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In math.stackexchange there is an interesting post regarding Modus ponens .

It seems to me that mathematician's "standard answer" shows a sort of annoyance with McGee's objections.

I'm not convincd that mathematicians and philosophers speaks completely different languages: it seems to me that the difference must be located into a different "sensibility" towards the relation between natural language and formalization.

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 *intersection dem = the empty set).

I assume that the set rep has only two elements R and A (i.e. rep = { R, A }, and A is different from R).

I assume only one "obvious" axioms, translating the "rule of the game", using a single predicate win :

win(dem) or 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 :

not win(rep) is equivalent to win(dem).

So we have the "tirvial" :

lnot win(rep) or win(rep).

But due to the fact that the only republican candidates are R and A, the last amount to :

not win(rep) or [win(R) or win(A)] --- (A).

Note : we are not using the → connective (i.e. theconditional) 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) → win(A).

I introduce now what I'll call Shoenfield rule (from Joseph Shoenfield, Mathematical Logic (1967), page 28 :

from A and (not A or B), infer 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 or A ) from A

Contraction Rule : infer A from (A or A)

Cut Rule : infer (B or C) from (A or B) and (not A or C).

With the Cut Rule and the (only) propositional axiom : not A or A, we can derive the Lemma 1 : irom A or B, infer B or A.

Now we prove Shoenfield rule :

(1) --- A - assumption

(2) --- B or A --- from (1) by Expansion

(3) --- A or B --- from (2) by Lemma 1

(4) --- not A or B - assumption

(5) --- B or B --- from (3) and (4) by Cut

(6) --- B --- from (5) by Contraction.

Up to now, nothing new : all is trivial (classical) propositional logic.

Now, we go back to (A) :

not win(rep) or (win(R) or win(A))

and add the premise :

win(rep);

by Shoenfield rule we conclude the "obvious" :

win(R) or win(A).

Nothing has gone wrong ... We only has used standard rules for propositional connectives in a classical framework, with the use of or in a situation where the alternative are mutually exclusive.


Assuming that the previous argument is "sound", may we say that it gives support to the claim that there is no "contradiction" between truth-fucntional analysis of connectives and natural language ?

I think that 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 [see Vann McGee, A Counterexample to Modus Ponens (1985) and Robert Fogelin & W.Sinnott-Armstrong, A defense of Modus Ponens (1986)] about the modus ponens seems to me the "old" dissatisfactions about the translation of "if ... then" in term of the truth-functional connective → .

This one is blind about the nuances of natural language (that relevant logic try to recover). In the same way, when I use "or" 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.

closed as unclear what you're asking by DBK, Hunan Rostomyan, iphigenie, Thomas Klimpel, ChristopherE Mar 9 '14 at 22:36

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    I'm a little worried too about the 'what do you think' aspect of this question (i.e., that this is the only question-mark in the post, making it read something like a poll...) --Also the headline is really not very helpful in terms of specifying exactly what you'd like explained to you (what precisely your question is). Is there any chance I might be able to persuade you to clarify this and make it clearer what the concern is here, particularly the headline? – Joseph Weissman Feb 22 '14 at 18:08
  • -1 I fail to see a question. It's more like a blog post or an answer (?) that should be posted over at math.se. – DBK Feb 26 '14 at 4:01
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Looking through McGee's list of counterexamples to modus ponens on 462-463, I think what you are saying is fundamentally correct. Or to put it another way, McGee is objecting to drawing an equivalence between how the conditional operator works and how if-then works in natural language. But I think he's mis-locating the problem.

I have found students are often uneasy with the truth table for the conditional operator:

    A   B   A --> B
    T   T      T
    T   F      F
    F   T      T
    F   F      T

For this, students are often bothered by the truthfulness of the operator when the antecedent is false. They wonder legitimately why a false antencedent should be able to give us a truth value for the entire claim.

What McGinn seems to have missed which many of my students understood is that if-then != -->. One must keep in mind what is meant by the natural language before translating it into symbolic language. And that means sometimes that one must translate it to something else. In other words, I think McGinn is creating his own problems by imagining a strict correlation where there is a strong one. I place the problem not with modus ponens but with thick-headed symbolization.

  • This sounds like a confusion between modus ponens and modus tollens, and thus a confusion between sufficient condition and necessary condition. – labreuer Feb 22 '14 at 18:58
  • I'm no longer a student but it still makes me rather uneasy (unlike the rather different conditional in probability theory) - but you said that, or at least something consistent with it. That's the uneasiness, right there ;) – Lucas Mar 16 '14 at 20:42

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