| 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 is derived..."
from
syl 15. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most
of the proof is in alsyl 1615. There are a legion of sources for Barbara,
including http://www.friesian.com/aristotl.htm 1615,
http://plato.stanford.edu/entries/aristotle-logic/ 1615, and
https://en.wikipedia.org/wiki/Syllogism 1615. (Contributed by David A.
Wheeler, 24-Aug-2016.) |
