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.
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.”