New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  barbara GIF version

Theorem barbara 2301
 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, http://plato.stanford.edu/entries/aristotle-logic/, and https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.)
Hypotheses
Ref Expression
barbara.maj x(φψ)
barbara.min x(χφ)
Assertion
Ref Expression
barbara x(χψ)

Proof of Theorem barbara
StepHypRef Expression
1 barbara.min . 2 x(χφ)
2 barbara.maj . 2 x(φψ)
3 alsyl 1615 . 2 ((x(χφ) x(φψ)) → x(χψ))
41, 2, 3mp2an 653 1 x(χψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557 This theorem depends on definitions:  df-bi 177  df-an 360 This theorem is referenced by:  celarent  2302  barbari  2305
 Copyright terms: Public domain W3C validator