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I am attempting to analyze Euclid's proof demonstrating that the interior angles of a triangle are equal to two right angles (book 1, prop 32). In particular, I'm looking for a way to convert the arguments into a more standard syllogism or syllogisms and find the major, minor, and middle premises, and to fit the premises into recognizable A, E, I, and O formats. Does anyone know how I might go about doing this?

So far I have attempted portraying it as a series of conditional syllogisms, for example "If a line can be drawn through a given point parallel to a given line [I.31] then line CE can be drawn through the point C parallel to the straight line AB." However, I am still not sure if the major and minor terms are evident in this example.

Appreciate any help y'all might have, thanks for your help!

For your reference, here is the entirety of prop 32:

Let ABC be a triangle, and let one side of it BC be produced to D; I say that the exterior angle ACD is equal to the two interior and opposite angles CAB, ABC, and >the three interior angles of the triangle ABC, BCA, CAB are equal to two right angles.

For let CE be drawn through the point C parallel to the straight line AB. [I. 31]

Then, since AB is parallel to CE,

and AC has fallen upon them, the alternate angles BAC, ACE are equal to one another. [I. 29] Again, since AB is parallel to CE,

and the straight line BD has fallen upon them, the exterior angle ECD is equal to the interior and opposite angle ABC. [I. 29] But the angle ACE was also proved equal to the angle BAC;

therefore the whole angle ACD is equal to the two interior and opposite angles BAC, ABC. Let the angle ACB be added to each;

therefore the angles ACD, ACB are equal to the three angles ABC, BCA, CAB. But the angles ACD, ACB are equal to two right angles; [I. 13]

therefore the angles ABC, BCA, CAB are also equal to two right angles. Therefore etc.

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    As is well known, Syllogistic uses only unary relations, i.e. predicates. In the above proof we have equality: "ACD is equal to the two interior and opposite angles BAC, ABC" as well as ternary relations: "Let CE be drawn through the point C parallel to the straight line AB."
    – Mauro ALLEGRANZA
    Commented Mar 19 at 6:48
  • On historical attempts to 'syllogize' Euclid by Herlinus, Clavius, Jungius, Vagetius, Leibniz, and why they did not work, see De Risi, pp. 28-29. On the positive side, you can analyze Euclid's proof into a chain of inferences, but some of them will be more complex than syllogisms to handle binary and ternary geometric relations. To do so, look at Hilbert's proofs in his Foundations of Geometry that supplement Euclid's axioms to fully formalize Euclidean geometry.
    – Conifold
    Commented Mar 19 at 10:53
  • See Fabio Bertato. ATTEMPTS to FORMALIZE as SYLLOGISMS the Propositions of the ELEMENTS of EUCLID (2018)
    – Mauro ALLEGRANZA
    Commented Mar 19 at 15:50
  • And Joseph Novak, A Geometrical Syllogism (1978)
    – Mauro ALLEGRANZA
    Commented Mar 19 at 15:56

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I think you are wasting your time on this exercise. Aristotle's logic is too weak for the task you propose. Before the invention of predicate logic, some scholars attempted to express Euclid's geometry in formal logic using Aristotelian syllogisms, but it doesn't work. Those who tried it found that there were inevitabe gaps that have to be filled in with inferences from diagrams.

One of the main limitations of Aristotle's logic is that it can express properties of things, but not relations between things. The A, E, I and O propositions are all statements of properties. But they cannot express relations like 'is taller than' or 'is adjacent to' or 'likes', etc. Also, those propositions are limited to one quantifier only. They can have a single occurrence of 'all', 'some' or 'no', but that's all. They cannot express things like, "every boy loves some girl", or "there is some girl that every boy loves", nor demonstrate the logical relations between those.

This is important because in geometry we have to be able to express relations. For example, "point A lies between point B and point C" makes use of a three-place relation, 'lies between'. Similarly, 'is parallel to' and 'is congruent with' are relations. A ratio is a relation.

There are many other limitations to Aristotelian logic. It can be thought of as being approximately a fragment of first order predicate logic with the restriction of monadic predicates only and a single quantifier per sentence. Compared with the full generality of first order predicate logic, it is extremely weak. Aristotelian logic can only do a tiny, tiny fraction of what can be achieved in predicate logic, which is why it is hardly used by logicians any more.

There is a way to extend Aristotelian logic called relational syllogistic. You may be able to use that, but I'm not sure what the value would be. The initial motivation for the invention of predicate logic was to provide a good way to represent mathematical propositions and proofs. It would be better to use that.

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    Great answer. (filler so it let's me post the comment)
    – David Gudeman
    Commented Mar 19 at 6:27

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