  This paper explores trivalent truth conditions for indicative conditionals,
examining the "defective" table put forward by de Finetti 1936, as well as
Reichenbach 1944, first sketched in Reichenbach 1935. On their approach, a
conditional takes the value of its consequent whenever its antecedent is True,
and the value Indeterminate otherwise. Here we deal with the problem of
choosing an adequate notion of validity for this conditional. We show that all
standard trivalent schemes are problematic, and highlight two ways out of the
predicament: one pairs de Finetti's conditional (DF) with validity as the
preservation of non-False values (TT-validity), but at the expense of Modus
Ponens; the other modifies de Finetti's table to restore Modus Ponens. In Part
I of this paper, we present both alternatives, with specific attention to a
variant of de Finetti's table (CC) proposed by Cooper 1968 and Cantwell 2008.
In Part II, we give an in-depth treatment of the proof theory of the resulting
logics, DF/TT and CC/TT: both are connexive logics, but with significantly
different algebraic properties.
