Make Electronics

Experiment 19: Learning Logic

184 Chapter 4

BAckground

From    Boole   to  Shannon

George Boole was a British mathematician, born in 1815, who did something

that few people are ever lucky enough or smart enough to do: he invented an

entirely new branch of mathematics.

Interestingly, it was not based on numbers. Boole had a relentlessly logical

mind, and he wanted to reduce the world to a series of true-or-false statements

which could overlap in interesting ways. For instance, suppose there is a couple

named Ann and Bob who have so little money, they only own one hat. Clearly,

if you happen to run into Ann and Bob walking down the street, there are four

possibilities: neither of them may be wearing a hat, Ann may be wearing it, or

Bob may be wearing it, but they cannot both be wearing it.

The diagram in Figure 4-51 illustrates this. All the states are possible except the

one where the circles overlap. (This is known as a Venn diagram. I leave it to you

to search for this term if it interests you and you’d like to learn more.) Boole took

this concept much further, and showed how to create and simplify extremely

complex arrays of logic.

No one

wears

the hat

Ann wears

the hat

Bob wears

the hat

Both

wear thehat

Figure 4-51. This slightly frivolous Venn diagram illustrates the various possibilities

for two people, Ann and Bob, who own only one hat.

Another way to summarize the hat-wearing situation is to make the “truth table”

shown in Figure 4-52. The rightmost column shows whether each combination

of propositions can be true. Now check the table in Figure 4-53. It’s the same

table but uses different labels, which describe the pattern you have seen while

using the NAND gate.

Boole published his treatise on logic in 1854, long before it could be applied

to electrical or electronic devices. In fact, during his lifetime, his work seemed

to have no practical applications at all. But a man named Claude Shannon

encountered Boolean logic while studying at MIT in the 1930s, and in 1938 he

published a paper describing how Boolean analysis could be applied to circuits

using relays. This had immediate practical applications, as telephone networks

were growing rapidly, creating complicated switching problems.