The traditional connectives for negation (not), conjunction (and), disjunction (or), implication (if...then) and equivalence (if and only if) allow an efficient representation of logical relationships as they appear in language. In electronics there are three other relationships normally used (and implication is normally not used): negated conjunction (NAND), negated disjunction (NOR), and exclusive disjunction (XOR).
The idea of a truth table to represent logical relationships was itself the creation of one of Bertrand Russell's students, Ludwig Wittgenstein. Still another suggestion to reduce the number of connectives required came from H.M. Sheffer, who showed that "alternative denial" (the NAND relationship) could be used to express every possibility. Sheffer used a stroke ( | ) to do this, and we still refer to this as the Sheffer stroke.
These would be the expressions for several of the sixteen possible truth tables:
| p | q | p | p (same as ~p) | p | q ( same as ~(p & q)) | (p | q) | (p | q) (same as p & q) |
| true | true | false | false | true |
| true | false | false | true | false |
| false | true | true | true | false |
| false | false | true | true | false |
We could just as easily have used the NOR relationship (symbolized with a dagger). What is interesting is that the idea of negation is already built in, so we can get by with just one symbol. In our conventional notation we still need the tilde (or curl) but could otherwise make use of just one more connective.
Experiments:
1. What would be the stroke equivalent of p v q ?
2. Let's imagine we can use the symbol ! for the dagger. What would be the stroke equivalent of p ! q ?
3. Suppose we used both the stroke and the dagger. In conventional notation, what would be the equivalent for p|(p!q) ?
4. Negation is a unary connective and all the rest are binary connectives. With infix notation it would be difficult to represent a ternary relationship, but we could do this with postfix notation. Imagine the connective * and the expression pqr* . What would be one truth table you could suggest (remember that with three variables you will need eight lines)? What would be the equivalent expression in conventional notation?