Specifically for the material implication if possible. Who was the first to use a truth table for this and justify its validity?
You can see:
- Irving Anellis, The Genesis of the Truth-Table Device (2004)
as well as:
- Irving Anellis, Peirce's Truth-functional Analysis and the Origin of the Truth Table (2012).
Before Bertrand Russell (Harvard logic course: 1914) and Ludwig Wittgenstein (Russell and Wittgenstein's manuscript dated 1912; see also: Tractatus (1921), 4.31 and 4.442 for material implication), Charles Sanders Peirce and his followers must be credited.
In Whitehead and Russell's Principia (page 115), we have a description of the truth-functional semantics of connective : not, if..., then..., and, or (but not in tabular form).
The modern tabular form is present in E.Post, Introduction to a General Theory of Elementary Propositions (1921), §2 Truth table development, with explicit reference to W&R's Principia (see footnote 6).
The verbal" description of conditional is in Frege's Begriffsschrift.
Philo, for example, said that the conditional is true when it does not begin with a true proposition and finish with a false one, so that a conditional, according to him, is true in three ways and false in one way. For when it begins with a true one and finishes with a true one, it is true, as in “If it is day, it is light.” And when it begins with a false one and finishes with a false one, it is again true – for example, “If the earth flies, the earth has wings.”