Autodoxastic Conditional Reasoning: The Monotonic Case

Ramsey’s test for conditionals seems to be in conflict with the so-called Thomason conditionals. A Thomason conditional is a conditional in which either the antecedent or the consequent is a statement about the reasoning agent’s own beliefs. Several authors have pointed out that resolving the apparent conflict is to be sought by abandoning the belief revision interpretation of the Ramsey test in favor of a suppositional interpretation. We formalize an AGM-style notion of supposition, showing that it is identical to revision for agents who are not autodoxastic—agents who do not reason about their beliefs. We present particular realizations of supposition in terms of revision and identify the relations between the conditionals supposition and revision give rise to. 1 Ramsey, Thomason, and Moore In this paper, we attempt to consolidate two issues that, apparently, are in tension: the Ramsey test for conditionals and the so-called Thomason conditionals. The Ramsey test grounds the plausibility of conditionals (roughly, sentences of the form “If P then Q”) in a process of belief change. In an often quoted excerpt from [1], Robert Stalnaker gives a procedural interpretation of the Ramsey test: First, add the antecedent (hypothetically) to your stock of beliefs; second, make whatever adjustments are required to maintain consistency (without modifying the hypothetical belief in the antecedent); finally, consider whether or not the consequent is then true. Researchers in artificial intelligence (AI) and philosophical logic will immediately recognize the process Stalnaker is referring to as one of belief revision [2]. This is certainly a welcome interpretation of the Ramsey test, since belief revision has been thoroughly investigated both in AI (for example [3, 4]) and ? To appear as Haythem O. Ismail and Aya S. Mahfouz , Autodoxastic Conditional Reasoning: The Monotonic Case. In Beigl et al. (eds.), Modeling and Using Context: Proceedings of the 7th International and Interdisciplinary Conference (CONTEXT 2011), Springer-Verlag, Berlin, 2011. 2 Haythem O. Ismail and Aya S. Mahfouz philosophy. Unfortunately, interpreting the Ramsey test as implying a process of belief revision has been dealt at least two blows. First, Peter Gärdenfors [5] proved that the belief revision interpretation of the Ramsey test is inconsistent with a minimal set of harmless demands on a logical theory. We do not address this problem here; but see [6]. Second, a number of authors, since 1980 [7] till the turn of the decade [8–11], have pointed out that the Ramsey test, à la Stalnaker, provides counter-intuitive judgments of some conditionals—the so-called Thomason conditionals [7]. Thomason conditionals are conditionals in which either the antecedent or the consequent are statements about the reasoning agent’s own beliefs. In particular, they come in four main forms: TC1. If φ, then I believe φ. TC2. If I believe φ, then φ. TC3. If φ, then I do not believe φ. TC4. If I do not believe φ, then φ. Given the Stalnaker (belief-revision) interpretation of the Ramsey test, one should accept TC1 and TC2, and reject TC3 and TC4; otherwise, one would succumb to accepting a Moore-paradoxical sentence: M1. φ and I do not believe φ. M2. Not φ and I believe φ. But these judgments are not always correct, as several examples attest. (1) ? If Sally is a spy, then I believe that Sally is a spy. (2) ? If I believe that Sally is a spy, then Sally is a spy. (3) If Sally were deceiving me, I would believe that she was not deceiving me (because she is so clever). (4) Even if I were not to believe that Sally is a spy, she would be a spy (my misconceptions do not change the facts). A number of authors have attempted to reconcile the Ramsey test with such data [8–11]. The main idea is that Stalnaker’s interpretation of the Ramsey test is not exactly faithful to Ramsey’s real proposal. However, except for Willer [10] who presents a theory within the framework of update semantics, these proposals are largely informal. In this short report, we propose to stay as close as possible to the AGM belief revision tradition [2]. We introduce an AGMstyle belief change operator—supposition—that, we claim, adequately accounts 1 Strictly speaking, the agent’s own beliefs need not explicitly appear in a conditional to make the same “Thomason-effect”, but even in such cases, beliefs of the agents are assumed to establish the relevance of the antecedent to the consequent. 2 This is a classical example attributed to Richmond Thomason (hence, “Thomason conditionals”), but appears in print in [7]. Autodoxastic Conditional Reasoning: The Monotonic Case 3 for Thomason conditionals if taken to be the belief change operator implicit in Ramsey’s test. Unlike [10], however, we retain the common epistemic reading of the Ramsey test side by side with the suppositional one. Hence, we distinguish two classes of conditionals, based on these two readings. 2 Autodoxastic Agents We call an agent that can reason about its beliefs an autodoxastic agent. Why is it paradoxical for an autodoxastic agent to hold beliefs such as M1 and M2? Intuitively, in normal situations (which we assume throughout), an agent cannot hold that φ is true and simultaneously fail to believe it; or hold that φ is not true and, nevertheless, believe it. But things are not that simple, or otherwise how would one reply to Chalmers and Hájek’s remarks [8] about the oddity of unconditionally accepting TC1 and TC2? The two pairs (M1,M2) and (TC1,TC2) both make sense in some contexts of reasoning, and make no sense in others. In some contexts of reasoning—regular contexts—the contexts within which the agent normally and usually reasons, what the agent takes to be true is identical to what it believes. In such contexts, TC1 and TC2 are totally acceptable, while M1 and M2 are impossible. In non-regular contexts, the opposite is true: M1 and M2 are possible while TC1 and TC2 are usually not acceptable. What are these non-regular contexts? These are hypothetical contexts of reasoning, in which the agent entertains what the world would be like under one or more suppositions. But once the agent is endowed with the capacity for hypothetical reasoning, it may suppose anything about the world, even things about its own beliefs and their incompatibility with how the (hypothetical) world is. Certainly, the agent may entertain worlds in which φ is true, but it does not believe it. Hence, in non-regular contexts M1 and M2 are not paradoxical at all. In addition, TC1 and TC2 are clearly irrational, in general. Thus, unlike in TC1 and TC2, the “if” in TC3 and T4 does not mean “if I accommodate φ (respectively, ¬φ) in my regular context” (the belief revision, epistemic reading); rather, it means “if I suppose φ (respectively, ¬φ) in a non-regular context” (the suppositional reading). This being said, our task then is to formalize the distinctions between (i) regular and non-regular contexts, (ii) revision and supposition, and (iii) epistemic and suppositional conditionals. 3 We use “autodoxastic” rather than “autoepistemic” not just to be different from Moore [12]. In Moore’s autoepistemic logic, only what correspond to our regular contexts are considered. Thus, for an autoepistemic agent, knowledge and belief are equivalent. In our case, since we consider non-regular contexts of reasoning, using the term “autoepistemic” is at least misleading; we stand by “autodoxastic” then. 4 Haythem O. Ismail and Aya S. Mahfouz 3 Truth is in the “I” of the “B” Holder Suppose we have a logical language with an operator B for belief and a constant I that denotes the reasoning agent; the agent whose beliefs are represented by sentences of the language. What characterizes a regular context of reasoning? According to our discussion so far, regular contexts of reasoning should admit TC1 and TC2. Thus, we may label a context “regular” if it admits the following B-schema (after Tarski’s T-schema).