[Revised] Difficulties for the 'simple theory of conditionals'

Adam Rieger (2006) gives a simple theory of conditionals. It has simple truth-conditions: those of the material conditional. In addition, it has three simple assertion conditions.

A → C is assertable by S only if:

1. S knows that (A ⊃ C)


2. S doesn’t know that A and S doesn’t know that ~A


3. S doesn’t know that C and S doesn’t know that ~C. (2006: 233-234)

On the one hand, this account seems to avoid the paradoxes of material implication. The paradoxes result from the fact that either the (known) falsity of the antecedent or the (known) truth of the consequent will guarantee the truth of an associated material conditional. Knowing the truth-value of either the antecedent or consequent of a conditional, however, undermines its assertability on Rieger’s analysis; thus ruling out the paradoxes as problems for his simple theory.

On the other hand, one can state the paradoxes as single propositions, e.g., P → (Q → P) and ~P → (P → Q). In such cases, the associated sentences may be assertable on the simple theory. Let P = ‘Steve is in Paris’ and let Q = ‘Steve is in London.’ Suppose that I don’t know where Steve is. Then, given the material truth-conditions for conditionals, I don’t know whether the conditional, ‘If Steve is in London, then Steve is in Paris,’ is true. This means that the compound conditional, ‘If Steve is in Paris, then if he is in London, he is in Paris,’ is assertable. I cannot imagine a situation in which this claim would be usefully uttered, but it meets the conditions of assertability for the simple theory. It meets condition I because the associated material conditional is true and I know it is true. It meets conditions II and III because I don’t know whether the antecedent, ‘Steve is in Paris,’ or the consequent, ‘If Steve is in London, then he is in Paris,’ is true.

There are at least three possible replies. First, Rieger notes that his modest goal does not account for conditionals in unasserted contexts (2006: 237). There is available a second possibly stronger reply within the simple theory. For, if the import/export equivalence holds, then one could argue that these single statement versions of the paradoxes violate condition II. Consider the imported version of the problematic sentence: ‘If both Steve is in Paris and he is in London, then he is in Paris.’ I may not know the truth-value of the atomic components, but I surely know that the antecedent is false. Hence, this would not be assertable on the simple theory. This response only works for instances of the paradox in which the atomic components are contraries or contradictories. When the atomic components are neither contraries nor contradictories Rieger may wish to reassert that the simple theory was not constructed to handle unasserted or embedded conditionals.

A third reply depends on Rieger’s response to Jackson’s third objection. Jackson claims that if the material conditional gives the truth conditions for indicative conditionals, then ((P → Q) ∨ (Q → P)) is true but not generally assertable. Rieger, following Bennett (2005), claims that this objection isn’t about conditionals. Rather, it is a problem for the general assertion conditions of logical truths (2006: 236). Likewise, perhaps the problem with P → (Q → P) and ~P → (P → Q), is a problem regarding the assertion conditions of logical truths, not conditionals.

Unfortunately, this response isn’t adequate – not even to Jackson’s objection. These sentences are logical truths only if the material analysis gives the correct truth-values for indicative conditionals. Moreover, there is nothing in the simple account to rule out asserting these claims. The difficulty arises in association with the material analysis. By invoking any other theory of truth for indicative conditionals, be it Stalnaker’s (1968), Davis’s (1979) or Lycan’s (2005) , one would avoid this difficulty. Since the problem wouldn’t arise if conditionals had different truth conditions, this is a problem about the analysis of conditionals and not about the general assertion conditions for logical truths.

The claims of the previous paragraph undermine confidence in Rieger’s first assertion condition. Yet, as was noted, there are a number of easy fixes for this – though since these fixes employ non-material truth conditions, they wouldn’t accord with Grice’s defense or Rieger’s stated objectives. This raises the question of whether the other two assertion conditions are necessary for typical conditionals. Is there a class of normal conditionals whose typical assertion patterns do not require that the assertor not know the truth-value of either the antecedent or consequent?

One possibility is the class of conditionals known as Dutchman conditionals. For example, ‘If Analysis accepts this paper, then I’m a Dutchman.’ Such conditionals are assertable because the assertor invites the hearer to use modus tollens to conclude that Analysis won’t accept the paper. If this were a class of real conditionals rather than circuitous denials, say, it would be a stretch to call them normal. Thus, we can leave them aside.

Causal conditionals are yet another possibility. We assert such conditionals only in instances where the conditional correctly expresses the causal connection between antecedent and consequent, regardless of the truth-value of those elements. For example, suppose I watch a Rube Goldberg machine with my daughter. We both see that the unnecessarily intricate workings have as a last step a wooden hand feeding a bowl of fish. Because I can’t visualize as well as my daughter I don’t understand the connection between some of the components, especially at the end of the process. I might think that a bowling ball smashing an egg is tangential to a wooden arm feeding the bowl of fish. But my daughter’s keen sense tells her: ‘The fish are fed only if the bowling ball breaks the egg.’ I have seen every element of the machine work, but I have failed to appreciate one of the connections. In such a case, I will know both that the fish are fed and that the bowling ball broke the egg. Suppose I trace, perhaps in thought, the machine’s workings. When at last I understand its workings, I exclaim: ‘Ah, the fish get fed only if the bowling ball breaks the egg.’ To assert such a conditional requires that I know, or at least believe, that the proper connection obtains between antecedent and consequent. I’m not required to be uncertain about the truth-values of the components. For it doesn’t matter in such cases whether the antecedent is true, say, because the key feature for assertion is the causal connection.

Consider also what we might call explanatory conditionals such as are uttered by attorneys in the context of trials. Often an attorney, whether defense or prosecution, will assert conditionals for which the assertor knows the truth or falsity of (at least one of) the components. A defense attorney could assert: ‘If Mr. Jones called his sister at 10:00 from Uxbridge, then he wasn’t at Hornchurch at 10:15, as the prosecution has alleged.’ To support this assertion, the attorney might produce a map of the Tube. Uxbridge is the easterly end of the Metropolitan line, while Hornchurch is near the westerly extremity of the District Line – a ride of almost two hours with at least one exchange. Does it matter whether the attorney knows whether Mr. Jones called his sister at 10:00 from Uxbridge? If the simple theory is correct, then the attorney errs in uttering the conditional because she knows that the antecedent is true. Moreover, it wouldn’t be enough to add a condition regarding the intended targets of the assertion – that the targets of the assertion not know the truth of either element of the conditional. For, suppose that the first step in the attorney’s defense was to get the jury or judge to believe that Mr. Jones called his sister from Uxbridge at 10:00. Even on this supposition, the assertion of the conditional is still acceptable. Again, we don’t require uncertainty regarding the truth-value of either antecedent or consequent for acceptable assertion. Instead, we require both antecedent and consequent be possible.

Neither causal nor explanatory conditionals fit the model of Grice’s bridge convention example. Suppose some bridge player has a convention expressed by the conditional, ‘If I have a red king, I also have a black king.’ (Grice 1989: 60, quoted in Rieger 2005: 234) Such conditionals are outside the realm of the simple theory. Thus, the class of conditional for which II and III don’t apply is larger than first appeared. Are conditions II and III simply general, but accidental, features of typical conditional assertions? Consider: ‘The butler stole the vase; and if he stole the vase, then it probably isn’t the first time he’s stolen from the family.’ Does this contrast with: ‘If the butler stole the vase, then it probably isn’t the first time he’s stolen from the family’?

Take any acceptably uttered conditional using the criteria of the simple. One can discover or invent a different context in which that same conditional ought to be acceptable even though the assertor knows the truth or falsity of antecedent or consequent. This doesn’t disprove the necessity of Rieger’s conditions II and III – though it suggests as much. Perhaps the epistemic openness of the antecedent and consequent are accidental features of conditional utterances.

Here is a modified simple theory. A conditional, A → C is assertable for a speaker S only if:

I* S believes/knows A → C

II* ◊A

III* ◊C.

Let the truth conditions of ‘→’ be given by Davis’s possible worlds account (1979), though any non-material account would work. In addition let ‘◊’ be an alethic rather than epistemic modality. With these new conditions, we can avoid all of the prior problem cases. First, we don’t have the worries about the paradoxes of material implication because truth isn’t given in terms of material conditionals. This answers cases like P → (Q → P) and ~P → (P → Q), as well as Jackson’s ((P → Q) ∨ (Q → P)). Moreover, it answers these problems directly in terms of truth conditions rather than begging off the problems as ancillary to the present concerns. Second, the problem regarding the uncertainty of both antecedent and consequent goes away because we require of our indicative conditionals only that their components be (known to be) possible.

Finally, let’s examine the other supposed problem cases for Grice. If the modified theory gets them right, then it is preferable to the unmodified simple theory.

(1) If the sun goes out in ten minutes’ time, then the Earth will be plunged into darkness in about eighteen minutes’ time.

(2) If Reagan runs, Carter will win.

(3) If Reagan doesn’t run, Carter will win.

(4) The sun will come up tomorrow, but if it doesn’t, it won’t matter.

(5) The sun will come up tomorrow, but if it doesn’t, that will be the end of the world.

According to Jackson, example (1) is a problem for assert the stronger because the denial of the antecedent is at least as strong as the conditional and should therefore be asserted in its place. The simple theory handles this case because the chance that the sun goes out, though slim, undermines a speaker knowing that it won’t. So, the simple theory gets this answer correct. The modified theory handles (1) in terms of truth. Sentence (1) is true on this account because the closest world in which the sun goes out in ten minutes is also a world in which the Earth is plunged into darkness in eighteen minutes. For the same reason that the simple theory can rule out knowing that the antecedent is true, the modified theory can rule in this possibility. Thus, the modified theory gets (1) right too.

For sentences (2) and (3), the assertor believes that Carter will win. Jackson claims that both are in fact assertable, though neither would be assertable under assert the stronger. Rieger claims that this pair isn’t a problem because neither conditional is assertable on it’s own, though their conjunction may be. Here the modified theory diverges from the simple theory. The modified theory can have both (2) and (3) true, though given the actual results of the 1980 elections, it turns out that only one of these conditionals is true. Rieger rightly claims that there are assertable even if conditionals corresponding to (2) or (3). Still, it isn’t hard to describe a context in which both (2) and (3) are separately assertable.

Jim: Carter will win if Reagan doesn’t run.
John: What about if Reagan runs.
Jim: In that case too, Carter will win if Reagan runs.

Perhaps Jim’s second conditional sounds better with even if. Still, it is assertable on the interpretation of the modified theory in which the epistemic status of the assertor requires only belief, though it is unassertable if the requirement is knowledge. Hence, the modified theory can handle such pairs.

Finally, the last pair, (4) and (5), are problems for assert the stronger because when considered as material conditionals, they are equally strong. But (4) seems less assertable to Jackson than does (5). Rieger concedes an inability to distinguish (4) from (5), though he argues that such sentences include either a subjunctive element or uncertainty regarding the antecedent. In the former case, they are outside the purview of the simple theory. In the latter case, the simple theory handles them with ease.

As was the case with the previous examples, the modified theory can handle cases like (4) and (5) in terms of truth. One could argue that (4) is false on such a theory, while (5) is true. Hence, according to I*, one can assert (5) but not (4). This means that the modified theory is at least as good as the simple theory regarding the problem cases. And, since it also answers the paradoxes of material conditionals more directly, the modified theory is preferable.

Part of the problem with theories of assertion that depend upon knowledge is that the actual truth-value of the uttered proposition plays too big a role in its acceptability for the speaker. This is not, of course, to advocate the willy-nilly utterance of falsities. Rather, what seems to make a sentence acceptable for utterance is that the speaker must believe the sentence. For example, if I falsely believe that Jackson is Welsh, I can assert: Jackson is Welsh. When a suitably informed interlocutor informs me of my error, I withdraw my assertion – though not just because it is false, but that I have come to believe it is false. Insofar as belief is weaker than knowledge, we can be sure that those sentences we utter because we know them, will still meet this condition.

Regarding conditionals, then, the theory of assertion requires, for a speaker S, A → C is assertable for that speaker only if S believes that A → C. This puts the burden of conditional analysis back onto truth conditions.

The modified truth conditions for conditionals are: A → C is true iff Pr(C|A) is high and Pr(~A|~C) is high. These probabilities could be spelled out in terms of possible worlds as well. In that case, A → C is true just in case, both, the closest A-world is also a C-world, and the closest ~C-world is a ~A-world.

With these new truth conditions, we can avoid all of the prior problem cases. First, we don’t have the worries about the paradoxes of material implication because truth isn’t given in terms of material conditionals. This answers cases like P → (Q → P) and ~P → (P → Q), as well as Jackson’s ((P → Q) v (Q → P)). Moreover, it answers these problems directly in terms of truth conditions rather than begging off the problems as ancillary to the present concerns. Second, the problem regarding the uncertainty of both antecedent and consequent goes away because we require of our indicative conditionals that the antecedent makes the consequent more likely and that the denial of the consequent makes the denial of the antecedent more likely.

Finally, let’s examine the other supposed problem cases for Grice. If the modified theory gets them right, then it is preferable to the unmodified simple theory.

(1) If the sun goes out in ten minutes’ time, then the Earth will be plunged into darkness in about eighteen minutes’ time.

According to Jackson, example (1) is a problem for assert the stronger because the denial of the antecedent is at least as strong as the conditional and should therefore be asserted in its place. To apply the simple theory handles we note that the chance that the sun goes out, though slim, undermines a speaker knowing that it won’t. This means that this sentence passes condition (II) of the simple theory. However, as the simple theory requires that the speaker know that (A ⊃ C), there is nothing about this conditional that guarantees that A ⊃ C is true. Thus, the sentence isn’t assertable on the simple theory – contra intuition.

The modified theory handles (1) in terms of truth. Sentence (1) is true on this account because the closest world in which the sun goes out in ten minutes is also a world in which the Earth is plunged into darkness in eighteen minutes. Also, because the closest world in which the sun isn’t plunged into darkness in 18 minutes’ time is also a world in which the sun doesn’t go out eight minutes before, the modified theory offers an explanation for the truth of the conditional that accords better with what a normal assertor would give. Thus, the modified theory is better for sentences like (1).

For sentences (2) and (3), the assertor believes that Carter will win. Jackson claims that both are in fact assertable, though neither would be assertable under assert the stronger.

(2) If Reagan runs, Carter will win.
(3) If Reagan doesn’t run, Carter will win.

Rieger claims that this pair isn’t a problem because neither conditional is assertable on it’s own, though their conjunction may be. Here the modified theory diverges from the simple theory. The modified theory can have both (2) and (3) true, though given the actual results of the 1980 elections, it turns out that only one of these conditionals is true. Rieger rightly claims that there are assertable even if conditionals corresponding to (2) or (3). Still, it isn’t hard to describe a context in which both (2) and (3) are separately assertable.

Jim: Carter will win if Reagan doesn’t run.
John: What about if Reagan runs.
Jim: In that case too, Carter will win if Reagan runs.

Perhaps Jim’s second conditional sounds better with even if. Still, it is assertable on the interpretation of the modified theory in which the epistemic status of the assertor requires only belief, though it is unassertable if the requirement is knowledge. Hence, the modified theory can handle such pairs.

Finally, the last pair, (4) and (5), are problems for assert the stronger because when considered as material conditionals, they are equally strong.

(4) The sun will come up tomorrow, but if it doesn’t, it won’t matter.
(5) The sun will come up tomorrow, but if it doesn’t, that will be the end of the world.

Sentence (4) seems less assertable to Jackson than does (5). Rieger concedes an inability to distinguish (4) from (5), though he argues that such sentences include either a subjunctive element or uncertainty regarding the antecedent. In the former case, they are outside the purview of the simple theory. In the latter case, the simple theory needs to explain why someone who is uncertain about the antecedent would know that the conditionals associated with that antecedent are true (or false). Such an explanation won’t be easy to develop.

As was the case with the previous examples, the modified theory can handle cases like (4) and (5) in terms of truth. One could argue that (4) is false on such a theory, while (5) is true. This means that the modified theory is at least as good as the simple theory regarding the problem cases. And, since it also answers the paradoxes of material conditionals more directly, the modified theory is preferable.

References
Bennett, J. 2003. A Philosophical Guide to Conditionals. Oxford: Oxford University Press.
Clark, M. 1971. Ifs and hooks. Analysis 32: 33–39.
Davis, W. 1979. Indicative and subjunctive conditionals. The Philosophical Review v. 88 n. 4: 544-64.
Dretske, F. 1971. Conclusive reasons. The Australasian Journal of Philosophy 49: 1–22.
Dudman, V. 1992. Probability and assertion. Analysis 52: 204–11.
Gazdar, G. 1979. Pragmatics: Implicature, Presupposition, and Logical Form. New York: Academic Press.
Gettier, E. 1963. Is justified true belief knowledge? Analysis 23: 121–23.
Goldman, A. 1976. Discrimination and perceptual knowledge. The Journal of Philosophy 73: 771–91.
Grice, P. 1989. Studies in the Way of Words. Cambridge, Massachusetts: Harvard University Press.
Jackson, F. 1979. On assertion and indicative conditionals. The Philosophical Review 88: 565–89.
Lewis, D. 1996. Elusive knowledge. The Australasian Journal of Philosophy 74: 549– 67.
Lycan, W. 2001. Real Conditionals. Oxford: Clarendon Press.
Moore, G. E. 1962. Commonplace Book 1919–1953. London: Allen & Unwin.
Nozick, R. 1981. Philosophical Explanations. Cambridge, Massachusetts: Harvard University Press.
Stalnaker, R. 1968. A theory of conditionals. Studies in Logical Theory, American Philosophical Quarterly Monograph Series, No. 2: 98–112. Oxford: Blackwell.
Williamson, T. 2000. Knowledge and its Limits. Oxford: Oxford University Press.

Even If

For class I assigned Chapter 17 of Bennett's A Philosophical Guide to Conditionals on Even If conditionals. One of the elements I want to examine is the so-called consequent entailment problem.

For which of the following examples do we think the conditional entails its consequent?

(1) You will fail the class even if you get an A on the final paper.

(2) Steve won't pay retail even if his life depends on it.

(3) Professor Prude will fly off the handle even if the class misbehaves a little.

The question is whether the cases that seem to count against consequent entailment are really cases of even if?

The alternative, of course, is that even attaches to some other element of the sentence.

A standard example:

(4) He'll lose his contract even if he is arrested once.

The suggestion:

(4') He'll lose his contract if he is arrested even once.

These are the same examples that seem to count in favor of even if having if as a truth condition.

(5) He'll lose his contract if he is arrested once.

What do you think?

--Ian

only if and contraposition

Just a quick question: Do you think that any of the apparent counterexamples to modus tollens/contraposition would work if the conditional were stated using "only if" as the conditional marker?

1. If it rained, it didn't rain hard.
1a. It rained only if it didn't rain hard.

Try out any of the other apparent problems for MT/contraposition with only if in place.

The logically equivalent paraphrase in (1a) doesn't seem to have the same plausibility as (1), that is, I can't give a good context for using 1a.

I'm curious if any of you have thoughts about this phenomenon.

--Ian

Bennett's Logic of Indicative Conditionals

In Chapter 9, Bennett codifies the logic of indicatives by endorsing some inferences (as probabilistically valid), and rejecting others.

As regards the acceptable inferences, Bennett allows for restricted modus ponens. The restriction deals with what will count as the second premise in a canonical mp-inference. The unrestricted version allows anything that entails the antecedent of the conditional to be part of the inference. The restricted version allows only the antecedent as a premise.

There are four (other) rejected inference forms:
(1) Or-to-if: (P ∨ Q) ∴ (¬P → Q)
(2) Contraposition: (P → Q) ∴ ¬Q → ¬P
(3) Transativity: (P → Q), (Q → R) ∴ (P → R)
(4) Antecedent Strengthening: (P → Q) ∴ ((P ∧ R) → Q)

Bennett derives what he calls the "Security Thesis".

Security Thesis: If X is an argument whose conclusion is an indicative conditional A → C, and if what results from replacing → by ⊃ throughout X is a classically valid inference, then X is probabilistically secure to the extent that P(A) is high. (Bennett, 141)

One rejects or-to-if because accepting it would entail the ⊃-analysis of → (if one accepts the propositional analysis of →). To see this, recall that, at least according to Bennett (p.24) that → is at least as strong as ⊃. This means that (P → Q) ∴ (P ⊃ Q). To get the other direction, that (P ⊃ Q) ∴ (P → Q):

1. (P ⊃ Q)
2. (¬P ∨ Q) [1 Def of ⊃ and ∨]
3. (P → Q) [2 Or-To-If]
So, (P ⊃ Q) ∴ (P → Q)

Since (P → Q) entails (P ⊃ Q), and (P ⊃ Q) entails (P → Q), (P → Q) ≡ (P ⊃ Q).

I leave it to the reader to finish the proofs for the other rejected inferences.

Reading Update

Let's read chapter 9, The Logic of Indicative Conditionals (pp. 127-151), of Bennett's APGTC for the next round of posts. I'd like to see both comments and posts from some of you non-posting, non-commenting fellows.

Chapter 10 starts Bennett's discussion of subjunctive/counterfactual conditiionals. I don't want to start it until we've discussed indicatives a bit more.

No more meetings. The summer session is over, at least for me, so from here on out, let the blog be our meeting place.

--ijd

Bradley's Preservation Condition

Bradley (2000, p.220) offers the following as a "preservation condition" for conditionals:

If Pr(A) > 0, but Pr(B) = 0, then Pr(A → B) = 0.

He goes on to explain that all this means is "that one cannot be certain that B is not the case if one thinks that it is possible that if A then B, unless one rules out the possibility that A as well." (op cit)

Is this condition correct for all real conditionals (by "real" I simply mean to focus attention on conditionals that we find, as opposed to those we make up)? Moreover, is this condition similar to Jackson's robustness-condition for conditional assertion? In reading and rereading his explanation of the preservation condition I've come to wonder how different this is from Jackson's account as both can be explained in terms of modus ponens.

Recall that for Jackson (1987, pp. 28ff.), a conditional is assertible just in case it both has high enough probability on its own, and is robust with respect to its antecedent. This kind of robustness is explained by appealing to whether or not one's subjective probability regarding a conditional's acceptability is affected negatively were one to come to believe the antecedent. This rules out as unacceptable those conditionals whose truth depends solely upon the falsity of their antecedents. We would be wary of empolying such conditionals in modus ponens inferences.

Now look at the preservation condition and its explanation. The reason that you can't both doubt B and accept (even just a little that) A → B, unless you discount A, is that A and (A → B) would force B's truth (by modus ponens).

What is curious is that this preservation condition, unlike robustness wrt antecedent, seems to facilitate modus tollens as well. Accepting (even just a little that) (A → B) and denying that B, forces that you deny A (by modus tollens). If the preservation condition is true of all real conditionals, then Jackson's robustness condition is too narrow for assertibility. Instead, we would have to add two-sided robustness to Jackson's account (which I have advocated for other reasons).

References
Bradley, R. (2000), "A preservation condition for conditionals," Analysis 60.3, pp. 219-222.
Jackson, F. (1987) Conditionals Basil Blackwell.

Meeting Place

Meeting: Tuesday @ 8. Same place as last week. See you there.

--Ian

ps: here are a series of true conditionals.

1. If you go to the meeting tonight, you won't find food served. (Simple conditional?)
2. If you brought food to the meeting tonight, no one will be angry if you eat it yourself. (Compound conditional?)
3. Your host will have sodas, water, and beer if you're interested. (Biscuit conditional?)