Monday, February 11, 2013

Logic Geometrically

(1) Let terms be represented by lines.

(2) Where two lines are parallel, the corresponding terms exclude each other. For two lines to be parallel is for no part of one to be part of the other.

(3) Where two lines intersect, the terms at least partly overlap. For two lines to intersect is for part of one to be part of the other.

(4) Part of part of a line is part of that line.

(5) Lines that intersect a line in whatever way are not thereby assumed to intersect its parallels at any point. We should think of the lines as finite line segments or arcs.

(6) We can then give geometrical representation to each of the four families of categorical propositions.

A (Every S is P): the whole line S is at least part of the line P
----(----)S----P

[There's no easy way to represent this in this post. I simply draw a line for P, and then put parentheses on the line, labeled S, to indicate that the whole line S is at least part of P

E (No S is P): the whole line S is parallel to the line P
----------------S

----------------P

I (Some S is P): the line S intersects the line P
\
  \
    \
------------P
       \
         \S

[This is obviously also difficult to represent simply by a keyboard, although it is easy to draw. One thing that we have to be careful about is that the geometrical representation of logic used here does not care about any angles beyond whether they are equal to 0. Thus an I proposition, despite its appearance in the above representation, is consistent with an A proposition: the A proposition is just the I proposition with a zero degree angle between the subject and the predicate, so that the subject line is at least part of the predicate line. As such it might be more convenient to represent an I proposition as being one of two possible diagrams, the one just given and the diagram for A propositions. If we do this, the above diagram is actual for the exclusive rather than the standard particular: it tells us not that Some S is P but that Only some S is P. I haven't decided which is the handier way of doing it.]

O (Some S is not P): the line S intersects a line parallel to P
------------P

\
  \
    \
------------nonP
       \
         \S

(7) Conversion is built into the representation, as are some of the other immediate inferences.

Obversion of A: If the whole line S is part of the line P, the whole line S is parallel to any line parallel to P.
Obversion of E: If line S is parallel to the line P, the whole line S is at least part of a nonP line.
Obversion of I: If the line S intersects line P, at least part of the line S intersects a line parallel to nonP.
Obversion of O: If the line S intersects a line parallel to P, at least part of the line S is at least part of a nonP line.

Contraposition of A: If the whole line S is part of the line P, the whole line nonS is part of a nonP line.
Contraposition of O: If the line S intersects a line parallel to P, the line S intersects a nonP line.

(8) The Barbara Syllogism (Every M is P, Every S is M; therefore Every S is P):

----(----(----)S----)M----P

The Celarent Syllogism (No M is P; Every S is M; Therefore No S is P)

----(----)S----M

------------P

The Ferio Syllogism (No M is P; Some S is M; therefore Some S is not P)

------------P

\
  \
    \
------------M=nonP
       \
         \S

The Darii Syllogism (All M is P; Some S is M; therefore Some S is P)

  \
    \
      \
----(----)M----P
          \
            \S

(Also difficult to represent with the keyboard, although very easy to draw: just draw line P, then mark at least part of P M, and draw line S intersecting line M.)

All other syllogisms are reducible to these in the standard way; we could even draw out our diagrams of these other syllogisms and reason geometrically to the relevant First Figure syllogism.

(9) This is similar to a kind of diagramming developed by George Englebretsen (as he notes, we have reason to think that Aristotle actually used line diagrams at least occasionally in his work on logic, but we don't have a precise idea of how he used them); but he doesn't press the fact that you could reason like Euclid with them beyond some obvious ways, whereas I think the geometrical element can be taken quite far. The geometry wouldn't be Euclid's, but it would be a genuine geometry, and it would directly represent the entire syllogistic apparatus. We can then read categorical syllogisms as construction-instructions. For instance:

Given: Every dog is a mammal; some pets are dogs.
To Prove: Some pets are mammals.

Proof:

  \
    \
      \
----(----)dog----mammal
          \
            \pet

Line dog is part of line mammal. Line pet intersects line dog. But part of part of a line is part of that line. Therefore line pet intersects line mammal.

Therefore from Every dog is a mammal and some pets are dogs we have proven that some pets are mammals, QED.

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