| Description: "Barbara", one
of the fundamental syllogisms of Aristotelian logic. All
𝜑 is 𝜓, and all 𝜒 is 𝜑, therefore all 𝜒 is
𝜓. In Aristotelian notation, AAA-1:
MaP and SaM therefore SaP.
For example, given "All men are mortal" and "Socrates is
a man", we can
prove "Socrates is mortal". If H is the set of men, M is the
set of
mortal beings, and S is Socrates, these word phrases can be represented
as ∀𝑥(𝑥 ∈ 𝐻 → 𝑥 ∈ 𝑀) (all men are mortal) and
∀𝑥(𝑥 = 𝑆 → 𝑥 ∈ 𝐻) (Socrates is a man) therefore
∀𝑥(𝑥 = 𝑆 → 𝑥 ∈ 𝑀) (Socrates is mortal). Russell and
Whitehead note that "the syllogism in Barbara [barbara 2687] is derived
from [syl 18]" (quote after Theorem *2.06 of [WhiteheadRussell]
p. 101). Most of the proof is in alsyl 1926. There are a legion of
sources for Barbara, including http://www.friesian.com/aristotl.htm 1926,
http://plato.stanford.edu/entries/aristotle-logic/ 1926, and
https://en.wikipedia.org/wiki/Syllogism 1926. (Contributed by David A.
Wheeler, 24-Aug-2016.) |
