Diagrammatic Logic of Existential Graphs: A Case Study of Commands Ahti-Veikko Pietarinen Department of Philosophy P.O. Box 9, FI-00014 University of Helsinki
[email protected]
Abstract. Diagrammatic logics have advantages over symbolic cousins. Peirce thought that logical diagrams (Existential Graphs, EG) are capable of “expression of all assertions”, as our reason is no longer limited to the “line of speech” (MS 654). This paper points out one such value: the economy resulting from combining multi-dimensional diagrams with multi-modal features. In particular, EGs are well-suited for representing and reasoning about non-declarative assertions, such as questions (interrogatives, vert), commands (e.g., imperatives, vair) and the compelled (potent). An advantage over symbolic-logical counterparts is multidimensionality that entitles recognition of non-declarative moods in an instantaneous fashion: there is no need to attune to the phonetics of expressions. The paper suggest an application of diagrammatic logic of commands to the cases where (i) minimal reaction time to commands is of essence, (ii) a full comprehension of the meaning of imperatives (‘search for their objects’) is needed, and (iii) an effective discrimination of commands from other non-declarative moods is critical. Keywords: diagrammatic logic, existential graphs, multi-dimensionality, multi-modality, tinctures, commands.
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Diagrammatic Logic of Existential Graphs
Peirce’s diagrammatic logic of Existential Graphs (EGs, [1,5]) is both visual (iconic) and formally rigorous. The expressive power goes up to higher-order modal logics. EGs can represent any assertion that has propositional content. 1.1
Multi-dimensionality
EGs are scribed on a multi-dimansional manifold, the Sheet of Assertion. For example, in the theory of Beta graphs, which corresponds to the theory of predicate logic with identity, the sheet is 4-dimensional. 1.2
Modality
EGs capture modalities in terms of a broken cut [3,5]. A broken cut does not compel the interpreter to admit that the graph P within its enclosure is true: G. Stapleton, J. Howse, and J. Lee (Eds.): Diagrams 2008, LNAI 5223, pp. 404–407, 2008. c Springer-Verlag Berlin Heidelberg 2008
Diagrammatic Logic of Existential Graphs: A Case Study of Commands
P possibly, not P
P
P
necessarily P
possibly P
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P possibly necessarily P
These correspond to the modal-logical formulas 3¬P , ¬3¬P , 3¬¬P and 3¬3¬P , respectively. Different modalities are represented by different colours or tinctures. 1.3
Multi-modality: Tinctured Graphs
Different modes of being, or different kinds of universes (actualities, possibilities, ignorance, power, futurity, intention, etc.) are represented as follows: P
11 00 00 11 P 00 11 00 11
11 00 P 00 11
1111 0000 0000 1111 P 0000 1111
11 00 00 11 00 11 P 00 11 00 11
111 000 P 000 111
11 00 00 11 P 00 11 00 11
Ordin.actuality
Subj.possibility Interrog.mood Metaph.necessity Imperative
Spec.actuality
Obj.possibility
11 00 00 11 00 11 P 00 11 00 11
Freedom/ Ability
111 000 000 111 000 111 P 000 111 000 111
Purpose/ Intention
11111 00000 00000 11111 00000 11111 P 00000 11111 00000 11111
The compelled
Peirce’s examples was “There is a Turk who is the husband of two different persons” (CP, 1906): husband Turk husband
111 000 000 111 000 111 000 111
That is, it would be contrary to what is known (subjective, epistemic possibility), by the one who scribes the graphs, that the two individuals are identical. Peirce confesses “that the use of the Tinctures is, in practice, perplexing” (MS 300: 40). Nevertheless, with tinctures, he anticipated alethic modal logic, epistemic logic, erotetic logic, deontic logic, belief-desire-intention logic, and the logic of commands. We study the last case. 1.4
Non-declarative Assertions
Since non-declarative assertions have propositional content, they can be represented and reasoned about by means of EGs.
2
Diagrammatic Logic of Commands
Commands are species of imperatives. Commands have propositional content, since their (immediate) objects are found in the intentions of those asserting the commands.1 In case the command is actually complied with, the (dynamic) object is in the excecution subsequent to the command. 1
Witness Peirce: “You may say, if you like, that the Object is in the Universe of things desired by the Commanding Captain at that moment. Or since the obedience is fully expected, it is in the Universe of his expectation” (CP 8.178).
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2.1
A.-V. Pietarinen
The Logic of Commands in EGs
Rescher [4] interprets commands in a tense-logical sense (‘Do now!’; ‘Do always!’) and proposes a notion of validity and inference. In EGs commands may be represented using colours and tinctures (vert). To exclame ‘If you love me, kiss me!!’ is to scribe: love me
11111 00000 00000 11111 kiss me 00000 11111 00000 11111 ‘Help all victims of war!!’ and ‘Help implies the existence of war’ do not entail ‘Help the existence of war!!’ (Good Samaritan Paradox thus resolved):
1111111 0000000 000 help war 111 0000000 1111111 victim 111 000 0000000 1111111 help 000 111 0000000 1111111 000 111 0000000 1111111
=⇒
111111111 000000000 victim help war 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111
The notion of satisfiability and validity is by possible-worlds semantics. Tinctures denote different sets of accessible worlds corresponding to different modalities. To reason about command graphs is to perform illative transformations, including cross-modal transformations (e.g., from the compelled to imperatives). Reasonings are iconic; they capture our ‘Moving pictures of thought’ [3]. Such graphs may be combined with tensed modalities, to deal with commands such as ‘Do p whenever (= now and henceforward) X is the case!’2
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Explanations and Applications
Such graphs are very amenable to applications. The logic is diagrammatic, and hence recognition of non-declarative moods is instantaneous. There is no reading or hearing sentences, vital to quick but reasoned human responses. Reaction time to commands is of essence in numerous applications. Time span also depends on two other qualities: Comprehension of imperatives (‘search for their objects’) means movement from immediate object of a command to its dynamic object. The objects are sets of possible worlds. Immediate objects are sets of possible worlds compatible with what the Commander intends. Dynamic objects are sets of possible worlds compatible with what the Executor brings about. Moreover: (i) Intersection of the two sets is larger for iconic than non-iconic representations. (ii) Movement from immediate to dynamic object is more effective when the iconic logic of commands is used than with linear symbolic sentences. (iii) Empty intersection is bypass, hence the logic is three-valued. 2
The logical form of ‘Whenever X happens, p happens’ is beyond the expressive power of first-order logic, and so a corresponding modification to graphs is needed [2].
Diagrammatic Logic of Existential Graphs: A Case Study of Commands
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Discrimination An effective and quick discrimination of commands from other non-declarative moods is critical whenever having to decipher the line of speech is too time consuming. Because iconicity, in EGs such discrimination is more effective and less error prone than symbols.
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Conclusions
It is to be expected that representing the logical structure of commands using iconic and diagrammatic notation of EGs comes close to the actual workings of our cognitive processes about their interpretation, reasoning, and excecution.
References 1. Peirce, C.S.: Collected Papers. Harvard University Press, Cambridge (1931-1958) 2. Pietarinen, A.: Peirce’s diagram logic in IF persp. In: Blackwell, A.F., Marriott, K., Shimojima, A. (eds.) Diagrams 2004. LNCS (LNAI), vol. 2980, pp. 91–97. Springer, Heidelberg (2004) 3. Pietarinen, A.-V.: Signs of Logic. Springer, Dordrecht (2006) 4. Rescher, N.: The Logic of Commands. Routledge, London (1966) 5. Roberts, D.D.: The Existential Graphs of C.S. Peirce. Mouton, The Hague (1973)
