The Algebra of Logic, by Louis Couturat potx

2 + 3 = 5
(a + b)c = ac + bc ;
<
<
≤
∈
a b a < b
a b
b a
b
a b
b a
a b
a b a < b
a b
a b a b
a b <
a < b
a b <

<
=
a = b,
a < b b < a
a = b a < b b < a
=
a = b a b b a
a b
a = b a b b a
a b
a = b b = a
a < b b < a
a > b b < a
<
a < b a
b b a
a b
a < a,
a
a a
a a
a = a,
a
a < a, a < a,
a = a.
a
a
(a < b)(b < c) < (a < c).
a b b c a c
a b b c a c

(a < b) (b = c) < (a < c),
(a = b) (b < c) < (a < c),
(a = b) (b − c) < (a = c).
(a < b) (b < c) (c < d) < (a < d),
(a = b) (b = c) (c = d) < (a = d).
a b p
p < a, p < b,
x
x < a, x < b,
x < p.
a b s
a < s, b < s,
s < x.
p s
ab a + b
p s
p
a b s
<
a b a b
a b
a b
ab < a, ab < b,
(x < a)(x < b) < (x < ab),
a < a + b, b < a + b,
(a < x)(b < x) < (a + b < x).
(x < ab)(ab < a) < (x < a),
(x < ab)(ab < b) < (x < b).
(x < ab) < (x < a)(x < b).
(a < a + b)(a + b < x) < (a < x),

(b < a + b)(a + b < x) < (b < x).
x + y x y
(a + b < x) < (a < x)(b < x).
(x < ab) = (x < a)(x < b),
(a + b < x) = (a < x)(b < x).
x ab
a b x a + b
a b
ab = ba,
(ab)c = a(bc),
a + b = b + a,
(a + b) + c = a + (b + c).
a = aa, a = a + a.
aa < a,
(a < a)(a < a) = (a < aa)
(aa < a)(a < aa) = (a − aa)
a < a + a,
(a < a)(a < a) = (a + a < a),
(a < a + a)(a + a < a) = (a = a + a).
a + ab = a, a(a + b) = a.
(a < a)(ab < a) < (a + ab < a),
a < a + ab,
(a + ab < a)(a < a + ab) = (a + ab = a).
(a < a)(a < a + b) < [a < a(a + b)],
a(a + b) < a,
[a < a(a + b)][a(a + b) < a] = [a(a + b) = a].
(a) (ab)
(a + b)
(a < b) < (ac < bc), (a < b) < (a + c < b + c).
ac < c,

(ac < a)(a < b) < (ac < b),
(ac < b)(ac < c) < (ac < bc).
c < b + c,
(a < b)(b < b + c) < (a < b + c).
(a < b + c)(a < b + c) < (a + c < b + c).
(a = b) < (ac = bc), (a = b) < (a + c = b + c).
(a < b)(c < d) < (ac < bd),
(a < b)(c < d) < (a + c < b + d).
(ac < a)(a < b) < (ac < b),
(ac < c)(c < d) < (ac < a),
(ac < b)(ac < d) < (ac < bd).
(a < b)(b < b + d) < (a < b + d),
(c < d)(d < b + d) < (c < b + d),
(a < b + d)(c < b + d) < (a + c < b + d).
(a = b)(c < d) < (ac < bd),
(a = b)(c < d) < (a + c < b + d).
(a = b)(c = d) < (ac = bd),
(a = b)(c = d) < (a + c = b + d).
(a < b) = (a = ab) (a < b) = (a + b = b)
(a < b) < (a = ab), (a < b) < (a + b = b).
(a < a)(a < b) < (a < ab),
(a < b)(b < b) < (a + b < b).
ab < a, b < a + b,
(a < ab)(ab < a) = (a = ab)
(a + b < b)(b < a + b) = (a + b = b).
(a = ab) < (a < b), (a + b = b) < (a < b).
(a − ab)(ab < b) < (a < b),
(a < a + b)(a + b = b) < (a < b).
(a = b) = (ab = a + b).
(a = b) = (a < b)(b < a),

(a < b) = (a = ab), (b < a) = (a + b = a),
(a = ab)(a + b = a) < (ab = a + b).
(ab = a + b) < (a + b < ab),
(a + b < ab) = (a < ab)(b < ab),
(a < ab)(ab < a) = (a = ab) = (a < b),
(b < ab)(ab < b) = (b = ab) = (b < a),
(ab = a + b) < (a < b)(b < a) = (a = b).
(a < b) = (a = ab)
(b < c) = (b = bc) b
a = abc a ab ab = abc
ab < c a ab a < c
ac + bc < (a + b)c, ab + c < (a + c)( b + c).
(a < a + b) < [ac < (a + b)c],
(b < a + b) < [bc < (a + b)c];
[ac < (a + b)c][bc < (a + b)c] < [ac + bc < (a + b)c]
(ab < a) < (ab + c < a + c),
(ab < b) < (ab + c < b + c),
(ab + c < a + c)(ab + c < b + c) < [ab + c < (a + c)(b + c)].
(a + b)c < ac + bc, (a + c)(b + c) < ab + c,
(a + b)c < ac + bc.
(a + b)c = ac + bc
(a + b)(c + d) = ac + bc + ad + bd,
(a + c)(b + c) = ab + c.
(a + c)(b + c) = ab + ac + bc + c,
ac + bc + c = c.
(a + c)(b + c) < ab + c,
ab + ac + bc = (a + b)(a + c)(b + c),
(a + b)(a + c)(b + c) = (a + bc)(b + c) = ab + ac + bc.
x
0 < x.

x
x < 1.