The Weak Logic Argument

Adding impossible worlds, then, has a real impact on the non-vacuist’s coun- terfactual logic. Generally speaking, since impossible worlds can be among the

closest worlds and any connection between two formulas can be severed at an impossible world, the logic that emerges is very weak.

This can be turned into an argument against the non-vacuist. Williamson puts the problem as follows:

We may also wonder what logic of counterfactuals [non-vacuists] envisage. If they reject elementary principles of the pure logic of counterfactual conditionals, that is an unattractive feature of their position. (Williamson 2007: 174)

Call this theweak logic argument. It is based on a perceived lack of adequacy or mismatch between the non-vacuist counterfactual logic, on one hand, and (1) Ordinary speaker’s validity intuitions; and (2) Ordinary speaker’s deductive behavior, in the other.

I gave an intuitive gloss of what I mean by “validity intuition” above. Some- times we have the intuition that a sentence simplyfollows from a given set of sentences. We can have such an intuition even if we have no particular intuition about the truth value of the sentences involved. An intuition of validity is an intuition about therelation between the truth-value of sentences, not about the actual truth-value of the sentences themselves.

By the expression “deductive behavior” I mean something like the follow- ing. In ordinary discourse we can take certain attitudes regarding particular assertions. Roughly speaking, we can assent to them, we can deny them or we can suspend our judgement on them.7 _{It seems clear that there are certain}

normative principles that rule the way in which we can assent, deny or suspend judgement. A relatively plausible one is the one that says that we cannot assent to a propositionp while denying a logical consequence of p. Insofar as this is a good principle, rationality would demand that we obey it. If it is pointed out to us that we are in a position where the collection of our assertions and denials violates it, we must recognise that something needs to be changed (even if we do not know immediately what). This assigns a particular role to logic in governing our linguistic practice. Counterfactuals, of course, are no exception. Now consider the following dialogue:

A: If I had spent more money in my campaign, I would have won the election. The only reason I lost was that my opponent had much more resources than me.

B: But, wait. If you had spent more money, your opponent would also have spent more money, right? After all, he was finantially supported by all the oil tycoons! If you think that resources were the key to this election, spending more of your hard-earned money would not have made a difference.

A: I guess you are right... I need to understand what was my mistake, though.

Here is a natural way to understand this interaction. A agrees that the following two things are true under the counterfactual supposition of A’s spending more money: a) The candidate with more money would still have won; b) B would still have spent more money than A. The conclusion that B would still have won follows logically from a) and b). So A is forced to agree he needs to retract his initial assertion, it is not the case that if A had spent more money, she would have won the election. It seems that A and B are adhering to (Closure) as stated above. Though it could have occurred to A to refute one of the two counterfactuals B is considering (maybe the important thing for A was to have funding above a certain threshold, therefore making a) false under the counterfactual supposition), it does not seem like rejecting the conclusion that “If A had spend more money in the campaign, B would still have won” after

accepting a) and b) was an option at all. And the natural way to understand this is that there was a logical inference involved and that logical consequences of what has been assented to must not be denied.

If this is true, if ordinary intuitions regarding what follows from what are violated and if certain normative principles governing the assertions people as- sent to in ordinary discourse lose their justification, then this is a cost for the non-vacuist. How can the non-vacuist respond?

One option would be to say that the cost of violating intuitions of validity is so much smaller than violating intuitions of truth-value that the cost can just be accepted. Validity is, after all, a somewhat theoretical notion, one that needs to be explained patiently to students taking a first-year logic class. This can be seen to erode some of our confidence in validity intuitions. And even if the connection of logical validity and what can be assented and denied is a particularly nice explanation of our linguistic behavior, it might not be the only one.

Given that the issue of the relative weight of different factors in the cost- benefit analysis is a particularly murky one, it is not easy to assess this reaction. My sense is that this can vary with different people’s intuitions. For the purposes of this chapter, the important part about this strategy is that it agrees that there is a cost associated with having a weak logic. In the endgame, this cost should be compared with the cost of whatever alternative strategies there are. Since I have no clear idea on how to do this, I will leave this worry to one side.

The traditional reaction of non-vacuists, first suggested in Nolan (1997) and adopted by Berto et al. (2017), is different. They want to build a counterfactual logic using impossible worlds that is more robust than what initial appearances suggest. They do this by imposing a condition on admissible similarity metrics. We will present and evaluate their efforts in the next sections.