I would like to "reopen" the previous post regarding Modus ponens because, frankly speaking, I'm not satisfied with some (most of ?) answers by the mathematicians community.
Disclaim: I'm not aiming to "unravel the mystery", but I'm not convincd either that mathematicians and philosophers speaks completely different languages.
This is my argument, in two steps : a "mental experiment" followed by some considerations about formalization and natural language.
The experiment I'm trying is based on a reformulation of McGee's first example (see Vann McGee, A Counterexample to Modus Ponens (1985)), regarding the US presidential election of 1980.
I'll neglect the aspects regarding "belief" and the nuances connected to verbal tense (see the paper of Robert Fogelin & W.Sinnott-Armstrong, A defense of Modus Ponens (1986)), also because I'm not a native english speaker.
I assume as domain of the problem a non-empty universe (call it $US$) where there are only two mutually exclusive subsets : $rep$ and $dem$ (so that : $rep \cap dem = \emptyset$).
I assume that the set $rep$ has only two elements $R$ and $A$ (i.e. $rep = \{ R, A \}$, and $A \ne R$).
I assume only one "obvious" axioms, translating the "rule of the game", using a single predicate $win$ :
$win(dem) \lor win(rep)$.
The first consideration - we will discuss it later - is that the above condition is really a "XOR": "a republican will win or a democrat will win, but not both".
We have also :
$\lnot win(rep) \equiv win(dem)$.
So we have the "tirvial" :
$\lnot win(rep) \lor win(rep)$.
But due to the fact that the only republican candidates are $R$ and $A$, the last amount to :
$\lnot win(rep) \lor [win(R) \lor win(A)]$ --- (A).
Note : we are not using $\rightarrow$ in this argument; if we would use it, with the classical truth-functional semantics, the sub-formula between the square brackets would amount to : $\lnot win(R) \rightarrow win(A)$.
I introduce now what I'll call Shoenfield rule (from Joseph Shoenfield, Mathematical Logic (1967), page 28 :
if $\vdash A$ and $\vdash \lnot A \lor B$, then $\vdash B$.
The above rule is proved in Shoenfield's system using three of the four "propositional" primitive rules [page 21 : the last one, the Associative Rule, is not used in the proof below] :
Expansion Rule : infer $B \lor A$ from $A$
Contraction Rule : infer $A$ from $A \lor A$
Cut Rule : infer $B \lor C$ from $A \lor B$ and $\lnot A \lor C$.
With the Cut Rule and the (only) propositional axiom : $\lnot A \lor A$, we can derive the Lemma 1 : if $\vdash A \lor B$, then $\vdash B \lor A$.
Now we prove Shoenfield rule :
(1) --- $\vdash A$
(2) --- $\vdash B \lor A$ --- from (1) by Expansion
(3) --- $\vdash A \lor B$ --- from (2) by Lemma 1
(4) --- $\vdash \lnot A \lor B$
(5) --- $\vdash B \lor B$ --- from (3) and (4) by Cut
(6) --- $\vdash B$ --- from (5) by Contraction.
Disclaim: nothing new; all is trivial (classical) propositional logic.
Now, we go back to (A) :
$\lnot win(rep) \lor (win(R) \lor win(A))$
and add the premise :
$win(rep)$;
by Shoenfield rule we conclude the "obvious" :
$win(R) \lor win(A)$.
Nothing has gone wrong ... We only has used standard rules for propositional connectives in a classical framework, with the use of $\lor$ in a situation where the alternative are mutually exclusive.
Question : Is the previous argument "sound" ?
The above argument, assuming it is "sound" suggests to me some considerations about formalization and natural language.
The "regimentation" that symbolic logic - from Frege on - has deliberately imposed on natural language has been greatly fruitful; this does not imply that the richness of natural language can be wholly "explained away" with formalization.
The dissatisfaction of McGee about the modus ponens seems to me the "old" dissatisfactions about the translation of "if ... then" in term of the truth-functional connective $\rightarrow$.
This one is blind about the nuances of natural language (that relevant logic try to recover). In the same way, when I use $\lor$ in a context where the alternatives are mutually exclusive, I "loose" some presuppositions (some implicit information that the speaker aware of the context knows).
This does not means that the rule of logic are "wrong"; neither that philosopher does not know logic. Aristotle and Leibniz and Peirce and Frege and Russell were all philosophers.
In conclusion, I think that there is no "contradiction" between the way mathematical logic formalize truth-functional connectives and natural language.