Similarities between Euclidean Theorems and Syllogisms

Charlie Ma

An Euclidean theorem is similar to an syllogism in many aspects; they are usually interchangeable. They share the same fundament principle which is deriving a conclusion from a few assumptions. The details coincide so consistently that it can be said that Euclidean theorems are syllogisms

The premises of a syllogism is the assumption aspect. There are unlimited number of premises in an syllogism however there is a minimum of two. The premises are combined to form a conclusion. The premises are statements which may be deemed true or false. If the premises are all true, then the conclusion must be true. If any one, all of, or combination of premises are false, then the conclusion may be either true or false. For example:

Premise 1: All dumbbells made of steel are objects which are heavier than woods of the same volume.
Premise 2: All tables at Bob’s place are objects which are crafted from only wood.
Conclusion: All dumbbells are objects which are heavier than tables at Bob’s place.

When both premises are true, the conclusion must be true. There are two possibilities if one or both the premises are false:

Case 1: One table is produced entirely from plastics.
Result: Since coincidentally the plastics are also less dense than steel, conclusion remains true.

Case 2: One table is produced from Steel.
Result: Since steel cannot be heavier than itself, the conclusion is false.

Case 3: A special type of wood is found to be denser than steel, and one of the tables is produced from the special wood.
Result: Since one of the tables is produced from that special type of wood, the conclusion becomes false.

Case 4: A special type of wood is found to be denser than steel, and one of the tables is produced from plastics.
Result: Since none of the tables are produced from that special type of wood, the conclusion remains true (the plastic table is irrelevant to the conclusion.)

The validity of the conclusion is independent of the truthfulness of the premises. The conclusion is only valid when it is appropriately derived from the premises. For example:

Case 1:
Premise 1: All dumbbells made of steel are objects which are heavier than woods of the same volume.
Premise 2: All tables at Bob’s place are objects which are crafted from only wood.
Conclusion: All dumbbells are objects which are heavier than tables at Bob’s place.
Validity: Valid, since if all premises are true, the conclusion is true.

Case 2:
Premise 1: All chairs are blue.
Premise 2: All people become blue when they choke.
Conclusion: All people become chairs when they choke.
Validity: Invalid obviously, since even if both premises are true, the conclusion is not necessarily true.

The axioms of Euclidean theorems is the assumption aspect, much like the premise to the syllogism. There are also unlimited number of axioms, with minimum of two, analogous to the premises. There is a minimum since nothing can be derived from a single statement; this is the reasoning for both cases, and is the reason a minimum is required for both. The axioms are combined to form a theorem. For example:

Axiom 1 (Premise 1):It is possible to draw one and only one straight line from any point to any point.
Axiom 2 (Premise 2): From each end of a finite straight line it is possible to produce it continuously in a straight line by an amount greater than any assigned length.
Axiom 3 (Premise 3 ): It is possible to describe one and only one circle with any center and radius.
Axiom 4 (Premise 4 ): All right angles are equal to one another.
Axiom 5 (Premise 5 ): Through a given point not on a given straight line, and not on that straight line produced, no more than one parallel straight line can be drawn.
Theorem (conclusion): In any parallelogram the opposite sides and angles are equal to one another, and the diagonal bisects the area.

The axioms are statements which are assumptions which may actually be true or false. If the axioms are all true, then the theorem must be true. If any one, all of, or combination of axioms are false, then the theorem may be either true or false. For example, if axiom 4 was false and there could be different right angles, then the theorem would be false. The theorem would still remain valid much like in the case of syllogisms. A good example of the validity is the mutual existence of Euclidean geometry, hyperbolic geometry, and elliptic geometry. Under hyperbolic and elliptic geometry, the fifth premise is untrue but that does not make the theorem invalid. The truthfulness and validity of the theorem, and the dependents are identical to the syllogisms and the examples.

Syllogisms and Euclidean theorems have vast similarities. Axioms and premises are almost interchangeable much like theorems to conclusions. They followed the same rules in truthfulness and validity. Even virtually all the foundations are identical. Saying syllogisms and Euclidean theorems are similar is an understatement.