MP is just Modus Ponens, telling us that if a material conditional and its antecedent are theorems, then so is the consequent.
The axiom MP, for modus ponens states a connection between the material conditional and the more general notion of conditionality encoded by '>'.
At least in languages where the conditional is the material conditional, and so A ⊃ B is equivalent to ¬A ∨ B, DLiar is equivalent to the Curry sentence 'DLiar is true ⊃ 1 = 0'.
Mott 1973 and Chellas 1974 (and Chellas 1980) offer influential analyses of the puzzle by combining a non-material conditional and a unary deontic operator to form a genuine componential compound, p ⇒ OBq, for representing conditionals like 3) above); DeCew 1981 is an important early critical response to this sort of approach.
OB-K, which is the K axiom present in all normal modal logics, tells us that if a material conditional is obligatory, and its antecedent is obligatory, then so is its consequent.[13] OB-D tells us that p is obligatory only if its negation isn't.
More specifically, first-order modal languages will have variables, individual constants, n-place predicates, atomic formulas such as 'Pacb' and 'x=y', and the usual boolean, quantified, and modal formulas involving the familiar logical constants '~' (negation), '→' (material conditional), '∀' (the universal quantifier), and '□' (the necessity operator).
It is now virtually universally acknowledged that Chisholm was right: the sort of conditional deontic claim expressed in (3) can't be faithfully represented in SDL, nor more generally by a composite of some sort of unary deontic operator and a material conditional.
As noted in the case of the sea battle, when rewriting in the formal mode captures the sense of what is being said, and when the formulations "if p, q" and "p only if q" seem idiomatically equivalent, then an inferential interpretation will be in order, von Wright's equivalences will hold, and the material conditional may well give a reasonable account of such cases.
But given the truth-conditions for the material conditional "→", that just amounts to saying p is true at all those i-accessible worlds (if any) where d is true, which in turn holds iff p is true at all the i-accessible worlds falling within DEM, i.e., at their intersection (which is non-empty by strong seriality).
Some Stoic presentations of the argument recast it in a form which replaced all the conditionals, 'If A then B', with 'Not(A and not-B)' to stress that the conditional should not be thought of as being a strong one, but rather the weak Philonian conditional (the modern material conditional) according to which 'If A then B' was equivalent to 'Not(A and not-B)'.
Given their acceptance of the principle of bivalence and their presentation of the argument as invoking a material conditional, they blocked the sorites by claiming some one conditional to be false (since not true) and that there comes a point in any sorites series where the relevant predicate ceases to apply and its negation does.
