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.
