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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.

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    $\begingroup$ Your first derivation is sound. On the assumption that a republican (i.e. one of $R$, $A$) wins, you correctly derive that $R$ wins or $A$ wins. - For the rest I scrolled through the linke McGee question and could not understand the problem. If I accept (a) and (b) I personally am unable to not accept (c). $\endgroup$ – Hagen von Eitzen Feb 21 '14 at 22:00
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    $\begingroup$ where does (4) --- $\lnot A \lor B$ come from? $\endgroup$ – Willemien Feb 21 '14 at 22:22
  • $\begingroup$ @Willemien From assumption. $\endgroup$ – frabala Feb 21 '14 at 23:39
  • $\begingroup$ @frabala which assumption?, also why the tag philosphy (not philosophy) $\endgroup$ – Willemien Feb 22 '14 at 11:39
  • $\begingroup$ @Willemien At that part, he tries to prove that if $\vdash A$ and $\vdash\neg A\vee B$, then $\vdash B$. So, formulas $A$ and $\neg A\vee B$ are given by assumption. $\endgroup$ – frabala Feb 22 '14 at 11:52
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TLDR version: in the earlier thread, Nate Eldredge and Mathieu Vidal had it basically right. In this thread, Confutus’s obervation about people “living in a realm of uncertainty,” as opposed to the realm of mathematics, is quite on point.

Long, read-if-you-like version.

Logic deals with two kinds of truths, namely necessary and contingent. And it does two kinds of things with those truths: on the one hand it talks about what must be the case, if something else is the case; and on the other hand it talks about what a person should ideally accept, given that s/he asserts or accepts something else.

When it comes to the necessary truths of mathematics, we know from Goedel that there exist truths that can’t be proven. Nonetheless, there’s no difference between determining the truth of something and judging its assertibility. In this context, the material conditional treatment of “if…then” (translating “If A, then B” as “Either B or else not-A”) works just fine, and McGee’s modus ponens counterexample doesn’t get any traction.

But philosophers (and linguists) also deal with contingent truths and statements about them. Here things get messy, because a gap opens up between what seems to be true (on our best understanding of how truth works) and what people are prepared to accept or affirm. Material implication starts to fail as a model of “if…then,” because (to boil McGee’s example down a bit) one might be quite prepared to say, “Either Reagan or Anderson will win,” since Reagan is way ahead in the polls and a Reagan win will make the statement true; and at the same time one would be reluctant to say, “If Reagan doesn’t win, Anderson will,” because almost surely if Reagan loses it will be to Carter.

There are two obvious ways to deal with this problem. One can say that contingent if…then statements don’t have truthvalues at all, just assertibility conditions. Ernest Adams is the best-known proponent of that view. Or one can say that for if…then statements, the truth conditions are those of the material conditional but assertibility conditions are another thing. Frank Jackson is the leading exponent of this approach. There are many other maneuvers to be tried, but those are the two obvious ones.

Either way, one then ends up with the question whether assertibility relationships among statements, including conditional statements, can be codified--and if so, how. If we try this, we end up with something one could call a logic of assertibility. If you like, you can balk at calling this logic, but it has the structure of a set of symbols, and a set of rules for how to form statements, and a set of entailment relations, so it’s certainly logic-like.

So then McGee should be understood to be pointing out that in a logic of assertibility, Modus Ponens is valid only with restrictions. He doesn’t actually talk about a logic of assertibility, but his focus is on the realm of asserting, accepting, and so forth. He writes: “How do we account for this discrepancy [between standard logic and the given examples]? The simplest diagnosis is …. that we have noticed that, when we accept a conditional and accept its antecedent, we are prone to accept the consequent, as well. We have supposed that this pattern held universally.... However, [etc.]” (p. 468).

Two footnotes:

McGee is actually taking the position that not only doesn’t the material conditional entail the plain-English “if...then” (lots of people would agree with him on that) but that English “if…then” doesn’t entail the material conditional, either. That is an unusual view, but he makes an interesting case.

McGee discusses a rule known as import-export, according to which “If A, then if B, then C” is equivalent to “If A and B, then C.” He offers a proof that if you accept unrestricted modus ponens AND import-export, you have to accept that the material conditional is the right model of English if...then. His Reagan/Anderson example and the other examples are offered to motivate the choice of rejecting unrestricted modus ponens and keeping import-export.

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The Schoenfield rule is equivalent to the classical two valued definition of the material conditional. This is perfectly adequate for classical two-valued logic.

People live in a realm of uncertainty, and experience leads them to intuitive ideas about valid reasoning in an environment of uncertainty. Natural language reflects that experience. However, natural language and intuition are often inconsistent and unsatisfactory. Doubts about whether the the material conditional is adequate or whether modus ponens is always a valid rule of inference in a context of uncertainty continue to resurface. Putting conditions on a problem that allow it to be restricted to a classical two valued solution does little to resolve such doubts.

There are partial successes with systems that reject one or other of the conclusions of classical logic, but the cirumlocution required means that the systems are highly technical and the essential ideas are buried in formal systems and symbolism which are not easy to work with, interpret, or even discuss intelligibly. The fact that the problems are perceived to be difficult has led to the development of ever more complex formal systems.

My experience and experimentation with going back to early work in the subject, such that of Lukasiewicz, Kleene, Heyting, C.I. Lewis, and Heyting and rethinking it with better tools for examining mathematical structure than they had, has led me to conclude that skepticism about the excluded middle, the adequacy of the material conditional, and the problems with modus ponens are both closely related and justified in the wider world of uncertainty, however well they work in classical two-valued logic. If the conditional "if P then Q" or the implication it may express may be doubtful, (does Q really follow from P?), there is little reason to expect conclusions using modus ponens to be valid.

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