Symbolic logic ordinarily works with just two values, true and false (or on and off or 1 and 0). The only statements that can be symbolized, then, are those which can properly be described as true or false, regardless of whether we know which they are. A sentence such as "Jane studies" is either true or false (assuming there is someone named Jane that we can be talking about in whatever context--real or fictitious--we have in mind). A sentence such as "Jane might be studying" is not exactly in the same category since the corresponding negative sentence "Jane might not be studying" is not properly a contradiction--both could be true. Also, a sentence such as "Jane ought to study" can be both true or false depending on the angle from which her situation could be looked at (it could be true from the angle--or POV--that she needs to study to pass her test and false from the angle that by staying home to study she could lose her job). For that reason we exclude probability statements and most judgments about decisions from what we can symbolize in a strict deductive system.

We begin with the idea that we have very basic ideas--statements--that could be expressed as short sentences. Sometimes we refer to these as atomic sentences and think of other expressions as built out of them just as molecules are built out of atoms. A code expresses the smallest possible set of these, which means we usually express them in present tense and without a word such as "not." Every other statement then will be composed of some combination of these atomic sentences, and whether a given statement is true or false depends on whether its basic parts are true or false (what it means to say they are truth-functional).

There are only a handful of useful logical relationships, which we indicate through our signals A,B,C,M,N,O. Suppose we start with a code that reads as follows:
....P: Logic is easy
....Q: Logic is fun
....R: Logic is interesting
We can then have truth-functional sentences expressing
negation: PN (logic is not easy)
conjunction: PQA (logic is easy and fun)
disjunction: PQNO (either logic is easy or it is not fun)
implication: PQC (if logic is easy then it is fun)
equivalence: PQB (logic is easy if and only if it is fun--and notice that there are four words used to express this, not just only if)
mutual exclusion: PQM (logic is either easy or fun but not both)

With a truth table we can show all the possible ways each of the above sentences could work out to be true or false.

P Q || PN | PQA | PQNO | PQC | PQB | PQM
T T.......F.........T...........T.........T........T.........F
T F.......F.........F...........T.........F........F..........T
F T.......T.........F...........F.........T........F..........T
F F.......T.........F...........T.........T........T..........F
We can also build still longer strings that are also truth functional, such as PQAPNO (either logic is both easy and fun or it is not easy) and by looking at its truth table (TFTT, if we read its truth-table series horizontally) we find that it is equivalent to the string PQC. This last relationship could then be expressed through one more string: PQAPNOPQCB. Also, since with a four-line truth table only sixteen permutations of T and F are possible, we recognize that every propositional string with just two variables must be equivalent to one other basic string. The possible series are indicated below, and the shortest (those with just one signal) can be seen as expressing the meaning of the signal involved.

TTTT -- PPNO (or any tautology)
TTTF -- PQO
TTFT -- QPC
TTFF -- P
TFTT -- PQC
TFTF -- Q
TFFT -- PQB
TFFF -- PQA
FTTT -- PQAN
FTTF -- PQM (=PQBN)
FTFT -- QN
FTFF -- PQNA (=PQCN)
FFTT -- PN
FFTF -- QPNA(=QPCN)
FFFT -- PQON
FFFF -- PPNA (or any contradiction)

Looking at the list of series you might want to note how one half in a sense is a mirror image of the other. Also you might have begun to wonder whether there is still another, more mathematical pattern in this development. Look ahead to the section on algebraic notation for more on this.

Use the programs PLN2 (or PLN2-colors, for the circle-box game) to look at truth tables for individual strings.

Return to variables and signals.

Go ahead to deductive reasoning.

Go back to the starting page.