Logico-Philosophicus and Battleship

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Tractatus Logico-Philosophicus and Battleship Darren Jackson

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INTRODUCTION

Understanding Wittgenstein’s conception of the relationship between language and the world in Tractatus Logico-Philosophicus is difficult; understanding the game Battleship, not so much. In this paper, I argue that there are significant structural similarities between the Tractatus and Battleship, which make comprehending this relationship easier. It’s my contention, that if you know how to play Battleship, then you implicitly understand important aspects of the Tractatus.

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BATTLESHIP

Battleship is a game of strategy for two players.1 The game is played on four 10 x 10 grids, two for each player. The individual squares of the grid are identified by a letter and a number as in Figure 1; here, the yellow square indicates E5.

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Figure 1

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Battleship, the plastic board game by Milton Bradley that most of us are familiar with, has its origins in the French pencil and paper game L’Attaque, played during World War I. Although the different versions of the game are formally isomorphic, the diagrams used in this paper will more closely resemble the older pencil and paper version of the game.

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Each player has five ships and each ship is one square wide and a specific number of squares long. The aircraft carrier is five squares long; the battleship, four; the submarine, three; the destroyer, three; and the patrol boat, two. The objective of the game is to “sink” all of your opponent’s ships before they sink all of yours. Before the game begins, both players secretly place their ships on one of their two grids as demonstrated in Figure 2.

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Figure 2

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The exact positions of the ship can be given in terms of alphabetic-numeric coordinates as follows: aircraft carrier, E9-I9; battleship, F3-F6; submarine, C1-E1; destroyer, B5-B7; and patrol boat, I2-J2. Players take turns trying to guess the secret locations of their opponent’s ships. The guesses must be confined to one square per turn, which means that “C2” and “G7,” for example, are valid guesses, whereas “E8-E10” and “G1-I1” are not. These guesses are recorded by each opponent on their remaining grid.

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Let’s assume that Figure 2 represents the placement of my ships. My opponent begins her assault with “C8,” and I retort, “miss,” as her guess fails to identify the location of any of my ships. I mark her miss with an X on my grid as in Figure 3.

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Figure 3

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She, in turn, records the miss on her grid with a red square as in Figure 4.

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Figure 4

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On her next turn, my opponent guesses “F4,” and I reluctantly respond, “hit.” I update my grid to reflect the “hit” as in Figure 5.

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She marks the “hit” on her board with a green square as in Figure 6.

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Let’s assume that after a few more rounds our respective grids look like Figures 7 and 8.

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At this point my opponent guesses “F6,” and I respond by saying, “You sunk my battleship!” The sinking of the battleship is reflected on my and my opponent’s grid in Figures 9 and 10, respectively.

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The black outline around the green squares indicates that my opponent has “sunk,” i.e., correctly described or located the position of, one of my ships. As I stated earlier, the objective of the game is to sink all your opponent’s ships before they sink all of yours.

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If my opponent were to accomplish this feat without any additional misses, her grid would look like Figure 11.

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WORLD

To anyone who has played Battleship, the above explanation is likely obvious. What is not obvious, but potentially instructive, is how the Tractatus can be understood in terms of this game. To be clear, the analogy relies not so much on the playing of the game, i.e., taking turns trying to “hit” and “sink” the opponent’s ships, but rather on the rules and structure of the game itself. The players in this heuristic are LANGUAGE and WORLD. At the heart of the Tractatus is the assumption that language and the world share the same form, what Michael Morris, calls the “same-form assumption” (17). It is this similarity of form that makes it possible for language to meaningfully represent or picture the world. Wittgenstein says that “there must be something identical in a picture and what it depicts, to enable the one to be a picture of the other at all” (TLP 2.161). What language and the world share is logical form. Wittgenstein states: “What any picture, of whatever form, must have in common with reality, in

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order to be able to depict it - correctly or incorrectly - in any way at all, is logical form, i.e., the form of reality” (2.17) In Battleship, what LANGUAGE and WORLD have in common is the 10 x 10 grid. This grid is the logical form of the game Battleship. The grid consists of 100 squares. These squares fulfill the same function in Battleship that objects fulfill in the Tractatus. Wittgenstein characterizes objects as simple, unalterable, and combinatory. To say that they are simple is to say that they are not composite, i.e., they are indivisible (2.02). In Battleship, E5, for example, cannot be divided like the series of squares E5E7. To say that they are unalterable is to say that they are unchanging (2.027 and 2.0271). E5 will always be E5, and there will always be the same 100 squares. Taken together, these squares, i.e., objects, constitute the grid, i.e., logical form. As Wittgenstein states: “Objects are just what constitute this unalterable form” (2.023). Wittgenstein refers to these objects taken together as the substance of the world (2.021) and equates this substance with the unalterable logical form, saying: “Objects, the unalterable, and the subsistent are one and the same” (2.027). To say that objects are combinatory is to say that they are essentially capable of combining with other objects. As Wittgenstein states, “there is no object that we can imagine excluded from the possibility of combining with others” (2.0121) Objects combine to produce states of affairs (2.072); indeed, a state of affairs is nothing more than a combination of objects (2.01), and it is essential to objects that they can combine with other objects. In Battleship, states of affairs are established by placing the various ships on the grid; it is this placement, together with the sizes of the various ships, that determine the possible combinations of objects, or states of affairs.

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Thus, for example, when WORLD places the aircraft carrier on the grid in Figure 12, it establishes a state of affairs that combines the five objects, E9, F9, G9, H9 and I9 in the state of affairs E9-I9. Player 1 (WORLD) A

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Figure 12

There are only two restriction on the placing of ships: first, the ships must be placed either horizontally or vertically — diagonal placement is not permitted — and second, the placement of the ships cannot overlap. Given that the aircraft carrier is the largest ship in the game, consisting of five objects, it sets specific limits on the combinatory possibilities of objects. Thus, for example, the object E9 can only combine with the objects outlined in Figure 13.

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The outlined area represents the combinatory possibilities of the object, E9. In the game of Battleship, unlike in the actual world, there are only five possible states of affairs and these are represented by the placement of the five different ships on the grid. When WORLD places its ships as in Figure 2a, a world is created or actualized.

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The grid of objects itself is not a world, for it is only with the combination of objects in states of affairs, or facts, which are simply complex states of affairs, that we have a world. As Wittgenstein states, “The world is the totality of facts, not of things” (1.1). He continues saying that “the world is determined by the facts and by their being all the facts” (1.11). The world of WORLD is determined when all five ships are placed on the grid insofar as these are all the ships. Furthermore, the placement of the ships, i.e., the totality of facts, “determines what is the case and also whatever is not the case” (1.12, see also 2.04). What is the case, the totality of existing states of affairs, are the ships on the gird; what is not the case, the non-existence of states of affairs or negative facts, are the shipless squares on the grid.

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These are represented respectively by positive and negative signs in Figure 14.

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Figure 14

Anyone familiar with Battleship knows that the placement of the ships in Figure 14 is one among many and that a different configuration of ships results in a different world. Indeed, there are 30,093,975,536 different possible configurations of the five ships which means that there are over 30 billion possible worlds (Jestingrabbit)! What all these possible world share in common, however, is the grid, the logical form of Battleship. Wittgenstein states: “It is obvious that an imagined world, however different it may be from the real one, must have something - a form in common with it” (TLP 2.022). So although the states of affairs, i.e., the placement of the ships, may differ from game to game, the logical form, i.e., the 10 x 10 grid, is always the same. The grid gives all the possible arrangements of ships. This parallels Wittgenstein when he states; “If all objects are given, then at the same time all possible states of affairs are also given” (2.0124). He continues, “Objects contain the possibility of all situations [states of affairs]” (2.014). Before the ships are placed on the grid, any one of the 30 billion plus actual arrangements are

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necessarily possible; the grid is the space of possibility. Once the ships have been placed, one possible world is actualized.

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LANGUAGE

According to Wittgenstein, we picture facts or states of affairs to ourselves (2.1), and “a picture presents a situation in logical space, the existence and non-existence of states of affairs” (2.11). As stated earlier, logical space in Battleship is represented by the grid and the existence and nonexistence of states of affairs is represented respectively by the placement of the ships and the shipless coordinates on the grid. How then do pictures, specifically linguistic propositions, present situations in logical space? According to Wittgenstein, what makes a picture a picture is that the elements of the picture represent or stand for the objects pictured and that the arrangement of these elements mirrors the arrangement of objects pictured. Propositions are pictures that can be perceived by the senses (3.1). The elements that make up propositions are names that stand in determinate relations to one another (3.14 and 3.2-3.202). In Battleship, the names that make up propositions correspond to the alpha-numeric coordinates, i.e., objects, of the grid. Thus, for example, “E5” is a name that represents the object, E5. It’s important to note that although names makes up propositions, they themselves are not propositions; they are not pictures of the world. As Wittgenstein says, “what constitutes a picture is that its elements are related to one another in a determinate way” (2.14). Names, insofar as they are atomistic, representing only one object, do not have a determinate relational structure; in this sense, they are like points (3.144). In Battleship, propositions are made by combining names that represent the objects of the grid, for example, “E5-E6.” “One name,” according to Wittgenstein, “stands for one thing,

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another for another thing, and they are combined with one another. In this way the whole group like a tableau vivant - presents a state of affairs” (4.0311). By combining “E5” and “E6,” a proposition is created that can represent the state of affairs in Battleship, which is, for example, the patrol boat. I say “can represent” rather than “represents” because for Wittgenstein, “a proposition determines a place in logical space” (3.4), and “this place is a possibility: something can exist in it” (3.411). “E5-E6” is one of many possible places for the patrol boat to exist, for we can imagine it being placed by WORLD in any of the possible horizontal or vertical spaces on the grid. In the game of Battleship, the propositions of LANGUAGE are models of WORLD’s world as LANGUAGE imagines it (4.01). Propositions, however, do not describe the world simply as we imagine it; they are also either true or false. Wittgenstein states: “a picture agrees with reality or fails to agree; it is correct or incorrect, true or false” (2.21). The truth or falsity of propositions can only be ascertained by comparing them with the world. Wittgenstein says, “in order to tell whether a picture is true or false we must compare it with reality” (2.223, see also 4.05). Michael Morris suggests that “Wirklichkeit’, which is translated by Pears and McGuiness as ‘reality’ in the above quote, is more appropriately rendered as ‘actuality’. He states: “‘Die Wirklichkeit’ is what is actually the case - as opposed to what is merely possible” (52) “Actuality,” more accurately captures the contrast with possibility that Wittgenstein intended. In Battleship, what is actually the case, i.e., the actual world, is the placement of ships on the grid, whereas the grid itself represents possibility. Thus, in order to determine the truth or falsity of proposition in Battleship, we must compare LANGUAGE’s alpha-numeric propositions with the WORLD’s placement of the ships on the grid.

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Each “hit” moves LANGUAGE one step closer to formulating a true proposition and a true proposition is only formulated when LANGUAGE “sinks” one of WORLD’s ships. In sinking a ship, LANGUAGE correctly describes or pictures the placement of a ship, i.e., a state of affairs, in WORLD’s world. Wittgenstein states: “A proposition is a description of a state of affairs” (TLP 4.023). Given that there are only five ships, there can only be five states of affairs. When LANGAUGE “sinks” all of WORLD’s five ships, when it has won the game, it gives a complete description of WORLD’s world. As Wittgenstein says: “If all true elementary propositions are given, the result is a complete description of the world” (4.26, see also 4.023). To win a game of Battleship is to completely describe a world. Given that there are only five states of affairs in Battleship, there can only be five true propositions. Considering the arrangement of ships in Figure 14, the list of true propositions are: “C1-E1,” “I2-J2,” “F3-F6,” “B5-B7,” and “E9-I9.” LANGUAGE, if given enough turns, will always give a complete description of WORLD’s world as in Figure 15.

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Insofar as the determination of the existing states of affairs also determines all the non-existing states of affairs, or negative facts, as in Figure 16, a complete list of false propositions, for example “A1-A2,” “A1-A3,” “A1-A4,” etc., could also be given based on the five true propositions.

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“The world,” according to Wittgenstein, “is completely described by giving all elementary propositions, and adding which of them are true and which false” (4.26). Language, insofar as it can describe the world completely, exhausts the world. Thus, when Wittgenstein says that “the limits of my language mean the limits of my world” (5.6), he is highlighting the complete descriptive potential of language.

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CONCLUSION

By superimposing the grids in Figures 14 and 16 in Figure 17, the isomorphic relationship between WORLD and LANGUAGE in Battleship becomes clearly visible.

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Figure 17

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Insofar as WORLD and LANGUAGE share the same grid, they have the same logical scaffolding. It is this logical scaffolding that pervades both the world of WORLD and the propositions of LANGUAGE. The experience of actually playing a game of Battleship is similar to how Wittgenstein describes the relationship between a picture and what is pictured. He tells us that a picture is attached to reality by reaching right out to it (2.1511). The experience of sinking a ship in Battleship establishes a connection with the other player’s world; a connection that reaches right out to their grid. According to Wittgenstein, logic pervades both the world and language and “the force of a proposition reaches through the whole of logical space” (3.42, my emphasis). Language is able to meaningfully and completely describe the world, to reach through it or out to it, inasmuch as both language and the world are pervaded by logic (3.03-3.032 and 5.61). Similarly, ships can be sunk and games can be won in Battleship because

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both players share the logical space of the grid. It is the common grid, together with the two simple rules restricting diagonal and overlapping placement of the ships, which makes playing the game possible.


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Works Cited Jestingrabbit, Bill. “Battleship Permutations.” www.mathoverflow.net/questions/8374/battleshipMorris, Michael. Wittgenstein and the Tractatus. New York: Routledge, 2008. Print. Wittgenstein, Ludwig. Tractatus Logico-Philosophicus. Translated by D. F. Pears and B. F. McGuinness, New York: Routledge and Kegan Paul, 1974. Print.

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