# Infinity as a truth value

I seem to recall that Claude Shannon, in developing his symbolic theory of electrical networks, remarked in a footnote that for some purposes it would make more sense to represent truth by infinity rather than by 1 (consider the resistance of an open switch). Unfortunately I can no longer locate this passage in either Shannon's master's thesis or the published version. Can anyone help me find it?

Also, has anyone developed Shannon's remark at greater length?

NOTE: In giving a careful reading of both Shannon's master's thesis (available from https://dspace.mit.edu/handle/1721.1/11173) and his 1938 paper (https://www.cs.virginia.edu/~evans/greatworks/shannon38.pdf) after posting my question, I found that Shannon used 1 to represent an open switch and 0 to represent a closed switch, not the other way around. I gather that this was reversed at some point (so that electrical networks are nowadays seen as "admittance networks" rather than "impedance networks"). Can someone tell me how this reversal in came about?

Shannon's Master's thesis contains the following on page 4:

We shall limit our treatment to circuits containing only relay contacts and switches, and therefore at any given time the circuit between any two terminals must be either open (infinite impedance) or closed (zero impedance). Let us associate a symbol Xab or more simply X, with the terminals a and b. This variable, a function of time, will be called the hinderance of the two terminal circuit a-b. The symbol 0 (zero) will be used to represent the hinderance of a closed circuit, and the symbol 1 (unity) to represent the hinderance of an open circuit.

An open switch cannot conduct at all (infinite impedance). And a closed switch is idealized to be a perfect conductor hence has no impedance at all (zero impedance). Shannon provides this as motivator transition to a symbolic description of 0 and 1 where indeed infinite impedance is mapped to 1 and zero impedance is mapped to zero. Interestingly he transitions to a language of hinderance.

It is a particular reading of this passage that suggests that Shannon could have picked 0 and $$\infty$$ as binary symbols, but it's not explicitly expressed. Rather it appears to me that it goes the other way that Shannon looks to find a binary state and motivates it by impedance, where the 0 does match but the infinity doesn't and it is symbolically captured by the 1.

This is not irrelevant however, given that Shannon will give concrete serial and parallel gate realizations where indeed open switches map to $$1$$ and closed switches map to $$0$$, so an electrical engineer has a mental crutch using impedances to understand the logical functioning of the circuitry. Hence under this mapping the circuits will execute the underlying boolean logic.

It is worth noting that a closed line is less visually striking than the open (interrupted) one, hence picking the open one may well have expository value! We have duality, so either one works, but one just looks like a single line.

It seems to me that today binary representations are purely symbolic and essentially lifted off the electronic realization (unless one is building the very circuit, then you have to be aware of the signal level interpretation.) Indeed one can realize circuitry with either interpretation. For example NPN and PNP transistors switch on opposite signal states, but such an example relies on the discovery of the transistor which was of course after Shannon's thesis work.

Regarding infinity as a truth value: having read Shannon's writings yet again, I'm now convinced that my earlier understanding (that Shannon had contemplated using infinity in place of 1 as a truth value) was baseless.

As for the second part of my question, the story seems to complicated. Shannon himself acknowledged that there were two different ways to model switching circuits with Boolean logic, and he gave his reasons for picking the one that he chose. One source I found says that both conventions are in use today in different contexts (e.g., depending on whether one is discussing fixed-current or fixed-voltage systems).