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Logic of conditionals re Bill Mitchell’s quiz *May 4, 2011*

*Posted by larry in economics, Goedel, interpretation, Logic, material conditional.*

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In the most recent quiz by Bill Mitchell (http://bilbo.economicoutlook.net/blog/?p=14313&cpage=1#comment-17186), there was a brief discussion that did not clarify the problem under discussion. This problem was the nature of the logical of conditionals, in particular, the logic of the material conditional. Part of my response was not quite relevant and I would like to clear up any confusion I may have created here.

The problem was on how to interpret a conditional assertion made by Bill Mitchell. The commenters who dealt with this problem were Tom Hickey and MamMoTH. Mitchell asked the following question and whether it was true or false. In claiming Q 3 was false, Mitchell went on to show why this was the case without explicating the logic of the situation, no doubt because he thought it was obvious.

The question was:

If the stock of aggregate demand growth outstrips the capacity of the productive sector to respond by producing extra real goods and services then inflation is inevitable.

He then claimed that assertion was false. I am not here concerned with whether this assertion is false but with the logic of the situation. MamMoTH said:

According to the rules of logic, the correct answer to question 3 is

Truebecause False implies whatever is True. … @Tom, not sure what you mean, but a quick check on entailment with wikipedia to brush up some concepts shows that if S1 is inconsistent then S1 entails whatever.

Tom Hickey followed this with:

This is true of material implication but not formal implication (entailment). Most ordinary language arguments based on conditionals presume entailment.

While Hickey is right that a good many natural language arguments presuppose that the conditional being used is that of some kind of entailment, what MamMoTH says is not entirely right in the context of the material conditional. He appears to be confusing a logic of the conditional and a consequence of the definition of material implication with something else, perhaps Goedel’s incompleteness result.

In standard extensional (truth-functional) logic with the material conditional, the conditional is defined in such a way that ‘A implies B’ is true whenever A is false or B is true. All standard extensional systems are like this. However, in a non-standard logical system, particularly one with relevance conditions attached, this property is undesirable. Unfortunately, the term ‘entailment’ is loosely used by many, but the *locus classicus* for a logical system of entailment is Anderson and Belnap’s *Entailment: The Logic of Relevance and Necessity*. There are also logical systems in which contradictions, under certain conditions, are acceptable, but these are non-standard, too.

It seems to me that, in stating that ‘if S1 is inconsistent, then S1 implies whatever’, MamMoTH may have been implicitly referring to Goedel’s incompleteness theorem or a consequence of extensional logic with the material conditional, to wit, that if p & not-p then q, that is, that anything follows from a contradiction. I am assuming that what is meant by ‘if S1 is inconsistent’ is actually meant ‘if S1 is self-inconsistent’, for if S1 were self-inconsistent, then S1 might be of the form p & not-p, a logical contradiction. For if this were so, then S1 implying whatever would follow from its self-inconsistency. Simple inconsistency is not enough to obtain this result. Consider S1 = A or (B & not-B). Here a portion of S1 contradicts itself but S1 itself is not logically contradictory in the sense that an assignment of values to the variables of S1 will inevitably render S1 false. Since B & not-B is always false, then the value of S1 depends solely on A, which is not itself of the form, p & not-p. Therfore, ignoring B, if A is true, S1 is true, while if A is false, S1 is false. Is this happenstance sufficient to thereby render S1 inconsistent and thereby imply an arbitrary proposition q? If so, this is not the standard meaning of the notion of inconsistency.

Let us go on to the second matter. It is not true that anything follows from a falsehood, while it is true that anything follows from a logical contradiction, that is, a proposition that is always false. This situation can be confused with the definition of the material conditional. The material conditional is defined in such a way that ‘A (materially) implies B’ is false if and only if (iff) A is true and B is false. In setting up any deductive logical system, it is essential that no true statement ever lead to a falsehood. However, in the case of the material conditional, a falsehood leading to a falsehood is considered to be a true ‘material’ implication, though not generally in a logic of relevance or entailment. That is, ‘F implies F’ is true when ‘implies’ is material implication. There are good extensional mathematical reasons for this. However, a justification can be made that it should be the case that falsehoods should follow from falsehoods.

A problem arises because for material implication, ‘F implies T’ is also true, which appears to be unacceptable. Nevertheless, a justification can be given. One can argue that, in an extensional context like that of the material conditional, it is impossible to differentiate between truths and falsehoods arguing from a falsehood. Hence, it should occasion no surprise that both truths and falsehoods can arise equally from falsehoods.

One can get away with this seemingly counterintuitive result because no connection is assumed to exist between A and B in ‘A (materially) implies B’. There are natural language examples in which just such a situation exists. If Hitler is alive, then I’m a monkey’s uncle. While there is no connection between the antecedent and the consequent and both are viewed generally as false, the conditional itself is viewed as being true. Other examples where the antecedent is false and the consequent true are what are known as counterfactuals, a subset of subjunctive conditionals. An example is ‘if anyone jumps out of a 20 story window and falls to the ground, he will be squashed like a melon’. This is considered to be a true conditional even though the antecedent is false. However, this kind of conditional is non-extensional, hence, can not be considered to be a material implication. In such non-extensional contexts, strictly speaking, the term ‘entailment’ should be replaced by ‘implies’ in the sense of the material conditional, particularly since entailment is distinct from (material) implication in those cases where you need to make the distinction.

Assuming then that the material conditional is in play, this means that if the conditional in 3, A implies B, is false, this can only be because A is true and B is false. And this does not thereby imply an arbitrary proposition, q. It may be that Bill does not have the material conditional in mind. As for the question, I interpret it in such a way that there is an implied link between A and B which indicates that Bill is not intending that ‘implies’ stands for the implication of the material conditional, but rather some sort of relevance logic, say R. Unlike the standard extensional predicate calculus, there is more than one system, R. Until that has been specified, one must rely on the logical principles found in natural language and, hence, on some kind of informal reasoning with some math tacked on as an adjunct to the argument. There is nothing wrong in doing this, except that when this takes place in mathematics, what is not spelled out is generally clearly understood. This is not yet the case for logics of relevance and entailment.

Nevertheless, any standard mathematization of economics will be based on the extensional predicate calculus which relies on the material conditional. Keynes was aware of this problem and it was one of the reasons he felt that the mathematical tools of his time were inappropriate for formalizing an economic theory that included the ‘psychological states’ of the individual (such as animal spirits). This is still, pretty much, the case today. Except that we now have game theory, which is an advance on the mathematical tools available to Keynes – although the minimax theorem that von Neumann had already proved a few years before the publication of *The General Theory* would not, on its own, have been of much use to Keynes.

As for inconsistency and logic, there seems to be an implicit reference to Goedel’s incompleteness theorem. The theorem says that any logical system more complicated than arithmetic with addition will be incomplete in the sense that it will be able to say more than it will be able to prove using its own resources, if it is consistent. Goedel’s theorem is based on the system set out in Principia Mathematica and extensions of this system, which is pretty much all of standard mathematics. Goedel achieved this result by a quite tricky use of self-reference that is not self-contradictory. The upshot of Goedel’s result is that any sufficiently complicated formalized theory, if consistent, will be incomplete. There will be some truths it can not prove though it can formulate them. This result still appears to be counterintuitive over half a century later, even though Goedel was possibly the greatest logician of the 20^{th} century.

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