In this section, we will present Ramsey’s account of conditionals. On this account, conditionals are supported by certain rules Ramsey calls ‘variable hypotheticals’. First, we will exemplify what variable hypotheticals are by taking recourse to Ramsey’s cake example. This example illustrates the role of variable hypotheticals by presenting a situation, where two people disagree about the merely possible. We then move on to Ramsey’s analysis of conditionals. It has an inferential and a probabilistic aspect, to say the least. We will treat the inferential part before we return to Ramsey’s footnote. Finally, we will observe that there is a tension within Ramsey’s account: the inferential analysis applies to counterfactuals, whereas the degree of belief formulation does not. This internal tension sheds doubt on whether Ramsey provides a unified account of conditionals.
Variable hypotheticals
Ramsey’s account of conditionals in GPC centers on what he calls ‘variable hypotheticals’. A variable hypothetical is a rule Ramsey expresses by “If I meet a \(\phi \) I shall regard it as a \(\psi \)” (p. 241). Such rules express your inferential habits, that is inferences you are anytime prepared to make. For example, ‘all men are mortal’ is a variable hypothetical that expresses the belief ‘whenever you encounter a man, you judge him to be mortal’. To believe this variable hypothetical means to believe, for any x that turns up, if x is a man, x is mortal. Ramsey describes the logical form of a variable hypothetical by \(\forall x (\phi (x) \rightarrow \psi (x))\) (see p. 249). In this section, we follow Ramsey by characterising variable hypotheticals in contrast to finite conjunctions and material implications.Footnote 6
A variable hypothetical goes beyond our finite experience. There is, for instance, no way to conclusively establish that all women—past, present, and future—are mortal. In particular, we cannot foretell the future with certainty, and so we cannot conclusively know whether or not any future woman will be mortal. Formally speaking, a variable hypothetical ranges over an infinite domain, that is over an infinite number of individuals or events. Ramsey distinguishes variable hypotheticals from “the other kind” of “general propositions” (p. 237). The latter but not the former range over a finite domain and can be thought of as conjunctions.Footnote 7 The sentence ‘Everyone in Cambridge voted for this motion’, for instance, does not express a variable hypothetical. The sentence is meant to express a finite list or class of people who voted for the motion: Moore, Russell, Braithwaite, and Wittgenstein voted for it. The sentence is meant to range over a rather restricted domain, viz. a voting event at a certain time and place. ‘Everyone in Cambridge voted for this motion’ expresses that everyone present voted yes. By contrast, the sentence ‘Everyone in Cambridge votes for this motion’ understood as a variable hypothetical would express that anyone who ever has voted, is voting, or will be voting in Cambridge is voting for this motion. It would support the conditional ‘if she had been present, she would have voted yes’. But this is, of course, not a viable common sense interpretation of the initial sentence. Upon hearing the sentence, we naturally and automatically restrict the domain to a finite class of people in Cambridge.
Variable hypotheticals can, however, be applied to finite classes. All people are mortal can be applied to the finite class of people in a spatio-temporal region such as Cambridge. If we are inclined to do so, we obtain the inferential result that Moore, Russell, Braithwaite, and Wittgenstein are mortal. Importantly, variable hypotheticals go beyond any single application to a limited domain. They rather express beliefs that also apply to future, not yet existing instances. A variable hypothetical you believe encodes a general expectation you have. If you believe the variable hypothetical that all women are mortal, then you expect any woman you will meet to be mortal. The expectations encoded in variable hypotheticals guide how a believer, or epistemic agent, will change her beliefs.Footnote 8 The set of believed variable hypotheticals is thus “the system with which the speaker meets the future” (GPC, p. 241).
Let us say the motion was in fact passed unanimously. It follows that ‘if she was there, she voted for it’ in the sense of the material implication: either she was not there, or she voted for it. On the other hand, it is sensible to say ‘if she had been there, she would have voted against it.’ Interestingly, this conditional implies that she was not there given the fact that the motion was passed unanimously. What is expressed goes beyond what actually happened in the limited domain and touches on a mere possibility. To entertain the merely possible belief that she was there makes the agent believe that she would have voted against the motion. An agent may believe that the character of this particular woman was so that she would have voted so. To utter the conditional reveals thus a part of the system with which the agent meets the future.
Ramsey (GPC, pp. 246-7) illustrates what variable hypotheticals express and how they relate to conditionals using the following situation. Suppose you decide not to eat the piece of cake in front of you, because you believe doing so will upset your stomach. You act on the belief ‘if I eat the cake, I will have a stomach ache’. Let us also suppose, after you did not eat the cake, that I disagree with you by thinking ‘if you had eaten the cake, you would not have had a stomach ache’. This situation illustrates that the conditionals cannot be understood as material implications. We both know that you did not eat the cake. So we think the proposition expressed by the material implication ‘if you eat the cake, you will have a stomach ache’ is true. As you have no factually false belief from my point of view, on what do I disagree with you?
The cake example shows that we can disagree about the merely possible, about what would have happened had the facts been different from how they actually are. We both agree on the facts of the matter, that is you did not eat the cake and you had no stomach ache. The difference we have is not primarily about the facts we believe. Our disagreement comes from the different variable hypotheticals we adopt. You believe ‘if I had eaten the cake, I would have a stomach ache’, whereas I believe ‘if you had eaten the cake, you would not.’ According to Ramsey (GPC, p. 247), these “assertions about unfulfilled conditions” reflect that our beliefs are guided by different variable hypotheticals. If our systems with which we meet the future are not equivalent, it is logically possible that the systems imply different expectations for the future. Hence, we may have a disagreement, even if the actual future agrees with both of our non-equivalent belief systems.
There are two different ways in which we might disagree. In the example, you say ‘If I eat the cake, I will have a stomach ache’, and I say ‘No, you will not’. Thereby, I am not negating the conditional understood as material implication. The root of our disagreement is that you adopt certain variable hypotheticals (and background facts) which I do not believe. Technically, there are two ways in which I can disagree with you. Let us assume you adopt the variable hypothetical \(\forall x (\phi (x) \rightarrow \psi (x))\) that supports your conditional. First, I can disagree with you by believing the antithetical \(\forall x (\phi (x) \rightarrow \lnot \psi (x))\). Second, I can disagree with you by believing none of the mentioned variable hypotheticals. Thereby, I neither infer \(\psi \) nor \(\lnot \psi \) from \(\phi \). I simply do not comply to make any inference as to \(\psi \)’s status. To deny the inferability of \(\psi \) from \(\phi \) is one way to deny the conditional. We will return to this issue below. In particular, we will explain how variable hypotheticals ‘support’ conditionals on Ramsey’s view.
A final remark on variable hypotheticals is in order. According to GPC, pp. 248-9, laws have the form of variable hypotheticals. A variable hypothetical \(\forall x (\phi (x) \rightarrow \psi (x))\) is a causal law if any instance of \(\psi \) denotes events no earlier than the respective instantiation of \(\phi \).Footnote 9 For what follows, it is important to keep in mind the close link between variable hypotheticals and laws.
Conditionals
In this section, we present Ramsey’s account of conditionals and illustrate how it works, before we note two properties of Ramsey’s conditionals. In GPC, p. 248, he summarizes his account as follows:
‘If p then q’ means that q is inferrible from p, that is, of course, from p together with certain facts and laws not stated but in some way indicated by the context.
The meaning of Ramsey’s ‘if p then q’ is relative to a rational agent who believes certain facts and variable hypotheticals. Believing a Ramsey conditional implies that the corresponding material implication should be believed. To see this, assume Ramsey’s ‘if p then q’. Hence, the agent can infer q from the assumption of p together with her contextual information she believes to be true. There are two cases: if the assumption is true, she should believe p and so q as well; if the assumption is false, she should believe \(\lnot p\). Hence, believing Ramsey’s conditional implies that in one of the two cases \(\lnot p\) should be believed to be true, and in the other \(p \wedge q\) should be believed to be true. But this implies that the material implication \(p \rightarrow q\) should be believed to be true whenever Ramsey’s conditional is believed.Footnote 10
However, the meaning of ‘if p then q’ is not exhausted by the material implication. This becomes evident when considering the two paradoxes of the material implication, that is cases where p is known to be false and cases where q is known to be true. The first case has already been exemplified by the disagreement about whether eating a particular cake would induce a stomach ache. In the cake example, what you and I disagree on is not covered by a mere material implication. In the second case, where the consequent q is known to be true, uttering ‘if p then q’ may, for example, indicate an explanation for q. Suppose you and I know that he has a stomach ache. I ask you: “But why?” “Well,” you answer, “if he ate this cake, no wonder!” In both cases, the point to state a conditional is to convey more than a material implication is able to.
As we have just seen, conditionals can be used to express more than material implications.Footnote 11 To capture what conditionals express, in GPC, pp. 248-9, and in a more detailed way in MHP, pp. 237-9, Ramsey gives the following analysis of conditionals. ‘If p then q’ means that there is an r such that \((p \wedge r) \rightarrow q\) is an instance of a variable hypothetical \(\forall x (\phi (x) \rightarrow \psi (x))\). That is, for some individual constant a, \(p \wedge r\) is an instance of \(\phi \), in symbols \(p \wedge r = \phi (a)\), and q an instance of \(\psi \), in symbols \(q=\psi (a)\), and while p may be merely supposed and q merely inferred, r must be believed to be true. There are some further conditions on r. It must be a conjunction of propositions which do not contain \(\vee , \rightarrow , \exists , \lnot \), but may contain \(\forall \), and there are certain temporal restrictions on which events it can describe (see below).Footnote 12r can be conceived of as information imported from the actual context.Footnote 13 And, finally, the assumption of p and r must be compatible in the sense that \(\lnot (p \wedge r)\) is not an instance of any law.Footnote 14 This is Ramsey’s inferential analysis of conditionals in a nutshell.
Let us illustrate how Ramsey’s analysis works. An agent believes the conditional ‘If she had eaten the cake, she would have had a stomach ache’ if the agent believes a specific woman and a bad cake to be co-present, and the agent believes the variable hypothetical ‘if a woman eats a bad cake, she will get a stomach ache’. Believing this variable hypothetical is tantamount to infer that she will get a stomach ache upon supposing that the woman eats the cake (whether or not she actually does), at least in the situation described by r, viz. the co-presence of a woman and a bad cake.Footnote 15 The example illustrates how variable hypotheticals support conditionals. They do so by providing an inferential connection from the antecedent, in a certain context, to the consequent. And this works if the antecedent is already believed to be true or merely supposed. Notice the role of the contextual information r: if the agent does not believe that the cake is bad, the variable hypothetical ‘if a woman eats a bad cake, she will get a stomach ache’ is not triggered. Whether the conditional is believed thus depends on the contextual beliefs. If the beliefs about the actual context were different in the example, the conditional would not be believed. Even if some variable hypotheticals would support a conditional, the conditional is only believed when there is some appropriate information r imported from the context.
The r is ‘in some way indicated by the context.’ It can be made explicit though (maybe only to a limited degree). Let us try. To believe ‘If Catherine had eaten the cake, she would have had a stomach ache’ requires to believe a variable hypothetical \(\forall x (\phi (x) \rightarrow \psi (x))\), let us say ‘every woman who eats a bad cake will have a stomach ache’, and to believe a certain r. Let r be given by \(r_1 \wedge r_2\), where \(r_1\) stands for Catherine is a woman and \(r_2\) for a bad cake is present. Someone who disagrees might simply believe \(r_1 \wedge r_3\) instead of \(r_1 \wedge r_2\), where \(r_3\) stands for a good cake is present. Most disagreements seem to be more complex. What makes a bad cake? Well, you might believe that ‘all cakes that contain salmonella bacteria are bad cakes’ (\(r_4\)), and that ‘This cake contains salmonella bacteria’ (\(r_5\)). A disagreement similar to the more simple one arises when two agents believe the variable hypothetical under consideration and one believes \(r_1 \wedge r_4 \wedge r_5\), while the other believes \(r_1 \wedge r_4 \wedge \lnot r_5\). Of course, the disagreement might be due to other factors, or concern the variable hypotheticals involved. In any case, let us keep in mind that whether or not a certain conditional is believed depends on the information imported by r.Footnote 16
The compatibility of the antecedent p and r deserves further attention. \(\lnot (p \wedge r)\) is logically equivalent to \(p \rightarrow \lnot r\). Hence, the condition that \(\lnot (p \wedge r)\) is not an instance of any law requires that the assumption of p does not lawfully imply the negation of r.Footnote 17 We may illustrate this requirement by means of another example. Let p stand for ‘Sally can jump four meters high’ and q for ‘Sally jumps on the roof to fix it.’ Furthermore, let r be the conjunction ‘Sally weighs about 50kg and is within the human range of strength, she lives on Earth and the surface gravity of Earth is about 9.807 \(m/s^2\)’. Now, p and r are incompatible. For, if you assume p, you can infer, using common-sensical laws of how physical bodies behave, that \(\lnot r\). In other words, \(p \rightarrow \lnot r\) is instantiated by your system of variable hypotheticals.
What should you do when p and r are incompatible? Well, you just need to find some \(r'\). One strategy would be that you shrink the content of r to find \(r'\). While the conjuncts of r are required to be believed true, it is not required that any conjunct that is believed to be true is contained in r. Ramsey does not forbid one to assume an antecedent that violates our understanding of, for instance, mechanical laws. Rather the content of r is restricted by the believed laws. To be explicit, you could simply take away some of the components of r. Assuming that Sally can jump four meters high might then trigger a believed variable hypothetical or another that lets you infer that Sally is particularly strong or that the surface gravity of Earth is lower, or some other proposition that you actually do not believe, but which you would believe under the assumption. When assuming an antecedent that violates your variable hypotheticals, you encounter the problem that it is underdetermined which conjuncts of r you are supposed to give up, and also what exactly you are supposed to infer. It seems that Ramsey gives no answer to this problem of underdetermination, leaving the actual reasoning to the agent under consideration.Footnote 18
In the Sally example, you assume to be true p, while you believe \(\lnot p\) due to your beliefs in r and the common-sensical variable hypotheticals you believe. Nevertheless, p is assumed to be true, and so other beliefs incompatible with p must be removed from r. It is a tricky question which beliefs must go, but it depends on the variable hypotheticals believed by the epistemic agent. The content of r is thus restricted by the meaning of the assumption p, the believed laws, and, of course, the facts believed to be true in the first place.
We have said that there are certain temporal restrictions on which events r can describe. To clarify these restrictions, Ramsey (MHP, pp. 237-9) puts forth an example similar to the following. You know that it didn’t rain and that Mary came to a meeting. Consider the two conditionals:
-
(a)
If it had rained, she wouldn’t have come.
-
(b)
If it had rained, she would have come all the same.
Both conditionals have an antecedent p you believe to be false. The consequents are believed to be false and true, respectively. For conditionals of type (a), Ramsey (MHP, p. 238) says that “no further limitation on r is necessary” beyond being a true conjunction that is compatible with p. By contrast, there be a temporal restriction on r for conditionals of type (b). In these cases, r must be restricted to describing events no later than the ones described in the antecedent. r must be about events previous to Mary’s “actual or possible setting out” (p. 239). It can, for instance, not include her actual presence at the meeting now. Otherwise you would believe the consequent of (b) merely due to r without any need of the antecedent. You would trivially believe (b) in absence of the temporal restriction. Notice that the temporal restriction applies only when you believe the consequent to be true, or when you have a belief that lets you infer the consequent without using the antecedent.
Ramsey notes two properties of his conditionals. First, contraposition fails. ‘If she came to the meeting, she would have left it by now’ does not entail that ‘If she had not left the meeting by now, she would not have come to it.’ The issue here is that you can infer q from p given a suitable context and your variable hypotheticals, and at the same time p from \(\lnot q\) together with some different variable hypotheticals. If Mary did not leave the meeting by now, she is still there, which implies that she came to it. p and \(\lnot q\) might trigger different variable hypotheticals and they might be lawfully compatible with quite different information imported from the context. Although contraposition fails in general, there is an interesting observation to be made. For the same true r, \(p \wedge r \rightarrow q\) is logically equivalent to \(\lnot q \wedge r \rightarrow \lnot p\). Without contextual shifts a Ramsey conditional thus implies its contrapositive material implication. Hence, contraposition holds for a Ramsey conditional if the r remains the same and \(\lnot q\) is lawfully compatible with it. By contrast, contraposition may or may not hold for a Ramsey conditional when the context shifts.Footnote 19
Ramsey (GPC, p. 240) states the second property as follows: “the ordinary hypothetical [...] asserts something for the case when its protasis is true: we apply the Law of Excluded Middle not to the whole thing but to the consequence only”. A Ramsey conditional asserts its consequent under the assumption of the antecedent, independently of whether the antecedent is believed to be true or not. Supposing the antecedent either makes you infer the consequent, or else it does not. It depends on your system of variable hypotheticals whether or not you infer the consequent, its negation, or you simply infer nothing from the supposed antecedent.
The above means that Ramsey’s conditional does not validate the Law of Conditional Excluded Middle: it is not the case that you infer q or \(\lnot q\) from an arbitrary supposition p; you may infer neither. But this principle holds for any non-contradictory p: if you infer q from a supposition \(p \wedge r\), then you cannot consistently infer \(\lnot q\) from the same supposition. As a consequence, it is not the case that you can consistently believe both “if \(p_r\) then q” and “if \(p_r\) then \(\lnot q\)”, where \(p_r\) is short for \(p \wedge r\). By contrast, you can, for instance, consistently believe both ‘If Cesar had entered the Vietnam War, he would have used catapults’ and ‘If Cesar had entered the Vietnam War, he would have used the atomic bomb (and no catapults)’. The reason is that the respective r differs for the two conditionals: the first antecedent says something like ‘If Cesar had entered the Vietnam War and it is the time of Cesar and ...’ (\(p \wedge r_a\)), whereas the second antecedent says something like ‘If Cesar had entered the Vietnam War and it is the time of the Vietnam War and ...’ (\(p \wedge r_b\)). Where \(p_{r_a}\) abbreviates \(p \wedge r_a\) and similarly for \(p_{r_b}\), ‘if \(p_{r_a}\) then q’ and ‘if \(p_{r_b}\) then q’ are not the same conditional, and so you may consistently believe ‘if \(p_{r_a}\) then q’ and ‘if \(p_{r_b}\) then \(\lnot q\)’.
Ramsey’s footnote reconsidered
Let us return to the footnote in GPC and its interpretation. The context of the footnote is the cake example, where we adopt counterfactuals that are contrary “in a sense”. Before you decide not to eat the cake in front of you, we ask ‘If you eat the cake, will you get a stomach ache?’ Ramsey’s test requires to add (in a way to be specified) the antecedent hypothetically to our certain beliefs, or “stock of knowledge”, to see whether or not the consequent follows. If you infer the consequent and I infer its negation upon supposing the antecedent, we contradict each other ‘in a sense’, viz. the consequents contradict each other under the assumption of the antecedent.
The footnote continues with what sounds like a characterisation of its first sentence. In the cake example, we fix our degrees of belief in the consequent q given the antecedent p. “If p turns out false”, you have not eaten the cake, “these degrees are rendered void. If either party believes \({\overline{p}}\) for certain, the question ceases to mean anything to him except as a question what follows from certain laws or hypotheses.” (p. 247) The last two sentences of the footnote raise at least one issue: what does it mean that degrees of belief are rendered ‘void’? If we interpret the degree of belief in q given p by the conditional probability \(P(q \mid p)\), as Adams and Edgington suggest, then this degree of belief is undefined when \(\lnot p\) is believed for certain.Footnote 20 However, after we both know that you did not eat the cake, the question ‘if you had eaten the cake, would you have come to have a stomach ache?’ barely ceases to mean anything. After all, we can have a disagreement about this question. Of course, the degrees of belief are rendered void in the sense that nothing about actuality follows from your zero-probability ‘belief’ that you have eaten the cake and your belief if you had, you would have had a stomach ache. Importantly, this does not mean that nothing possible follows from the mere assumption of p. The question becomes “a question about what follows from certain” variable hypotheticals. On this reading, ‘void’ cannot mean undefined; after all, it is not undefined what follows from the variable hypotheticals. In Sect. 5, we will propose how to reconcile this tension induced by cases, where \(\lnot p\) is believed for certain, but the suppositional degree of belief in q given p is still meaningful.
The last two sentences of the footnote put the focus on counterfactuals. Above we stipulated with Chisholm that counterfactuals are conditionals whose antecedents are contrary to the facts. Ramsey’s account of conditionals is relative to an epistemic agent. Here counterfactuals are thus best understood as conditionals, where the agent believes the antecedent to be false. Ramsey (GPC, p. 247) calls such ‘counterdoxasticals’ “assertions about unfulfilled conditions”, or “unfulfilled conditionals” (p. 246) for short. As should be clear by now, Ramsey’s account covers counterfactuals. In fact, he does not make a substantial difference between indicative, subjunctive, and counterfactual conditionals. And in his prime example, the cake scenario, the unfulfilled conditionals are formulated in the subjunctive mood.
Ramsey (MHP, pp. 242-3) gives even three reasons why we are interested in what follows from unfulfilled conditions: counterfactual reasoning is a “fiction of peculiar interest because near to reality”, it is a “way of apportioning praise and blame”, and a “way of stating laws”.Footnote 21 Let us consider the last reason. According to GPC, p. 249 ‘If p had happened, q would have happened’ is a way to state laws. If the conditional is supported by a causal law, q describes events no earlier than p. In this case, Ramsey (MHP, pp. 240-1) speaks of “causal implications” which can be indicated by unfulfilled conditions.
In the cake example, we may even disagree before you do no eat the cake. The reason for this disagreement is that your degree of belief in q given p differs from mine (see GPC, p. 247). We have different variable hypotheticals of the form ‘If \(\phi (x)\), then probability a for \(\psi (x)\)’ (see p. 251). These probabilistic variable hypotheticals are certain conditional degrees of belief Ramsey (GPC, p. 207) calls “chances”. They do not have the logical form of \(\forall x (\phi (x) \rightarrow P(\psi (x))=a)\), or \(P(\phi (x) \rightarrow \psi (x))=a\) for all x and some number \(a \in [0,1]\). They have rather the form of a conditional probability, that is \(P(\psi \mid \phi )=a\). In Ramsey’s terms, a probabilistic variable hypothetical “clearly” does not express a degree of belief in \(\lnot \phi \vee \psi \), but it does express a degree of belief in \(\psi \) given \(\phi \) (see pp. 246, 251). To adopt different probabilistic variable hypotheticals gives rise to different degrees of expectation. In brief, you judge it very likely that you will get a stomach ache given that you eat the cake, whereas I think it is rather unlikely. Here we see that these “degrees of hypothetical belief” influence how we behave, whether or not we eat the cake for instance (p. 246).
After you have decided not to eat the cake, you believe for certain that you do not eat the cake. By interpreting your degree of belief in having a stomach ache given that you eat the cake as conditional probability, your degree of belief is mathematically undefined. Supposing that you ate the cake, when in fact you didn’t, has no meaning with respect to an actual stomach ache. You simply cannot change the past. After your decision not to eat the cake, the mere supposition of eating it is thus meaningless in a sense: whatever follows from your hypothetical action does not actually follow from that action. After all, you decided not to eat the cake. Counterfactuals are practically meaningless insofar the consequences of the merely supposed antecedents are not actual, at least not in virtue of those antecedents. Yet what follows from the believed variable hypotheticals (and some compatible facts) is of course not meaningless. The mere supposition is meaningful in the sense that it still informs us about the variable hypotheticals that guide your beliefs, and possibly, if your hypotheticals are not misleading, about what would have happened if the state of affairs had been different. So, if you believe the antecedent to be false, the counterfactual is not entirely meaningless. We can still have a dispute about it.
We have just argued that, after the decision not to eat the cake, the counterfactual becomes irrelevant for your practical deliberation about what actually follows from your merely hypothetical action. In general, your degree of belief in the consequent given the antecedent is ‘rendered void’ in the sense of becoming irrelevant for what actually follows from the antecedent. But your degree of belief is still relevant for what would have followed from the antecedent.Footnote 22 Or so we propose to read Ramsey.
In GPC, Ramsey indicates a connection between variable hypotheticals and probabilistic variable hypotheticals. He states: “A law is a chance unity” (p. 251). Suppose a law \(\forall x (\phi (x) \rightarrow \psi (x))\). Let us abbreviate \(p \wedge r\) by \(p_r\). Hence, for all \(p_r\) and q instantiating \(\phi \) and \(\psi \), respectively, \(P(p_r \rightarrow q)=1\). From this it follows that \(P(q \mid p_r) = 1\), if \(P(p_r) > 0\).Footnote 23 A law implies that the corresponding conditional probability equals 1, under the assumption that the probability of the antecedent does not equal 0. However, generalisations and conditional probabilities do in general not perfectly fit together. Aside the limiting cases of \(P(p_r \wedge \lnot q) = 0\) and/or \(P(p_r)=1\), the probability of a material implication does, in general, not equal the corresponding conditional probability.Footnote 24 This foreshadows a tension within Ramsey’s account of conditionals to which we turn next.
An internal tension
Consider the situation of the second half of the footnote: you believe for certain that ‘you did not eat the cake’—that is you believe for certain that \(\lnot p\). We are interested in the counterfactual ‘If you had not eaten the cake, you would have had a stomach ache’. On Ramsey’s inferential account, you believe ‘if p then q’ if there is a properly constrained r such that \((p \wedge r) \rightarrow q\) instantiates one of your variable hypotheticals \(\forall x (\phi (x) \rightarrow \psi (x))\). As outlined above, the inferential Ramsey Test applies straightforwardly to this situation. By contrast, the putative redescription of Ramsey’s inferential account in terms of degrees of belief cannot handle the present situation. On this degrees of belief account, you believe ‘if p then q’ if your degree of belief in q given \(p_r\) is appropriate. Now, if you believe \(\lnot p\) for certain, \(P(p_r)=0\). And so your conditional degree of belief \(P(q \mid p_r)\) is undefined. As outlined above, the degrees of belief account does not apply to the present situation.Footnote 25
Is it a problem for Ramsey’s account that the inferential account applies to the considered situation, whereas the degrees of belief account does not? There are at least two answers. The first says yes. Ramsey does not have a unified account for conditionals. Rather he proposes two accounts that do not fit together. The degrees of belief account does not merely refine the inferential account to include cases of uncertainty. As shown above, the probability of a material implication and the conditional probability come apart when the probability of the antecedent is non-extreme and the probability of the antecedent and the negation of the consequent is not zero. Hence, even if both accounts are applicable, they will have a different result in many cases that involve uncertainty. We could not find any textual evidence supporting that Ramsey put forth two unrelated accounts. Indeed, he thinks that degrees of belief “in the imaginary case” can have meaning (GPC, p. 248). And his inferential account is meant to apply to “hypotheticals in general” (ibid.). Ramsey’s test in the footnote seems to be part of his more general account of conditionals.
The second answer is no, there is no problem for Ramsey’s account of conditionals. There is only one account of conditionals to be found. However, in cases where you believe \(\lnot p\) for certain, you only use the inferential account. In cases of uncertainty, you use the degrees of belief account only. Since the two accounts do not give the same results in general, we need to decide on one. Ramsey seems to ‘clearly’ favour the degrees of belief account for cases of uncertainty. In brief, Ramsey has but one account of conditionals consisting of two sub-accounts that share the workload.
Even if you think there are two sub-accounts that share the work, an issue arises. We may, and sometimes do, utter conditionals that make a probabilistic statement, such as ‘If you had eaten the cake, it would have been very likely that you would have had a stomach ache’, or even ‘If you had eaten the cake, it is four times more likely than not that you would have had a stomach ache’. The issue is simple to express: which of the two accounts is meant to apply? The answer is tricky. It seems that Ramsey’s inferential account does not apply, because the conditionals express uncertainty.Footnote 26 Ramsey’s degrees of belief account cannot apply, because you assign the antecedent probability zero. Both sub-accounts do not apply. A unified account of conditionals, however, ought to apply to such cases. So either Ramsey does not provide a unified account for qualitative and quantitative conditionals after all, or there is a mistake in Ramsey’s account. Observe that the whole tension between the two sub-accounts arises only due to the artefact that a conditional probability is undefined for zero-probability antecedents. Perhaps, there is only a technical problem while the spirit of Ramsey’s account is dead-on. In Sect. 5, we will argue for this claim, in particular, that Ramsey’s interpretation of conditional degrees of belief by conditional probabilities is mistaken with respect to his inferential account of conditionals. First, however, we will compare Ramsey’s account of conditionals to Stalnaker’s possible worlds semantics.