Draft:Logical diagram

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A logical diagram is a diagram used to represent a logical proof, proposition or concept.[1]

History

Square_of_opposition,_set_diagrams

The development of diagrammatic logic began with the square of opposition diagram created by Boethius and Augustine[2] based on the 4 categorical propositions "Some S are P", "Some S are not P", "No S is P" and "All S are P". No further advancements in diagrammatic logic were made until the invention of Euler diagrams in the 18th century. Euler diagrams were intended to serve as visual representations of the various Aristotelian moods[3]. The next advance in diagrammatic logic came in the form of the Venn diagram in 1880[4].

Pierce's Role

See also: Existential graph
Charles_Sanders_Peirce

The most significant advancement in graphical logic in the 19th century was made by American logician and philosopher Charles Sanders Peirce who in 1882 created a system of graphical logic[5] referred to as existential graphs composed of three subsystems the alpha, beta and gamma graphs. Peirce was dissatisfied with the existing logical notation and wanted to create a system in which the signs resembled what they denoted.

Alpha graphs are isomorphic to Boolean algebra, beta graphs to first order logic with equality and gamma graphs are (partially) isomorphic to modal logic. Alpha graphs are both sound[6] and complete.

Modern developments

Semantic Tableaux

See also: Method of analytic tableaux

Semantic tableaux is a type of tree diagram which is used as a proof procedure for propositional and predicate calculus[7], a proof consists of splitting the given formula into several sub-formulae which are each either proven or refuted.

Conceptual Graphs

See also: Conceptual graph

Conceptual graphs are a graphical representation of first order logic formulae[8].

Euler and Venn diagrams

See also: Euler diagram
See also: Venn diagram

EulerDiagram

Euler and Venn diagrams are a notation for set membership and intersection.[9][10] Venn diagrams were created to be a more expressive version of Euler diagrams for the purpose of teaching[4] and would became widely used in probability, statistics, logic and set theory.

Advantages

Diagrammatic logic is seen as a more intuitive notation than symbolic logic and because of this is commonly used as a teaching tool[11]. Proofs in existential graphs specifically are much shorter than symbolic proofs.

Disadvantages

Diagrammatic logic has been seen as being overly cumbersome,[12] complex and not suited for practical use.

See also

References

  1. ^ "What is Logic Diagram and Truth Table?". online.visual-paradigm.com. Retrieved 2024-07-09.
  2. ^ Parsons, Terence. "The Traditional Square of Opposition". plato.stanford.edu. Retrieved 2024-07-10.
  3. ^ Shin, Sun-Joo; Lemon, Oliver; Mumma, John (2018), "Diagrams", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Winter 2018 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-07-10
  4. ^ a b Verburgt, Lukas M. (April 2023). "The Venn Behind the Diagram". Mathematics Today. Vol. 59, no. 2. Institute of Mathematics and its Applications. pp. 53–55.
  5. ^ Peirce, C. S., "[On Junctures and Fractures in Logic]" (editors' title for MS 427 (the new numbering system), Fall–Winter 1882), and "Letter, Peirce to O. H. Mitchell" (L 294, 21 December 1882), Writings of Charles S. Peirce, v. 4, "Junctures" on pp. 391–393 (Google preview) and the letter on pp. 394–399 (Google preview). See Sowa, John F. (1997), "Matching Logical Structure to Linguistic Structure", Studies in the Logic of Charles Sanders Peirce, Nathan Houser, Don D. Roberts, and James Van Evra, editors, Bloomington and Indianapolis: Indiana University Press, pp. 418–444, see 420, 425, 426, 428.
  6. ^ "Existential Graphs". www.jfsowa.com. Retrieved 2024-07-10.
  7. ^ Howson, Colin (1997). Logic with trees: an introduction to symbolic logic. London ; New York: Routledge. pp. ix, x, 24–29, 47. ISBN 978-0-415-13342-5.
  8. ^ Sowa, John F. (1984). Conceptual structures : information processing in mind and machine. Internet Archive. Reading, Mass. : Addison-Wesley. ISBN 978-0-201-14472-7.
  9. ^ Shin, Sun-Joo; Lemon, Oliver; Mumma, John (2018), "Diagrams", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Winter 2018 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-07-09
  10. ^ "What is Euler Diagram?". online.visual-paradigm.com. Retrieved 2024-07-10.
  11. ^ "ReadingQuest Strategies | Venn Diagrams". 2009-04-29. Archived from the original on 2009-04-29. Retrieved 2024-07-12.
  12. ^ Quine, W. V. (February 1935). "Collected Papers of Charles Sanders Peirce.--Volume IV: The Simplest Mathematics. Charles Hartshorne , Paul Weiss". Isis. 22 (2): 551–553. doi:10.1086/346931. ISSN 0021-1753.

External links

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