@ Doug Spoonwood: I am unable to date exactly the first use of the truth-tables among (professional) mathematicians, although Charles S. Peirce and Gottlob Frege could make up good approximations (implicit refs. to a famous paper published by Peirce in 1885 and to Frege's "Begriffsschrift" 1879). In particular, Peirce was aware [cca 1900-1902] of the truth-functional behaviour of ALL [two-valued] propositional connectives (including NAND and NOR, usually attributed to Henry M. Sheffer 1913), although he did not publish details, during his lifetime: his findings remained in MSS. On this, the SEP-reference given previously is rather misleading: Ludwig Wittgenstein comes later into the picture (some time during the early twenties or slightly before).
We know, however, for sure that truth-tables were popular already about 21 centuries before Frege and Peirce, among philosophers, more precisely they were found by Chrysippus of Sol[o]i, circa 279-206 BC, the founder of Stoic logic. Even Chrysippus was based on previous findings due to a 5th century BC philosopher, Euclides of Megara, a former pupil of Socrates, and the founder of the Megarian school (or, more likely, of one of his "Megarian" followers). For details, see e.g. the SEP-entry on the "Dialectical school", due to Susanne Bobzien, her SEP-entry on "Ancient logic", and, possibly, the Wikipedia entries on the Megarian school and the like.
As an aside, on should note that, unlike Aristotle and other members of Plato's Academy, Chrysippus and his followers were not particularly fond of mathematics (in the sense of the time).
I will hopefully publish, one of these days, a longer note on the (most plausible) "mathematical" method of construction behind the so-called "Stoic logic" (the "logic of Chrysippus"), including the truth-table stuff (as they could have figured it out, 22 centuries ago) [follow up academia.edu for would-be pre-prints].
@ Mitch, as of today, 20150323 [the answer is too long to fit the comment space] :
[ Mitch: ] Are you saying there is evidence that some Greek philosophers had the concept of truth functions (functions whose inputs and outputs are something like true and false) and graphical representation of truth tables (a tabular representation)? I don't doubt the first, but I do the latter. Also, the question I have is about the intellectual provenance of the truth-table display in modern mathematics, not the multiple possibly non-influencing reinventions across the world.
The first half of your question can be answered in the affirmative: Chrysippus and some later Stoics “had the concept of truth functions”, for all practical purposes, so to speak. In his / their terms, of course: they knew how to manipulate truth-functions in order to explain “semantically” complex propositions (the Stoic axiomata) in terms of simples, i.e. atomic propositions. (You don’t expect a second century BC philosopher to be conversant with our modern — set theoretical — concept, not even with the “old-fashioned” pre-Cantorian way of thinking of functions in traditional mathematics — functions as rules, say. Besides, as I said already, the Stoics were less interested in the mathematics of their times.) So, as regards the “intellectual provenance” (of the concept), the honors should go to Chrysippus, no doubt.
However, the second half of the question (concerning the actual representation of the truth-value assignments) cannot be answered properly, mainly because we have insufficient (textual or other kind of historical) evidence at hand, in order to say something definite about. Even the young Russell was a bit confused much later, around the turn of the xix-th century, about the concept of a “propositional function”, and Wittgenstein was not significantly more brilliant in this mater either…
Now, putting things in modern terms, it is likely the Stoics used something equivalent to our truth-tables (an ad hoc concrete representation, after all) only in order to justify semantically, so to speak, their proof-theoretic ruminations. Although nice to play with — a piece of kindergaten mathematics, after all —, the truth-functional explanation of the logical connectives was not central to their logical doctrines: the Stoics used to think about the binary connectives (connector = syndesmos in Greek) in terms of “polar oppositions” (or conflict = “maché”, in Greek; more or less: contradictions).
Examples: the pair of complex propositions reading (A AND B, A NAND B) in our jargon— both attested, btw, in the remnant Stoic fragments — makes up a “polar pair” (of opposites), and similarly for the “polar pairs” including IF (the material conditional) and its converse, reading approximatively SINCE. Actually the polar opposites of IF [IF A THEN B] and SINCE [A SINCE B] resp., are also attested in surviving Stoic fragments: the former would read “A mallon he B” [“A more than B”], in Chrysippus’ Greek — semantically = A AND (NOT B) —, while the polar opposite of SINCE [A SINCE B = IF B THEN A] would read “A etton he B” [“A less than B”], semantically = (NOT A) AND B. (The truth-functional interpretation of “mallon…” / “etton…” has been also confirmed, recently, by Prof. Bobzien, a major expert in Stoics doctrines, btw.) Lastly, the polar pair (A NOR B, A OR B) is also historically attested in late Stoic texts. So, we have “expressive completeness”, in fact (the biconditional IFF and the exclusive disjunction, XOR, can be explained in terms of the above. This can be also found in Stoic texts!) Whence, we know what is *(C) — the polar opposite of C — for any C. Moreover, “double negation” is granted for each form of (complex) C.
To anticipate the longer (technical) note promised earlier: Once the polar trick is well understood, it is relatively clear how to construct inconsistent sequences of propositions (elenchoi, or refutations in Greek) of the form A_1, …, A_n ||- Falsum, in the spirit of the Stoics, such that a valid entailment (syllogismos), A_1, …, A_n ||- C, can be accounted for in terms of … ||- Falsum by:
A_1, …, A_n ||- C iff A_1, …, A_n, *(C) ||- Falsum,
where Falsum is an arbitrary false proposition and *(C) is the polar opposite of C.
On this plan — granted the fact that we can analyse systematically complex *(C)’s into their “polar” components —, it is relatively easy to obtain something very similar to a Gentzen “natural deduction” system resp. a resolution / tableaux-like system (à la Hintikka, Beth, Smullyan and so on) for classical (i.e. Chrysippean) logic!
Well, I won’t go so far to claim that Chrysippus proved the completeness of his logic ( = classical logic ), relative to a would-be account of the classical connectives in terms of truth-tables, 22 centuries ago, but he was pretty close to it, anyway!