I don't believe that there is any reason to distinguish Peirce's Alpha graphs (propositional logic) from his Beta graphs (FOL).
In fact, Peirce himself (NEM 3:162-167) does not mention Alpha or Beta. He uses exactly the same rules of inference for both. I just reread it yesterday to see exactly what he says. On page 163, he uses a line of identity for "There is a phoenix" (fig 1) and "There is no phoenix" (fig 2). Immediately after that, he uses medads (no lines of identity) for "If thunder then lightening" (fig 3). and for "thunder and no lightening" (fig 4). On page 164, he wrote
A graph like "thunders" is called a "medad" as having no peg (though one might have made it mean "some time it thunders" when it would require a peg). A graph or graph instance having 0 peg is a _medad_. A graph or graph instance having 1 peg is a _monad_. A graph or graph instance having 2 pegs is a _dyad_. A graph or graph instance having 3 pegs is a _triad_.
In this letter of 1911, which is apparently a clean copy of MS 514, dated 1909, Peirce says nothing about distinguishing Alpha from Beta, and he lists the names for medad to triad without saying anything about blanks. In his earlier writings, Peirce was following the tradition of distinguishing propositional logic from quantified logic (FOL and HOL). But by 1909-1911, he seems to have decided that there is no reason to distinguish them. Any EG could mix medads with monads, dyads, triads, tetrads, pentads... I believe that NEM 3:162-167 is his final considered opinion: A proposition (medad) is just a relation without pegs. Any EG can mix medads with relations with one or more pegs, and there is no need to treat them as special cases. John
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