Supplement to Counterfactuals A. Indicative and Subjunctive Conditionals Historically, many philosophers have been tempted to assume that indicatives and subjunctives involve entirely different conditional connectives with related but substantially different meanings D. Lewis b; Gibbard ; Jackson ; J. Bennett

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Suppose there are two balls in a bag, labelled x and y. All you know about their colour is that at least one of them is red. Or: all you know is that they are not both red. Suppose you start off with no information about which of the four possible combinations of truth values for A and B obtains. You then acquire compelling reason to think that either A or B is true. In particular, you have no firm belief as to whether A is true or not.

You have ruled out line 4. The other possibilities remain open. Look at the possibilities for A and B on the left. You have eliminated the possibility that both A and B are false. So if A is false, only one possibility remains: B is true. The truth-functionalist call him Hook gets this right. Look at column ii. The non-truth-functionalist call her Arrow gets this wrong. Look at column v.

Eliminate line 4 and line 4 only, and some possibility of falsity remains in other cases which have not been ruled out. The same point can be made with negated conjunctions.

You rule out line 1, nothing more. Hook gets this right. Arrow gets this wrong. Here is a second argument in favour of Hook, in the style of Natural Deduction. Look at the last two lines of column i. Hook might respond as follows. How do we test our intuitions about the validity of an inference? The direct way is to imagine that we know for sure that the premise is true, and to consider what we would then think about the conclusion.

In this circumstance conditionals have no role to play, and we have no practice in assessing them. If our smoothest, simplest, generally satisfactory theory has the consequence that it does follow, perhaps we should learn to live with that consequence.

That needs investigating. We have seen that rival theories also have counterintuitive consequences. Natural language is a fluid affair, and we cannot expect our theories to achieve better than approximate fit.

There are some peculiarities, but as long as we are aware of them, they can be lived with. And arguably, the gain in simplicity and clarity more than offsets the oddities. The oddities are harder to tolerate when we consider conditional judgements about empirical matters.

The difference is this: in thinking about the empirical world, we often accept and reject propositions with degrees of confidence less than certainty.

We can, perhaps, ignore as unimportant the use of indicative conditionals in circumstances in which we are certain that the antecedent is false.

But we cannot ignore our use of conditionals whose antecedent we think is likely to be false. We use them often, accepting some, rejecting others. According to Hook, this person has grossly inconsistent opinions.

On the contrary, we would be intellectually disabled: we would not have the power to discriminate between believable and unbelievable conditionals whose antecedent we think is likely to be false. Arrow does not have this problem. I think B may be false, and will be false if certain, in my view unlikely, circumstances obtain. For example, I think Sue is giving a lecture right now. I reject that conditional. I think the consequent is true: I think a sufficient condition for the truth of the conditional obtains.

There are many ways of speaking the truth yet misleading your audience, given the standard to which you are expected to conform in conversational exchange. One way is to say something weaker than some other relevant thing you are in a position to say. Consider disjunctions. I am asked where John is. I am sure that he is in the pub, and know that he never goes near libraries. Another example, from David Lewis , p. Grice drew attention, then, to situations in which a person is justified in believing a proposition, which would nevertheless be an unreasonable thing for the person to say, in normal circumstances.

His lesson was salutary and important. He is right, I think, about disjunctions and negated conjunctions. But it is implausible that the difficulties with the truth-functional conditional can be explained away in terms of what is an inappropriate conversational remark. They arise at the level of belief. As facts about the norms to which people defer, these claims can be tested. A good enough test is to take a co-operative person, who understands that you are merely interested in her opinions about the propositions you put to her, as opposed to what would be a reasonable remark to make, and note which conditionals she assents to.

The Gricean phenomenon is a real one. Someone asks me whether the match will go ahead. I say something I believe, but I mislead my audience — why should I say that, when I think it will be cancelled whether or not it rains? This does not demonstrate that Hook is correct. Another example, due to Gibbard , pp. Intuitively this seems reasonable. The above examples are not a problem for Arrow. But other cases of embedded conditionals count in the opposite direction.

For Hook, Import-Export holds. Exercise: do a truth table, or construct a proof. Gibbard , pp. The antecedent of 2 entails its consequent. So 2 is a logical truth. So by Import-Export, 1 is a logical truth.

So 3 is a logical truth. Neither kind of truth condition has proved entirely satisfactory. How do you make such a judgement? You suppose assume, hypothesise that A, and make a hypothetical judgement about B, under the supposition that A, in the light of your other beliefs. A suppositional theory was advanced by J. Mackie , chapter 4. See also David Barnett But the most fruitful development of the idea in my view takes seriously the last part of the above quote from Ramsey, and emphasises the fact that conditionals can be accepted with different degrees of closeness to certainty.

Ernest Adams , , has developed such a theory. Similarly, we may be certain, nearly certain, etc. It is, at first sight, rather curious that the best-developed and most illuminating suppositional theory should place emphasis on uncertain conditional judgements.

If we knew the truth conditions of conditionals, we would handle uncertainty about conditionals in terms of a general theory of what it is to be uncertain of the truth of a proposition. But there is no consensus about the truth conditions of conditionals. It happens that when we turn to the theory of uncertain judgements, we find a concept of conditionality in use.

It is worth seeing what we can learn from it. The notion of conditional probability entered probability theory at an early stage because it was needed to compute the probability of a conjunction. Thomas Bayes wrote: The probability that two A simple example: a ball is picked at random. Call a set of mutually exclusive and jointly exhaustive propositions a partition.

The lines of a truth table constitute a partition. That is all there is to the claim that degrees of belief should have the structure of probabilities. Think of this distribution as displayed geometrically, as follows. Draw a long narrow horizontal rectangle. Divide it in half by a vertical line. Divide the left-hand half with another vertical line, in the ratio , with the larger part on the left.

Go back to the truth table. You are wondering whether if A, B. Assume A.

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## Indicative conditional

Suppose there are two balls in a bag, labelled x and y. All you know about their colour is that at least one of them is red. Or: all you know is that they are not both red. Suppose you start off with no information about which of the four possible combinations of truth values for A and B obtains. You then acquire compelling reason to think that either A or B is true. In particular, you have no firm belief as to whether A is true or not.

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## Indicative Conditionals

Therefore there are drawbacks with using the material conditional to represent if-then statements. One problem is that the material conditional allows implications to be true even when the antecedent is irrelevant to the consequent. The standard definition of implication allows us to conclude that, if the sun is made of plasma, then 3 is a prime number. Many people intuitively think that this is false, because the sun and the number three simply have nothing to do with one another. Logicians have tried to address this concern by developing alternative logics, e.

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