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Examples Applying Boole’s Algebra of Logic

Now we look at a selection of examples from Boole’s two books to seejust how his methods work.

Aristotelian Logic

Boole used the following translations for the categoricalpropositions:

Propositions		MAL (1847, p. 26)	LT (1854, p. 64)
\(A\)	All \(X\) is \(Y\)	\(x(1-y) = 0\)	\(x = vy\)
\(E\)	No \(X\) is \(Y\)	\(xy = 0\)	\(xy = 0\)
\(I\)	Some \(X\) is \(Y\)	\(v = xy\)	\(vx = vy\)
\(O\)	Some \(X\) is not \(Y\)	\(v = x(1-y)\)	\(vx = v(1-y)\)

Example 1. A conversion by limitation argument:

Argument	MAL (1847, p. 27)	LT (1854, p. 229)
All \(X\) is \(Y\)	\(x(1-y) = 0\)	\(x = vy\)
Some \(Y\) is \(X\)	\(x = vy\)	\(vx = vy\)

To justify this via the algebra of logic inMAL, Booleinvoked an argument based on solving \(x(1-y) = 0\) for \(x\). Ituses the method of division discussed below: from \(x = 0/(1-y) =(0/0)y + (0/1)(1-y)\) one has \(x = vy\). The reader ofMALmust wait till near the end of the book to learn the details of thismethod.

InLT Boole simply multiplied the first equation by \(v\) toobtain \(vx = vvy = vy\).

Example 2. A categorical syllogism:

(This example appears inMAL, but with the letters andpremises rearranged.)

Argument	MAL (1847, p. 34)	LT (1854, p. 231)
All \(X\) is \(Y\)	\(x(1-y) = 0\)	\(x = vy\)
All \(Y\) is \(Z\)	\(y(1-z) = 0\)	\(y = v'z\)
All \(X\) is \(Z\)	\(x(1-z) = 0\)	\(x = vv'z\)

InMAL Boole used a weak elimination procedure (whichhappened to work in this case) to justify this. The full-strengthelimination procedure, part of the General Method ofLTdescribed above, would proceed as follows:

\(x(1-y) + y(1-z) = 0\)	Reduction to a Single Equation
\([x(1-1) + 1(1-z)] [x(1-0) + 0(1-z)] = 0\)	Eliminate \(y\)
\((1-z)x = 0\)	Simplify (ordinary algebra)
\(x(1-z) = 0\)	Commute (ordinary algebra)

In Chapter XV ofLT Boole first gave a quick proof by simplysubstituting to obtain \(x = v(v'z) = vv'z\), a secondary form he usedto express “All \(X\) is \(Z\)”. Then he stated a generalresult designed to cover all syllogisms, justified by applying hisGeneral Method. Boole did not give full details of a proof of thisgeneral result, which are rather demanding. Here are the details forhow one applies Boole’s General Method to this particularexample.

\((x-vy)^2 + (y-v'z)^2 = 0\)	Reduction to a Single Equation
\([(x -v)^2 + (1-v'z)] [x + v'z] = 0\)	Eliminate \(y\)
\((2 - v - v'z)x = 0\)	Simplify (ordinary algebra + idempotent law)
\(x = (0/0)vv'z\)	Solve for \(x\) by division

So the conclusion equation would look like \(x = wvv'z\), instead of\(x = vv'z\) as given above.

Example 3. A hypothetical syllogism (theconstructive conditional syllogism):

Argument	MAL (1847, p. 56)	LT (1854)
If \(X\) then \(Y\)	\(x(1-y) = 0\)	—
\(X\)	\(x = 1\)	—
\(Y\)	\(y = 1\)	—

The ad hoc justification is: just substitute \(x = 1\) into the firstequation.

Using the General Method ofLT from Section 6.2 one has:

\(x(1-y) + (1-x) = 0\)	Reduction to a Single Equation
\(1-xy = 0\)	Simplify (ordinary algebra)
\(1(1-y) = 0\)	Eliminate \(x\)
\(y = 1\)	Simplify (ordinary algebra)

Example 4. Another hypothetical syllogism (thenon–exclusive disjunctive syllogism):

Argument	MAL (1847, p. 56)	LT (1854, p. 169,170)
\(X\) or \(Y\) is true	\(x + y - xy = 1\)	\(x(1-y) + y(1-x) = 1\)
\(X\) is false	\(x = 0\)	\(x = 0\)
\(Y\) is true	\(y = 1\)	\(y = 1\)

(Boole did not state this particular example inLT; theright column above gives the algebraic translations inLT ofthe propositions on the left.)

A substantial example fromLT

The following example in Chapter VII ofLT (pp.106–112) gives a better illustration of the workings ofthe General Method. Boole considered the following definition ofwealth [LT p. 106]:

Wealth consists of things transferable, limited insupply, and either productive of pleasure or preventive ofpain.

Then he used the following symbols to denote the 5 classes in thisproposition:

Symbol	Proposition
\(W\)	the class of things that constitute wealth
\(T\)	the class of things that are transferable
\(S\)	the class of things that are limited in supply
\(P\)	the class of things that are productive of pleasure
\(R\)	the class of things that are preventitive of pain.

He put the above definition of wealth in equational form [LTp. 106]:

\(w = st(p + r(1-p))\)

Six questions were posed and answered by Boole, the first two being:

1. “What is the expression for \(w\) in terms of \(p, s\) and\(t\) after eliminating \(r\)?” [LT pp. 106–107,112]

2. “What is the expression for \(s\) in terms of \(w, p\) and\(t\) after eliminating \(r\)?” [LTpp. 107–110]

Boole answered these two as follows, where intermediate steps thatBoole skipped over have been added — an asterisk (*) follows eachequation that is stated exactly as inLT:

1.	\(w - st (p + r- rp ) = 0\)	(*) write premiss as:something \( = 0\)
2.	\((w - st ) (w - stp ) = 0\)	(*) eliminate \(r\) in 1
3.	\(w^2 - wstp -stw + s^2 t^2 p = 0\)	Multiply out 2
4.	\(w - wstp - wst + stp = 0\)	(*) Simplify 3
5.	\((1 - st)w + pst(1-w) = 0\)	(*) expand 4 with respect to \(w\)
6.	\((1 - st)w = 0\), and \(pst(1-w) = 0\)	(*) terms in 5 are disjoint

Boole translated 6 as:

Wealth is limited in supply and transferable, andthat which is productive of pleasure and limited in supply andtransferable is wealth.

7.	\(w(1 - st - stp) + stp = 0\)	from 4, by ordinary algebra
8.	\(w = stp/(st+stp - 1) = stp + (0/0)st(1- p)\)	solve 7 for \(w\) by division

Boole translated 8 as:

“Wealth consists of all things limited in supply,transferable, and productive of pleasure, and an indefinite remainderof things limited in supply, transferable, and not productive ofpleasure.”

9.	\(s = w/(wt + wtp - tp) =\) \(wtp + wt(1- p)\ +\) \((1/0)w(1-t)p\ +\) \((1/0)w(1- t)(1- p)\ +\) \(0(1-w)tp\ +\) \((0/0)(1- w)t(1-p)\ +\)\((0/0)(1-w)(1-t)p\ +\) \((0/0)(1-w)(1-t)(1-p)\)	(*) finish division process
	\(0 = w(1-t)p\) \(0 = w(1-t)(1-p)\)	(*) 2 side conditions

Boole translated the expression for \(s\) and the two side conditions of9 as:

“All wealth transferable and productive ofpleasure—all wealth transferable, and not productive ofpleasure,—an indefinite amount of what is not wealth, but iseither transferable, and not productive of pleasure, or intransferableand productive of pleasure, or neither transferable nor productive ofpleasure.”