The one-to-one translation between those variables as Tarski calls them, ie the binary values, and the world of sentences and ideas is now being emphasized. This has implications far beyond the ability to do arithmetic like a pocket calculator, if we have a sufficiently powerful machine. Tarski again: "The meaning of these tables becomes at once comrehensible if we take the letters T and F to be abbreviations of 'TRUE SENTENCE' and 'FALSE SENTENCE'" (pages 40-41). Let's put it this way. If you testify in court and violate the rules of Truth Tables, almost certainly you will be in big trouble.
SAMPLE TRUTH TABLE
(matrix of combinatorial logic)
(p and q are variables/values/sentences/expressions)
p q >>>>> p AND q p OR q If p THEN q q IF AND ONLY IF p
(or vice-versa)
T T >>>>> T T T T
F T >>>>> F T T F
F F >>>>> F F T T
To the left of the hash marks (>>>>>) we have the truth values of the expressions or binary variables, the givens. In the first row it is given that p is true and q is true. So a bear is an animal (p) and a dog is an animal (q). We can establish that as a matter of fact or 'fit to a definition'. To the right of the hash marks we have the conclusions of this "inferential engine". The first says that with these givens, both bear AND dog must logically be animals. In other words, the inference is also true. The second inference says that at least one OR the other of the two must be an animal. The third inference says that with these givens, knowing that the bear is an animal, THEN we will also know the dog is an animal. The fourth says that with these givens, the dog (q) will be an animal IF AND ONLY IF the bear is an animal (p) and vice-versa. You can translate the symbols (variables) into words for the other examples. Tarski says the table above is fundamental and all others, the "derivative truth tables" can be contructed from them. In machine terms the hash marks can be considered as "logic gates".
FROM LOGIC MACHINE CONNECTIONS TO WIRING
Whether you agree with Tarski's set of fundamentals or provide another set, the main point is that all logicians agree there is a fundamental set and all others can be derived from them. Just recognize the fundamental pattern in the more complex matrix of positions and values in your machine and wire them the same way.
The wiring is only a matter of the series or parallel wiring most people are acquainted with from Christmas tree lights or high school electricity. Picture the AND gate as three locations on an Electric Abacus, each having a light bulb which can be turned on or off. The wiring must be such that when the p location is on and the q location is on, the resultant location, x, will also be on. In any other case, for any other COMBINATION of values for p and q, there is no wired connection to x. Likewise, for the OR gate, when the light bulb is on for p or the light bulb is on for q (ie either one is on) the light will also be on for x. And so on.
Just wire the Electric Abacus correctly and the inferences can be read infallibly. THE MACHINE DOES THE LOGIC AUTOMATICALLY WHEN THE SWITCHES ARE THROWN. It is wired to do logic correctly and only correctly. The user only has to input the results correctly and read the results correctly.
CAVEAT ON LOGIC AND TRUTH
There is nothing in the Truth Table or the logic machine itself which requires us to feed the truth as CORRECT FACTS, ie THE TRUTH, into the machine. We could enter, "The bear is an animal" and set that as T and "The dog is not an animal" as T. The machine would draw false conclusions using perfect inferential methods thereafter. It's truth FUNCTION would be correct. Hence the old saying from computing, "GIGO", ie "Garbage In Garbage Out". It is the responsibility of the machine users to manage the GIGO problem.
To obtain truthful conclusions we need TRUE INPUTS and correctly operating TRUTH FUNCTIONS.
With both conditions met, we have a true machine, as we might use an expression like true arrow.
Truth has to do with "how things really are", and those who seek universal truth can do no more than expand their horizons to approximate the "biggest picture" of that truth. Then there is metatruth or metalogic ....
http://www.geocities.com/ELECTRIC_ABACUS/page4