To avoid association of a letter with a word in this context, I choose letters A, B, C to denote three possibly different statements or propositions. The syllogistic major premise involves B, C; the syllogistic minor premise involves =>D, B. Then we have:

FIGURES OF THE SYLLOGISM

	I	II	III	IV
MAJOR PREMISE	BC	CB	BC	CB
MINOR PREMISE	DB	DB	BD	BD
CONCLUSION	DC	DC	DC	DC

Terms of the FIGURES are then put in moods a, e, i, o, representing, respectively, universal (all), existential (at least one), positive, negative (none). This allows for 4 x 4 x 4 x 4 = 256 possible syllogism, most of which are invalid or redundant.

When redundancies and invalidities are eliminated, FIGURE IV of the TABLE has vanished, and 6 VALID FORMS remain:
(I) a,a,a; e,a,e; a,i,i; e,i,o
(II) a,o,o
(IV) o,a,o. For example, the follwing oft-quoted (valid) syllogism is of the form aee:

				All men are mortal.
				Socrates is a man.
                                _____________________________
				Therefore, Socrates is mortal.

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In 1854 George Boole (1817-1864) published An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities. Known as The Laws of Thought, it presents Boole's algebraization of logic, solving by equations rather than syllogisms.

• Arbitrary letters denote set of objects, say, x, y, z.
• Let xy denote set of objects shared by sets x, y.
• Let x + y denote the set of objects belonging to x or to y or both.
• Let 0 denote the empty set; let 1 denote the universal set (set of all objects). Hence, x = 0 means set x has no members; and 1 - x denotes set of all objects not in set x.
• This algebra of logic has the rules or laws:
  • xy = yx; x + y = y + x. (Commutativity.)
  • x(yz) = (xy)z; x + (y + z) = (x + y)+ z. (Associativity<.)
  • x(y + z) = xy + xz. (Distributivity.)
  • x + 0 = x; 1x = x. (Identity.)
  • 2x = x + x = x; x² = xx = x. (Idempotency.)

All these laws except the last one (Idempotency) resemble laws in numerical algbra. The Idempotency law can be understood of we irst substitute zero, then substitue one in these equations:

• 2 * 0 = 0 + 0 = 0; 0² = 0 * 0 = 0; 1¹ = 1 * 1 = 1;
• and the other one, 2 * 1 = 1 + 1 = 1 can be understood since "1" denotes "everything".

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Using Boole's algebra of logic, we can write the syllogistic Figures with a, e, i, o as follows:

• S a P: s(1 - p) = 0.
• S e P: sp = 0.
• S i P: sp ¹ 0.
• S o P: s(1 - p)¹ 0.

The "Socrates" syllogistic second premise can be written either in e mood or in a mood. But it is convenient for the algebraic calculation to write it in the a mood. Then we have:

  				M(1 - P) = 0. [1]
				S(1 - M) = 0. [2]
                                _____________
                                S(1 - P) = 0.

From Boolean equation [1], we have: M - MP = 0 Þ M = MP. [3]

From Boolean equation [2], we have: S - SM = 0 Þ S = SM. [4]

Replacing from [3] in [4], we have: S = SM Þ S = S(MP) = SM(P) by associativity. Again, by commutativity, this becomes: S = MS(P). [5]

Replacing on the left of [5] from [4], we have: SM = MS(P), which, by commutativity, becomes: MS = MS(P). [6]

Eliminating M from both sides of [6], we have: S = SP. [7]

Subtracting SP from both sides of [7], we have: S - SP = 0, which, by distributivity, becomes S(1 - P) = 0, the form of the Syllogistic CONCLUSION.

Hence, we have derived the equation of the CONCLUSION from the equations of the PREMISES, so, THE ARGUMENT IS VALID.

STATEMENT LOGIC HAS BECOME AN ALGEBRA!