Epistemic Accounts of Indicative Conditionals

The standard use of the propositional calculus ('P.C.?) in analyzing the validity of inferences involving conditionals leads to fallacies, and the problem is to determine where P.C. may be ?safely? used. An alternative analysis of criteria of reasonableness of inferences in terms of conditions of justification rather than truth of statements is proposed. It is argued, under certain restrictions, that P. C. may be safely used, except in inferences whose conclusions are conditionals whose antecedents are incompatible with the premises in (...) the sense that if the antecedent became known, some of the previously asserted premises would have to be withdrawn. (shrink)

THE INDICATIVE CONDITIONAL. A PROBABILISTIC CRITERION OF SOUNDNESS FOR DEDUCTIVE INFERENCES Our objective in this section is to establish a prima facie case ...

It is now well known that, on pain of triviality, the probability of a conditional cannot be identified with the corresponding conditional probability [27]. This surprising impossibility result has a qualitative counterpart. In fact, Peter Gardenfors showed in [13] that believing 'If A then B' cannot be equated with the act of believing B on the supposition that A.

Recent work has shown that in spite of these negative results, the question 'how to accept a conditional?' has a clear answer. Even if conditionals are not truth-carriers, they do have precise acceptability conditions. Nevertheless most epistemic models of conditionals do not provide acceptance conditions for iterated conditionals. One of the main goals of this essay is to provide a comprehensive account of the notion of epistemic conditionality covering all forms of iteration.

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It is now well known that, on pain of triviality, the probability of a conditional cannot be identified with the corresponding conditional probability [25]. This surprising impossibility result has a qualitative counterpart. In fact, Peter Gärdenfors showed in [13] that believing ‘If A then B’ cannot be equated with the act of believing B on the supposition that A — as long as supposing obeys minimal Bayesian constraints. Recent work has shown that in spite of these negative results, the question (...) ‘how to accept a conditional?’ has a clear answer. Even if conditionals are not truth-carriers, they do have precise acceptability conditions. Nevertheless most epistemic models of conditionals do not provide acceptance conditions for iterated conditionals. One of the main goals of this essay is to provide a comprehensive account of the notion of epistemic conditionality covering all forms of iteration. First we propose an account of the basic idea of epistemic conditionality, by studying the conditionals validated by epistemic models where iteration is permitted but not constrained by special axioms. Our modeling does not presuppose that epistemic states should be represented by belief sets (we only assume that to each epistemic state corresponds an associated belief state). A full encoding of the basic epistemic conditionals (encompassing all forms of iteration) is presented and a representation result is proved. In the second part of the essay we argue that the notion of change involved in the evaluation of conditionals is suppositional, and that such notion should be distinguished from the notion of updating (modelled by AGM and other methods). We conclude by considering how some of the recent modellings of iterated change fare as methods for iterated supposing. (shrink)

Consider a rational agent X at certain point of time t. X's epistemic state can be represented in different ways.

Numerous triviality results have been directed at a collection of views that tie the probability of a conditional sentence to the conditional probability of the consequent on its antecedent. -/- In this paper I argue that this identification makes little sense if conditional sentences are context sensitive. The best alternative, I argue, is a version of the thesis which states that if your total evidence is E then the evidential probability of a conditional evaluated in a context where E is (...) salient is the probability of the consequent given the antecedent. The biggest challenge to this thesis comes from the 'static' triviality arguments developed by Stalnaker, Hajek and Hall. It is argued that these arguments rely on invalid principles of conditional logic and that the thesis is consistent with a reasonably strong logic that does not include the principles in question. (shrink)

In this paper I present a precise version of Stalnaker's thesis and show that it is both consistent and predicts our intuitive judgments about the probabilities of conditionals. The thesis states that someone whose total evidence is E should have the same credence in the proposition expressed by 'if A then B' in a context where E is salient as they have conditional credence in the proposition B expresses given the proposition A expresses in that context. The thesis is formalised (...) rigorously and two models are provided that demonstrate that the new thesis is indeed tenable within a standard possible world semantics based on selection functions. Unlike the Stalnaker-Lewis semantics the selection functions cannot be understood in terms of similarity. A probabilistic account of selection is defended in its place. -/- I end the paper by suggesting that this approach overcomes some of the objections often leveled at accounts of indicatives based on the notion of similarity. (shrink)

We present a semantic analysis of the Ramsey test, pointing out its deep underlying flaw: the tension between the “static” nature of AGM revision (which was originally tailored for revision of only purely ontic beliefs, and can be applied to higher-order beliefs only if given a “backwards-looking” interpretation) and the fact that, semantically speaking, any Ramsey conditional must be a modal operator (more precisely, a dynamic-epistemic one). Thus, a belief about a Ramsey conditional is in fact a higher-order belief, hence (...) the AGM revision postulates are not applicable to it, except in their “backwards-looking” interpretation. But that interpretation is consistent only with a restricted (weak) version of Ramsey’s test (in-applicable to already revised theories). The solution out of the conundrum is twofold: either accept only the weak Ramsey test; or replace the AGM revision operator ∗ by a truly “dynamic” revision operator ⊗, which will not satisfy the AGM axioms, but will do something better: it will “keep up with reality”, correctly describing revision with higher-order beliefs. (shrink)

A conditional takes the form ‘If A, then C’. On the truth-conditional view of conditionals, conditional statements state things with truth-conditions. On the suppositional view, conditional statements involve the expression of a supposition. I develop and defend a view on which conditional statements both state things with truth-conditions and express suppositions. On this view, something is fundamentally right about standard truth-conditional and standard suppositional views. Considerations in favor of conditional contents lead us to attribute truth-conditional contents to conditional statements; considerations (...) in favor of the suppositional view then lead us to an unexpected account of these contents. The resulting view has a number of benefits, including a unified treatment of conditional speech acts, a plausible account of our practice of ascribing truth-values to conditional statements, a simple explanation of the apparent equivalence between probabilities of conditionals and conditional probabilities, an intuitive treatment of ‘Gibbardian stand-offs’, a plausible logic of conditionals, and an explanation of why theorizing about conditionals has proved so difficult. (shrink)

According to the Ramsey Test hypothesis the conditional claim that if A then B is credible just in case it is credible that B, on the supposition that A. If true the hypothesis helps explain the way in which we evaluate and use ordinary language conditionals. But impossibility results for the Ramsey Test hypothesis in its various forms suggest that it is untenable. In this paper, I argue that these results do not in fact have this implication, on the grounds (...) that similar results can be proved without recourse to the Ramsey test hypothesis. Instead they show that a number of well entrenched principles of rational belief and belief revision do not apply to conditionals. (shrink)

The paper presents a non-monotonic inference relation on a language containing a conditional that satisfies the Ramsey Test. The logic is a weakening of classical logic and preserves many of the ‘paradoxes of implication’ associated with the material implication. It is argued, however, that once one makes the proper distinction between supposing that something is the case and accepting that it is the case, these ‘paradoxes’ cease to be counterintuitive. A representation theorem is provided where conditionals are given a non-bivalent (...) semantics and epistemic states are represented via preferential models. (shrink)

It is argued that indicative conditionals are best viewed as having truth conditions (and so they are in part factual) but that these truth conditions are ‘gappy’ which leaves an explanatory gap that can only be filled by epistemic considerations (and so indicative conditionals are in part epistemic). This dual nature of indicative conditionals gives reason to rethink the relationship between logic viewed as a descriptive discipline (focusing on semantics) and logic viewed as a discipline with a normative import (focusing (...) on epistemic notions such as ‘reasoning’, ‘beliefs’ and ‘assumptions’). In particular, it is argued that the development of formal models for epistemic states can serve as a starting point for exploring logic when viewed as a normative discipline. (shrink)

Vann McGee has proposed a counterexample to the Ramsey Test. In the counterexample, a seemingly trustworthy source has testified that p and that if not-p, then q. If one subsequently learns not-p, then one has reason to doubt the trustworthiness of the source and so, the argument goes, one has reason to doubt the conditional asserted by the source. Since what one learns is that the antecedent of the conditional holds, these doubts are contrary to the Ramsey Test. We argue (...) that the counterexample fails. It rests on a principle of testimonial dependence that is not applicable when a source hedges his or her claims. (shrink)

Frank Ramsey (1931) wrote: If two people are arguing 'if p will q?' and both are in doubt as to p, they are adding p hypothetically to their stock of knowledge and arguing on that basis about q. We can say that they are fixing their degrees of belief in q given p. Let us take the first sentence the way it is often taken, as proposing the following test for the acceptability of an indicative conditional: ‘If p then q’ (...) is acceptable to a subject S iff, were S to accept p and consider q, S would accept q. Now consider an indicative conditional of the form (1) If p, then I believe p. Suppose that you accept p and consider ‘I believe p’. To accept p while rejecting ‘I believe p’ is tantamount to accepting the Moore-paradoxical sentence ‘p and I do not believe p’, and so is irrational. To accept p while suspending judgment about ‘I believe p’ is irrational for similar reasons. So rationality requires that if you accept p and consider ‘I believe p’, you accept ‘I believe p’. (shrink)

In a recent pair of publications, Richard Bradley has offered two novel no-go theorems involving the principle of Preservation for conditionals, which guarantees that one’s prior conditional beliefs will exhibit a certain degree of inertia in the face of a change in one’s non-conditional beliefs. We first note that Bradley’s original discussions of these results—in which he finds motivation for rejecting Preservation, first in a principle of Commutativity, then in a doxastic analogue of the rule of modus ponens —are problematic (...) in a significant number of respects. We then turn to a recent U-turn on his part, in which he winds up rescinding his commitment to modus ponens, on the grounds of a tension with the rule of Import-Export for conditionals. Here we offer an important positive contribution to the literature, settling the following crucial question that Bradley leaves unanswered: assuming that one gives up on full-blown modus ponens on the grounds of its incompatibility with Import-Export, what weakened version of the principle should one be settling for instead? Our discussion of the issue turns out to unearth an interesting connection between epistemic undermining and the apparent failures of modus ponens in McGee’s famous counterexamples. (shrink)

How to accept a conditional? F. P. Ramsey proposed the following test in (Ramsey 1990).(RT) If A, then B must be accepted with respect to the current epistemic state iff the minimal hypothetical change of it needed to accept A also requires accepting B.

How to accept a conditional? F. P. Ramsey proposed the following test in . 'If A, then B' must be accepted with respect to the current epistemic state iff the minimal hypothetical change of it needed to accept A also requires accepting B. In this article we propose a formulation of , which unlike some of its predecessors, is compatible with our best theory of belief revision, the so-called AGM theory , chapters 1-5 for a survey). The new test, which, (...) we claim, encodes some of the crucial insights defended by F. P. Ramsey in , is used to study the conditionals epistemically validated by the AGM postulates. Our notion of validity is compared with the notion of negative validity used by Gärdenfors in . It is observed that the notions of PV and NV will in general differ and that when these differences arise it is the notion of PV that is preferable. Finally we compare our formulation of the Ramsey test with a previous formulation offered by Gärdenfors . We show that any attempt to interpret as delivering acceptance conditions for Ramsey's conditionals is doomed to failure. (shrink)

One of the main applications of the logic of theory change is to the epistemic analysis of conditionals via the so-called Ramsey test. In the first part of the present note this test is studied in the “limiting case” where the theory being revised is inconsistent, and it is shown that this case manifests an intrinsic incompatibility between the Ramsey test and the AGM postulate of “success”. The paper then analyses the use of the postulate of success, and a weakening (...) of it, generating axioms of conditional logic via the test, and it is shown that for certain purposes both success and weak success are quite superfluous. This suggests the proposal of abandoning both success and weak success entirely, thus permitting retention of the postulate of “preservation” discarded by Gärdenfors. (shrink)

In "Against the Indicative," AUSTRALASIAN JOURNAL OF PHILOSOPHY 72 (1994): 17-26, and more recently in "Classifying `Conditionals': the Traditional Way is Wrong", ANALYSIS 60 (2000): 147, V.H. Dudman argues that (a) `If Oswald didn't shoot Kennedy then someone else did' and (b) `If Oswald doesn't shoot Kennedy then someone else will' should not be classified together as "indicative conditionals." Dudman relies on the assumption that (a) is entailed by (c) `Someone shot Kennedy', whereas (b) is not entailed by (d) `Someone (...) will shoot Kennedy'. I argue that the same reasoning which shows that (d) does not entail (b) also shows that (c) does not entail (a). One upshot is that Dudman's and Mellor's respective interpretations of so-called past indicative conditionals cannot be correct. (shrink)

Peter Gärdenfors has proved (Philosophical Review, 1986) that the Ramsey rule and the methodologically conservative Preservation principle are incompatible given innocuous-looking background assumptions about belief revision. Gärdenfors gives up the Ramsey rule; I argue for preserving the Ramsey rule and interpret Gärdenfors's theorem as showing that no rational belief-reviser can avoid reasoning nonmonotonically. I argue against the Preservation principle and show that counterexamples to it always involve nonmonotonic reasoning. I then construct a new formal model of belief revision that does (...) accommodate nonmonotonic reasoning. (shrink)

Adams famously suggested that the acceptability of any indicative conditional whose antecedent and consequent are both factive sentences amounts to the subjective conditional probability of the consequent given the antecedent. The received view has it that this thesis offers an adequate partial explication of Ramsey’s test, which characterizes graded acceptability for conditionals in terms of hypothetical updates on the antecedent. Some results in van Fraassen may raise hope that this explicatory approach to Ramsey’s test is extendible to left-nested conditionals, that (...) is, conditionals whose antecedent is itself conditional in form. We argue that this interpretation of van Fraassen’s results is to be rejected. Specifically, we provide an argument from material inadequacy against a generalization of Adams’ thesis for left-nested conditionals. (shrink)

According to so-called epistemic theories of conditionals, the assertability/acceptability/acceptance of a conditional requires the existence of an epistemically significant relation between the conditional’s antecedent and its consequent. This paper points to some linguistic data that our current best theories of the foregoing type appear unable to explain. Further, it presents a new theory of the same type that does not have that shortcoming. The theory is then defended against some seemingly obvious objections.

Test for the rational acceptance of conditionals and it still incites much of the interest in conditional reasoning. For instance, the test has been considered as a good starting point for several formal semantics for conditionals. Furthermore, its ramifications have important implications for several disciplines, from logic and artificial intelligence to decision theory and psychology. This volume presents a small but fine sample of the state of the art of such multifarious area of research.

Abstract I analyse the relationship between the Ramsey Test (RT) for the acceptance of indicative conditionals and the so-called problem of decision-instability. In particular, I argue that the situations which allegedly bring about this problem are troublesome just in case the relevant conditionals are evaluated by non-suppositional versions, e.g. causal/evidential, of the test. In contrast, a suppositional RT, by highlighting the metacognitive nature of the evaluation of indicative conditionals, allows an agent to run a simulation of such evaluation, without yet (...) committing her to the acceptance of such conditionals. I conclude that a suppositional interpretation of RT is superior to its nonsuppositional counterparts and by briefly showing that a suppositional RT is compatible with a deliberational decision theory. (shrink)

Book synopsis: The Cambridge philosopher Frank Ramsey died tragically young, but had already established himself as one of the most brilliant minds of the twentieth century. Besides groundbreaking work in philosophy, particularly in logic, language, and metaphysics, he created modern decision theory and made substantial contributions to mathematics and economics. In these original essays, written to commemorate the centenary of Ramsey's birth, a distinguished international team of contributors offer fresh perspectives on his work and show how relevant it is to (...) present-day concerns. Each of the ten essays addresses fundamental and contentious issues, including success semantics, propositions, infinity, conditionals, conceptual analysis, decision theory, and intergenerational justice. They also shed light on the intellectual context in which Ramsey developed his thought, including his relationship with such leading thinkers as John Maynard Keynes, Bertrand Russell, and Ludwig Wittgenstein. The volume will be indispensable reading for anyone interested in the recent history of philosophy and economics, as well as for practitioners and students of logic, metaphysics, philosophy of mathematics, philosophy of science, philosophy of mind, decision theory, and welfare economics. (shrink)

Section 1 briefly examines three theories of indicative conditionals. The Suppositional Theory is defended, and shown to be incompatible with understanding conditionals in terms of truth conditions. Section 2 discusses the psychological evidence about conditionals reported by Over and Evans (this volume). Section 3 discusses the syntactic grounds offered by Haegeman (this volume) for distinguishing two sorts of conditional.

In this paper, I discuss conditionals as illocutionary speech acts whose interpretation depends upon the whole of the social context in which they are uttered and whose purpose is to affect the opinions and actions of others. I argue for a suppositional approach to conditional statements based in what philosophers call the Ramsey test and developing the psychological theory that conditionals elicit a process of hypothetical thinking in their listeners. By reference to the experimental psychological literature on conditionals, I show (...) that in general conditionals, even ones that are basic or abstract in nature, are not treated as truth-functional or material by ordinary people. Drawing upon the suppositional nature of conditionals and the influence of pragmatic implicature, I discuss uses of conditionals as advice, inducement, persuasions and dissuasion, arguing that speakers use conditionals to try to influence the beliefs and actions of their listeners by shaping their hypothetical thought about possibilities. (shrink)

There is an important class of conditionals whose assertibility conditions are not given by the Ramsey test but by an inductive extension of that test. Such inductive Ramsey conditionals fail to satisfy some of the core properties of plain conditionals. Associated principles of nonmonotonic inference should not be assumed to hold generally if interpretations in terms of induction or appeals to total evidence are not to be ruled out.

The book presents the results of the joint annual conference of the four Operations Research Societies DGOR, GM\OR, \GOR and SVOR, held in Vienna in 1990.

HANDBOOK OF DEFEASIBLE REASONING AND UNCERTAINTY MANAGEMENT SYSTEMS EDITORS: DOV M. ... and A. Hunter Volume 3: Belief Change Edited by D. Dubois and H. Prade HANDBOOK OF DEFEASIBLE REASONING AND ...

The purpose of this note is to formulate some weaker versions of the so called Ramsey test that do not entail the following unacceptable consequenceIf A and C are already accepted in K, then if A, then C is also accepted in K. and to show that these versions still lead to the same triviality result when combined with a preservation criterion.

The so called Ramsey test is a semantic recipe for determining whether a conditional proposition is acceptable in a given state of belief. Informally, it can be formulated as follows: (RT) Accept a proposition of the form "if A, then C" in a state of belief K, if and only if the minimal change of K needed to accept A also requires accepting C. In Gärdenfors (1986) it was shown that the Ramsey test is, in the context of some other (...) weak conditions, on pain of triviality incompatible with the following principle, which was there called the preservation criterion: (P) If a proposition B is accepted in a given state of belief K and the proposition A is consistent with the beliefs in K, then B is still accepted in the minimal change of K needed to accept A. (RT) provides a necessary and sufficient criterion for when a 'positive' conditional should be included in a belief state, but it does not say anything about when the negation of a conditional sentence should be accepted. A very natural candidate for this purpose is the following negative Ramsey test: (NRT) Accept the negation of a proposition of the form "if A, then C" in a consistent state of belief K, if and only if the minimal change of K needed to accept A does not require accepting C. This note shows that (NRT) leads to triviality results even in the absence of additional conditions like (P). (shrink)

This is a handbook article about indicative conditionals.

Conditional information can be equally asserted in the forms if p, then q (e.g., ?if I am ill, I will miss work tomorrow?) and q, if p (e.g., ?I will miss work tomorrow, if I am ill?). While this type of clause order manipulation has previously been found to have no influence on the ultimate conclusions participants draw from conditional rules, we used self-paced reading to examine how it affects the real time incremental processing of everyday conditional statements. Experiment 1 (...) revealed that clause order interacts with presuppositional congruency as readers hypothetically represent counterfactual statements. When if p, then q counterfactuals contained a presupposition that was incongruent with prior context, these statements took longer to read than when the presupposition was congruent, but for q, if p conditionals there was no such congruency effect. Experiment 2 revealed that reading times were influenced by the subjective probability of an indicative conditional regardless of clause order, with a penalty observed for low-probability statements relative to high-probability statements in both conditional clause orders. These data reveal a dissociation whereby clause order mediates the effect of suppositional congruency on reading times, but does not mediate the effect of subjective probability. (shrink)

This thesis is essentially a portfolio of four disjoint yet thematically related articles that deal with some semantic aspect or another of natural language conditionals. The thesis opens with a brief introductory chapter that offers a short yet opinionated historical overview and a theoretical background of several important semantic issues of conditionals. The second chapter then deals with the issue of truth values and conditions of indicative conditionals. So-called Gibbard Phenomenon cases have been used to argue that indicative conditionals construed (...) in terms of the Ramsey Test cannot have truth values. Since that conclusion is somewhat incredible, several alternative options are explored. Finally, a contextualised revision of the Ramsey Test is offered which successfully avoids the threats of the Gibbard Phenomenon. The third chapter deals with the question of where to draw the so-called indicative/ subjunctive line. Natural language conditionals are commonly believed to be of two semantically distinct types: indicative and subjunctive. Although this distinction is central to many semantic analyses of natural conditionals, there seems to be no consensus on the details of its nature. While trying to uncover the grounds for the distinction, we will argue our way through several plausible proposals found in the literature. Upon discovering that none of these proposals seem entirely suited, we will reconsider our position and make several helpful observations into the nature of conditional sentences. And finally, in light of our observations, we shall propose and argue for plausible grounds for the indicative/subjunctive distinction.distinction. The fourth chapter offers semantics for modal and amodal natural language conditionals based on the distinction proposed in the previous chapter. First, the nature of modal and amodal suppositions will be explored. Armed with an analysis of modal and amodal suppositions, the corresponding conditionals will be examined further. Consequently, the syntax of conditionals in English will be uncovered for the purpose of providing input for our semantics. And finally, compositional semantics in generative grammar will be offered for modal and amodal conditionals. The fifth and final chapter defends Modus Ponens from alleged counterexamples. In particular, the chapter offers a solution to McGee’s infamous counterexamples. First, several solutions offered to the counterexamples hitherto are all argued to be inadequate. After a couple of observations on the counterexamples’ nature, a solution is offered and demonstrated. the solution suggests that the semantics of embedded natural language conditionals is more sophisticated than their surface syntax indicates. The heart of the solution is a translation function from the surface form of natural language conditionals to their logical form. Finally, the thesis ends with a conclusion that briefly summarises the main conclusions drawn in its preceding chapters. (shrink)

Recent ideas about epistemic modals and indicative conditionals in formal semantics have significant overlap with ideas in modal logic and dynamic epistemic logic. The purpose of this paper is to show how greater interaction between formal semantics and dynamic epistemic logic in this area can be of mutual benefit. In one direction, we show how concepts and tools from modal logic and dynamic epistemic logic can be used to give a simple, complete axiomatization of Yalcin's [16] semantic consequence relation for (...) a language with epistemic modals and indicative conditionals. In the other direction, the formal semantics for indicative conditionals due to Kolodny and MacFarlane [9] gives rise to a new dynamic operator that is very natural from the point of view of dynamic epistemic logic, allowing succinct expression of dependence (as in dependence logic) or supervenience statements. We prove decidability for the logic with epistemic modals and Kolodny and MacFarlane's indicative conditional via a full and faithful computable translation from their logic to the modal logic K45. (shrink)

At the center of the literature on conditionals lies the division between indicative and subjunctive conditionals, and Ernest Adams’ famous minimal pair: If Oswald didn’t shoot Kennedy, someone else did. If Oswald hadn’t shot Kennedy, someone else would have. While a lot of attention is paid to figuring out what these different kinds of conditionals mean, significantly less attention has been paid to the question of why their grammatical differences give rise to their semantic differences. In this paper, I articulate (...) and defend an answer to this question that illuminates and unifies the meanings of both kinds of conditionals. The basic idea is that epistemic and metaphysical possibilities differ with respect to their interaction with time, such that there can be present epistemic possibilities with different pasts, while present metaphysical possibilities share the same past. The interpretation of conditionals is subject to a pragmatic constraint that rules out interpretations in which their consequents are directly settled by information used to build their domains. The past + future morphology on subjunctives, but not indicatives, is what allows them to receive a metaphysical interpretation in light of this pragmatic constraint. The resulting theory predicts several surprising features of indicatives and subjunctives, which I argue are correct. (shrink)

A proof by Allan Gibbard (Ifs: Conditionals, beliefs, decision, chance, time. Reidel, Dordrecht, 1981) seems to demonstrate that if indicative conditionals have truth conditions, they cannot be stronger than material implication. Angelika Kratzer's theory that conditionals do not denote two-place operators purports to escape this result [see Kratzer (Chic Linguist Soc 22(2):1–15, 1986, 2012)]. In this note, I raise some trouble for Kratzer’s proposed method of escape and then show that her semantics avoids this consequence of Gibbard’s proof by denying (...) modus ponens. I also show that the same holds for Anthony Gillies’ semantics (Philos Rev 118(3):325–349, 2009) and argue that this consequence of these theories is not obviously prohibitive—hence, both remain viable theories of indicative conditionals. (shrink)