Gibbardian Stand-Offs:

What They Mean for NTV and Subjectivity

Probably the largest controversy when it comes to the study of conditionals is ascribing a manner in which to assign truth values. While it seems that we should be able to say indicative conditional statements like, “If I add two and three, I get five,” are definitively and objectively true, we come across road-blocks when trying to assign truth values to other indicative conditionals. One of these ‘road-blocks’ is what Jonathan Bennett calls a Gibbardian stand-off (2003, p. 83). Through examining stand-off cases, various problems with assigning truth values to conditionals become clear, and objectivity is questioned. Two possible answers to resolving Gibbardian stand-offs are assigning conditionals subjective truth values, or assigning them no truth value at all. Both theories have their shortcomings, and while they make some progress in helping us to understand and include conditional statements in our logical language and concepts, neither fully patches up the hole created by Gibbarddian stand-off examples.

            Gibbardian Stand-Offs. Dorothy Edgington points out that, “Conditionals do not go into truth-functional contexts, or into each other, easily…” (1995, p. 284). In his book, Bennett discusses Allan Gibbard’s original example of a type of conditional situation where applying objectivized truth values is difficult, if not impossible (p. 83). I find Bennett’s reformulated example of a Gibbardian stand-off to be more clear in expressing the problem. It is:

Top Gate holds back water in a lake behind a dam; a channel running down from it splits into two distributaries, one (blockable by East Gate) running eastwards and the other (blockable by West Gate) running weswards. The gates are connected as follows: if east lever is down, opening Top Gate will open East Gate so that the water will run eastwards; and if west lever is down, opening Top Gate will open West Gate so that the water will run westwards. On the rare occasion when both levers are down, Top Gate cannot be opened because the machinery cannot move three gates at once.

            Just after the lever-pulling specialist has stopped work, Wesla knows that west lever is down, and thinks ‘If Top Gate opens, all the water will run westwards’; Esther knows that east lever is down, and thinks ‘If Top Gate opens, all the water will run eastwards’.                                       Bennett, 2003, p. 85

As Bennett points out, both Esther and Wesla have good reasons to believe their conditionals, and both are perhaps correct in believing them. However, it seems that their conditionals are contradictory, and intuitively, two contradictory statements cannot both be correct. When it comes to conditionals, the principle of Conditional Non-Contradiction [CNC] supports this claim. CNC states that it is not the case that ((A-->C) & (A-->notC)). It would be ridiculous to say that two people are both correct when the first one says, “If it rains today, the grass will be wet tomorrow,” and the second one says, “If it rains today, the grass will not be wet tomorrow.” This is what Wesla and Esther are doing when Wesla says, “If Top Gate opens, all the water will run westwards (not eastwards),” while Esther says, “If Top Gate opens, all the water will run eastwards (not westwards).”

            The horseshoe analysis (that the material conditonal  is the indicative conditonal) would allow for a rejection of the principle of Conditional Non-Contradiction. It seems wrong, though, to think that it would make sense for someone to accept two opposing conditionals at the same time. There are countless counter-examples to demonstrate this. I cannot coherently believe both “If I eat this mushroom, I will die,” and “If I eat this mushroom, I will live,” at the exact same time about the exact same mushroom, much like the same piece of paper cannot be both black and white at the same time and in the same respects. Furthermore, as Bennet shows, it is irrational to believe an account of conditional probability where two conditionals A-->C and A--> notC are both accorded a probability of >0.5 (p. 84). I choose to hold onto CNC, because the consequences of rejecting it far outweigh any problems it would seem to solve in assigning truth values.

            Perhaps the main issue with Gibbardian stand-offs lies in the antecedent of the conditional. As Dorothy Edgington writes, “The Gibbard phenomenon arises if and only if there are currently ascertainable facts which rule out A” (1995, p. 295). Bennett agrees with this and says that most cases where a conditional’s antecedent is false are based upon stand-off situations (p. 87).  Practically, it seems that we have little use for conditionals with false antecedents, so perhaps we should dismiss them out of hand as being unimportant. It might be appealing to say that in the Top Gate example, since Top Gate will not open, then the rest of the conditional doesn’t really give us any useful information since it talks about what will not come to pass. At best, one could infer that if both Wesla and Esther are speaking truthfully, then Top Gate must be closed. However, we should be able to decide whether Wesla or Esther was right in the hypothetical realm where it is the case that Top Gate opens, and the fact that it seems we cannot is troublesome. Furthermore, we often use conditionals of this sort in everyday speech in spite of (or perhaps because of) the fact that the situations they describe are at best hypothetical. Examples of this, such as “If Sarah Palin’s daughter has an abortion, then she is a hypocrite,” or “If Barack Obama is a Muslim, then he is a traitor,” don’t immediately appear useless to us in spite of the fact that their antecedents are false.

            Thus we come to an impasse. Because of CNC we can’t apply two truth values to contradictory conditionals like those described in Gibbardian stand-offs. We can’t conclusively state which one is true and which one is false because both conditionals have comparable if not exactly similar reasons for their assertion. We can’t say both are false because then we have to assign every conditional with a “false A [antecedent] whose falsity was causally determined by the antecedent state of the world” a false truth value (Bennett, p. 86).  This would make many conditionals false that were previously thought to be straight-forward. (An example Bennett give is, “If she fought off the infection, it will be such a relief”). The inability to objectively assign truth values to conditionals in stand-off situations leads to the conclusion that indicative conditionals do not have objective truth values.

            Subjectivity. Perhaps if conditionals do not have objective truth values, then they have subjective truth values. Subjective truth values are described by Bennett to come in two main types: those with self-description involved and those without. Subjective truth values involving self-description depend upon not only the sentence being expressed but also on the epistemic state of the person expressing it. For this theory, conditional propositions have normal truth values, “but what proposition it is depends not only upon A and C and the normal meaning of à but also on some unstated fact about…” the person asserting it (Bennett, p 89). So in order to understand the truth or falsity of a conditional assertion, it is necessary to take the relationship of three things into account: the antecedent, the consequent, and the belief system of the speaker.

            Both Bennett and Edgington have objections to the theory of subjective truth values with self-description involved. According to Bennett, this system gets rid of contradiction in stand-off situations, but it does not express what we really mean when we use conditional statements (p. 90). If we take the Top Gate example in this system, what Wesla meant by saying, “If Top Gate is open, the water will flow westwards,” has an unstated inclusion of, “In Wesla’s belief system,” which doesn’t conflict with Esther if Esther is thought to meant, “In Esther’s belief system, if Top Gate is open, the water will flow eastwards.” Bennett says that this misses the point (p. 90). When Esther asserts what she does she is not trying to convince the listener that she has a high conditional probability for (open-->eastwards), but that there is a high conditional probability for (open-->eastwards).

            Edgington’s objection is that by taking the belief system or context of the speaker into account, ‘truth’ can be applied to any conditional statement. If the speaker may factor anything about their beliefs or things they take for granted into the evaluation of truth and falsity, they can always say that their asserted conditional was true in its original context. Edgington’s example to show this is that of a rain dance, where it is taken for granted “that either we won’t do our rain dance, or it will rain” which automatically makes “If we do our rain dance, it will rain” true in the original context (p. 308). The truth of the statement is therefore not affected if we do a rain dance and no rain comes, since it was true for us in our belief system within the original context. This is counter-intuitive and shows that the only thing the theory of subjective truth values with self-description accomplishes is to change the semantics of what generally counts for ‘truth’ into a much looser term.

            Since subjective truth values with self-description seem to fail because of the self-description aspect, subjectivity may hold if self-description is removed.  Edgington uses Stalnaker’s definition of what defines a context-set or set of beliefs in her objection to subjective truth values with self-description (pp. 307-8). For Stalnaker, the presuppositions that define a context-set “will include whatever the speaker finds it convenient to take for granted, or to pretend to take for granted” (Stalnaker, 1975 pp. 141-2). Self-description in a context-set of this sort holds too much of a sway on determining the truth or falsity of the conditional to the point that objective facts (like the fact that it did not rain when we danced) are discounted or made unimportant.  Truth or falsity of a proposition is dependent upon context, as Edgington admits (p. 307), so perhaps by narrowing the definition of what a context-set is and taking out self-description we may find a type of subjectivity that more accurately solves the problems brought up by stand-offs.

            The most compelling argument for subjectivity of this sort is attributed to William Lycan and described by Bennett in his book. Lycan’s analysis is, “the truth of A-->C depends upon whether C obtains in a certain class of states of affairs in which A obtains, the class being partly defined by their being states of affairs that the speaker regards as real possibilities in a certain sense” (p. 92). This narrows down our definition of what can be taken into account when determining a context or belief set. Instead of arbitrarily including any belief the speaker has or takes for granted, the belief set is merely partly defined by the fact that the speaker regards certain affairs as possibilities. In this theory, it would be wrong to say the rain dance conditional was true merely because of its original context-set, since ultimately the consequent does not follow from the antecedent. The antecedent and consequent are taken into fuller account than they were when self-description was involved. This occurs when it is determined whether C obtains in the class of states of affairs in which A obtains, only partly taking into account the things that the speaker regards as possibilities.  

Bennett’s main objection to this account is only that Lycan does not make clear whether or not the speakers of these conditionals “are always talking about themselves…” (p. 92). He concludes that if this theory holds, asserting a conditional still appeals to the relationship between the antecedent, consequent, and belief system described earlier, and that when someone says, “You have convinced me,” they merely mean that they believe the speaker’s probability for C given A is high within the speaker’s belief system. I’m not sure this is completely correct as it could hold that the person who is convinced assimilates the speaker’s belief system into his or her own view and finds it to be correct. So when I have convinced someone that “If I dance, then it will rain,” I do not merely convince them that I believe the conditional is true, but that my belief system which affected my belief in the conditional is also correct. Therefore, they would believe the conditional probability of “If I dance, then it will rain,” is high, not simply that it is high for me.

In this case, however, it would not be enough to simply assert a conditional, as the simple assertion is not truly an argument capable of convincing anyone unless they were already predisposed to believing it. No one believes me when I say, “If I dance, then it will rain,” based simply on my statement of it; they would only believe it if they already held the possibility of dances bringing rain within their system of beliefs. However, if I were to make an argument for why I assert this conditional (the world around us had to have been created by gods who love us. If we honor them with a dance, they will bring us rain, etc.) and you accept my argument, you come to accept my conditional as true. The problem with this is that, once again, ‘truth’ is something that can be too easily changed depending on the context of the situation or the argumentation involved. Obviously, there is still contention in this account of subjectivity, and it does not completely save us from the problems brought up by stand-off situations.

NTV. It is clear that assigning any sort of truth values to indicative conditionals is difficult at best, impossible at worst. Because of this, many writers have taken to the notion that conditionals do not have truth values, a theory known as NTV. Bennett discusses the seemingly impossible task of embedding conditionals as support for NTV, since propositions or sentences with truth values should be able to be embedded within a larger sentence (p. 95). He continues on to discuss possible routes by which NTV is reached.

The argument Bennett brings forth for NTV has to do with the difference between expressing and reporting (p. 102). Earlier, it has been concluded there is something subjective to indicative conditionals, but the speaker of the conditional does not outright ‘report’ this subjectivity. He only ‘expresses’ his own belief set, and this expression is where we get the subjective element. Since conditional statements of this sort are not reports, Bennett believes this leads to a lack of truth values in conditionals. Though there is the objection that indicative conditionals do report about the speaker’s conditional probability of C given A, Bennett contests that this is not at all what conditionals are actually used for (p. 108). He comes back to the point that when someone says “If you eat that, you will get sick,” they do not actually mean “For me, within my belief set, I believe that if you eat that, you will get sick.”

This seems to be true, but it doesn’t change the fact that there are indicative conditional statements most of us would accept as objectively true or false. For example, it seems that statements like “If you add three and two, you will get five” must have a truth value of 1 or even things as immutable as addition of numbers could be thrown into question. Similarly to how Bennett objects to subjectivity with self-description on the grounds that it does not truly describe what is generally meant when conditionals are asserted, I would contend that NTV doesn’t describe the reality of the situation. Most people would find it peculiar if we told them their conditional statement had no truth value to it. The fact that we do use conditionals in every day speech in a way that ascribes truth or falsity (statements like, “If you touch me again, I will hit you,” express a probability of me hitting you, and whether or not I do when you touch me determines whether or not the statement is false) points to conditionals having some sort of truth value.

Furthermore, does simply denying the existence of truth values in conditionals help at all when it comes to Gibbardian stand-offs? Let’s assume that in the Top Gate example, neither Wesla’s nor Esther’s conditional statement has a truth value. Even if they have no truth values, it still seems the case that their statements are contradictory. One may not be right while the other is wrong, but they are still different in a directly opposing manner. Would NTV mean they simply exist as sentences at the same time? Unless we reject CNC (which it can plainly be seen is the wrong thing to do- believing two opposing conditionals at the same time and in the same respects would have fatal consequences for logic and knowledge in general) there seems to be no way to solve the stand-off. I find it impossible to truly think of Wesla and Esther’s statements as not being contradictory because they have no truth value, and trying to force myself to believe it is counterintuitive. I fail to see how refusing to assign either conditional a truth value explains or helps the situation at all; it merely complicates and confuses it. Stand-offs make it extremely difficult to come up with a working solution to assigning truth values, but I don’t think it is the best to simply give up trying.

            Making Conclusions. It would seem that neither subjectivity nor NTV offer complete and sufficient solutions to the problems illuminated by stand-off situations. Subjectivity fails when it comes to self-description by creating a triadic relationship between antecedent, consequent, and belief set that doesn’t describe what we really mean when we use conditionals. NTV fails in a similar manner because the majority of people do intend to report truth or falsity when using conditional statements. Furthermore, there are indicative conditionals that seem obviously true or false, and to deny their having truth values would be counterintuitive. It may be the case that each theory applies to different types of conditionals (for instance, allowing indicatives making moral judgments to ascribe to subjectivity but not NTV), but what the various types are and how to define them would be a pointless and arduous task since conditionals are the same in their basic aspects. However, it must be said that both theories make some positive strides towards understanding the concept of what conditionals mean in our language.

References

Bennett, Jonathan (2003). A philosophical guide to conditionals. New York: Oxford

University Press.

Edgington, Dorothy 1995. On Conditionals. Mind, 104(414), 235-329.

Stalnaker, R. 1975. Indicative Conditonals. Philosophia, 5, 269-86, reprinted in

Jackson, F. (ed.) 1991, 136-54. Page references to 1991.