Do Counterfactuals Violate Modus Ponens?

9 08 2008

There has been an intensive debate about whether modus ponens fails for indicative conditionals. Less attention has been paid to the question of whether similar examples can be constructed for counterfactuals as well. This is insofar surprising as McGee claimed that the Import/Export principle (which leads to the counterexamples for indicatives) holds also for counterfactuals. So, are there counterexamples to modus ponens for counterfactuals?

Let us recall the setting of McGee’s counterexample. There are three candidates for the 1980 election: the two republicans Reagan and Anderson, and the democrat Carter. The polls see Carter far behind Reagan, with Anderson a distant third. Prima facie, McGee’s counterexample can go counterfactual. Suppose I know about the polls but do not receive any relevant information afterwards, perhaps because I go on a safari trip or because I just don’t care. After the time of the election I consider the following argument:

(1) If a republican had won, then if it had not been Reagan, it would have been Anderson.

(2) A republican won.

(3) Therefore, if Reagan had not won, it would have been Anderson.

Given the polls, I will find the premises highly probable although I will dissent from the conclusion. This comes as a surprise: if an inference is classically valid, the uncertainty of the conclusion cannot exceed the sum of the uncertainties of the premises. This puts pressure on the validity of modus ponens for right-nested counterfactuals.

Posted by Moritz.

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“Might”-Counterfactuals and Reversed Sobel Sequences

19 05 2008

To which extent are counterfactuals context-dependent? Lewis suggested that we can do without a systematic dependence on context by combining an invariant similarity relation with a variably strict analysis of counterfactuals. Recently, this approach has been challenged partly by drawing attention to the phenomenon of reversed Sobel sequences: sometimes it seems as if the order in which two counterfactuals are uttered makes for a difference in truth-value. Philosophers who take this phenomenon to be semantic in nature have reacted to it by allowing the similarity relation to vary from context to context (for instance, have a look at von Fintel’s semantics for counterfactuals, which you can find here). In this note, I’d like to challenge the semantic analysis of reversed Sobel sequences by arguing that it does not square well with a plausible link between “would”-counterfactuals and “might”-counterfactuals.

Here is the phenomenon. In an initial context, the counterfactual

(1) If she had been at the concert, she would have seen Mick Jagger

may be truly asserted, or so it is assumed. Subsequently, the counterfactual

(2) If she had been at the concert and got stuck behind a group of tall people, she would not have seen Mick Jagger

may be accepted, too. All this is to be expected on Lewis’s account: strengthening the antecedent is not a valid rule of inference. But now suppose that (1) and (2) are uttered in reversed order: it seems that asserting (1) after (2) is not o.k. There is something odd about saying

(3) If she had been at the concert and got stuck behind a group of tall people, she would not have seen Mick Jagger, but if she had been at the concert, she would have seen Mick Jagger.

So, can the order in which these counterfactuals are uttered affect their truth-values?

 Posted by Moritz.

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A Problem for Supervaluationism and Relativism?

31 01 2008

Supervaluationism and Relativism are popular accounts of future contingents. Even though they differ quite radically, they agree, at least in their most common forms, in how they evaluate present utterances about the future. For instance, consider the sentence A = ‘The coin will come down heads’ as it is evaluated from the present time t. Both the supervaluationist and the relativist will say: A is (a) supertrue at t/(with respect to t) just in case there is no objective chance at t that A is false, (b) superfalse at t/(with respect to t) just in case there is no objective chance at t that A is true, (c) neither supertrue nor superfalse otherwise. 

In general, supervaluationism and relativism seem to fare better as accounts of future contingents than as theories of vagueness, since there is no equivalent of the problem of higher-order vagueness for future contingents. In this note, I will try to challenge this assumption by pointing to a problem which seems to arise for supervaluationism and relativism with respect to future contingents without being as problematic in the case of vagueness. 

Posted by Moritz. Read the rest of this entry »

A Counterexample to the Principal Principle

25 01 2008

In his paper ‘A Subjectivist’s Guide to Objective Chance’, Lewis proposes an intimate connection between subjective probabilities and objective chances: the Principal Principle. In Lewis’s eyes, this principle captures almost all there is to know about our conception of objective chances.

In a forthcoming paper entitled ‘Knowledge and Objective Chance’, Hawthorne and Lasonen mention in passing a counterexample to the Principal Principle (a draft of which you can find here). Essentially, they think that instances of the contingent a priori provide a source of potential counterexamples. This idea stands in an interesting relation to a recent paper of Williamson (‘Indicative versus Subjunctive, Congruential versus Non-Hyperintensional Contexts’), in which the modal status of statements involving objective and subjective probabilities is discussed (a draft of which you can find there). It seems to me that Williamson’s considerations may provide a deeper reason to explain why we should not expect something like the Principal Principle to hold. In the following note, I will present a structurally similar counterexample by way of relating it to Williamson’s claims about the modal nature of the two kinds of probability.

Posted by Moritz.

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Grades of Truth-functionality

21 11 2007

Truth-functionality is a notion at the heart of classic propositional logic. A standard explication of the notion runs roughly as follows:

An n-ary connective C is truth-functional iff it corresponds to a truth-function.

More precisely, C is truth-functional iff there is a mapping from the truth-values of any n sentences S1 … Sn to the truth-value of the concatenation of C with S1 … Sn.

According to that explication, the following operators are equally truth-functional:

T It is true that …

TF It is either true or false that …

EX It is expressible in English that …

The two latter operators are only special in corresponding to a constant truth-function, namely the function that maps both True and False unto True.

But one may sense an important difference between TF and EX: the former connective actually operates on the truth-values of the embedded sentence such that the truth-value of the complex sentence is a result of that operation. But the latter connective does not operate on the truth-value of the embedded sentence at all. That it corresponds to a constant truth-function is not the result of its being sensitive to the truth-value of the embedded sentence; in some sense, it does not depend on that truth-value.

A definition of truth-functionality that captures the described difference between TF and EX can be given in terms of the explanatory connective ‘because of’:

An n-ary connective C is truth-functional iff the concatenation of C with any sentences S1 … Sn has the truth-value it has because of the truth-values of S1 … Sn.

This definition will classify the standard connectices of propositional logic as truth-functional (a true negation, for instance, is true because the negated sentence is false, and a false negation is false because the negated sentence is true). And it also classifies TF as truth-functional (‘It is either true or false that snow is white’ is true because ‘Snow is white’ is true; had the latter been false, then the former had been true because of that.)

But EX does not count as truth-functional in the defined sense (‘It is expressible in English that snow is white’ is true, but not because ‘Snow is white’ is true).

While I do not think that these considerations show that the standard definition of truth-functionality is in any way flawed (it is a technical notion after all), the alternative definition captures differences between operators that might as well be associated with the term ‘truth-functional’.

(See the papers-section for a more detailed exposition of my proposal.)

Posted by Benjamin.

Knights and Knaves

24 10 2007

Most of you will know the knights and knaves of Raymond Smullyan’s logic puzzles, the notorious truth-tellers and liars (actually, ‘liar’ is not wholly adequate, but anyway). In a true classic among these puzzles, you have to determine which one of two paths to choose (one leads where you want to be, one leads to your doom). Recently, a new solution to the puzzle has been published. You can find it here.

Posted by Benjamin.