Sep 25-9 - Boole’s Logic

1. Set Theory

• Symbols:

∅ = empty set

ε = universe of discourse

U = union

∩ = intersection

X’ = complement

• ‘Laws of Set Theory’ – prove them!

X U Y = Y U X

X U (Y∩Z) = (XUY) ∩ (XUZ)

X U ∅ = X

X ∩ ε = X

X ∩ X’ = ∅

• Saying things with sets:

E.g., let ε = cars, X = red cars, Y = fast cars, Z = BMWs

X U Y –

Y ∩ Z –

X = ε –

Y = ∅ –

X ∩ Z = ∅ –

X ∩ Y ≠ ∅ –

X’ ∩ Y = ∅ –

X U Y = X –

2. Boolean Logic

• The world is composed of sets of things (the sets replace properties) and so reasoning about things is reasoning about sets – but reasoning is the algebraic manipulation of ideas. So logic is the algebra of sets.

• Translating set theoretic equations into algebraic ones – x, y, z …stand for sets, 1 = ε, 0 = ∅, + = U, ⋅ = ∩.

Translate our examples into algebraic form.

• The rules – p102-3 + substitution.

Translate our ‘laws’ into algebraic form – how do the rules of Boolean algebra differ from ordinary algebra? (That’s because it’s an algebra of set relations, not of numbers!)

• Prove:

x + 1 = 1

x + x = 1 (Q2, p102)

x.y = x ⇔ x.(1-y) = 0 ⇔ x + y = y – i.e., that these all say ‘All Xs are Ys’

• Boole’s logic of sets contains Aristotle’s logic of the syllogism:

Prove – Barbara, Cesare

3. Boolean Propositional Logic

• Suppose that the universe of discourse contains just one thing – ε = {‘TRUTH’}, so ∅ is the absence of truth. Further suppose that true sentences are names for ε, and false sentences names ∅. In algebraic terms, if x is a sentence (e.g., Bush is President):

x=1 means x is true

x=0 means x is false

• Now, x.y = 1 only if x=y=1; if x=0 or y=0 then x.y=0 – suggesting . has the logic of ‘and’.

What about +? (1-x)?

How might we express ‘if x then y’?

• Translate and give an algebraic proof of the following: “Either the Butler did not murder both Xavier and Yiorgos, or Agatha did not murder Zane. If the Butler murdered Xavier and Agatha is innocent of Zane’s murder, then Yiorgos murdered Zane. If Yiorgos murdered Zane then the Butler did not murder Yiorgos. Therefore, if the Butler murdered Xavier then he didn’t murder Yiorgos.”

4. Boole’s Psychologism:

• If logic is the rules of thought, and the rules of thought are as necessary as the laws of physics, then we could answer Glymour’s third question (how can there be deductively valid arguments?). But why can’t the laws of logic be understood as the ‘laws of the mind’ on the model of physics?

• What other short-comings does Boole’s theory have?