DE GRUYTER MOUTON Zeitschrift für Sprachwissenschaft 2016; 35(1): 89–96 Niki Pfeifer* Experimental probabilistic pragmatics beyond Bayes’ theorem DOI 10.1515/zfs-2016-0006 Introduction Quite recently, probabilistic approaches have become popular in formal epistemology and in the psychology of reasoning (Pfeifer and Douven 2014) This development has paved the way for new interdisciplinary approaches which are characterised by being formally elaborated (in a mathematical sense) and by investigations on their descriptive validity by means of experimentalpsychological methods It is natural that the success of such interdisciplinary approaches stimulates the development of new probabilistic approaches to study pragmatic phenomena, which are both formal and empirical The position paper by Franke and Jäger (this volume), henceforth F&J, is a strong example of this − in my view − fruitful and promising development F&J explain why Bayes’ rule is an important ingredient for investigating pragmatic phenomena from a probabilistic point of view No doubt that Bayes’ rule, which is based on the famous Bayes’ theorem, is important for probabilistic modelling From a probability-logical perspective, however, Bayes’ theorem can be conceived as one of many interesting theorems worth considering for formal model building in experimental pragmatics My comment advocates a probability-logical perspective on experimental pragmatics and draws the attention to selected theorems beyond Bayes’ theorem I argue that the research program outlined by F&J can be extended by exploiting existing formal and experimental results in probability logic I will illustrate how these results, which stem from single-agent contexts, can be adapted to study interactional (multiagent) contexts which are key to pragmatic phenomena What is probability logic? In a nutshell, probability logic studies uncertain argument forms − constructed from premises and conclusions − and investigates (deductive) probability propagation rules on how the uncertainty of the premises is transmitted to the conclusion (e.g., Hailperin 1996) Among the various approaches to probability, I advocate the coherence approach (for an *Corresponding author: Niki Pfeifer, Munich Center for Mathematical Philosophy, LMU Munich, E-mail: niki.pfeifer@lmu.de - 10.1515/zfs-2016-0006 Downloaded from De Gruyter Online at 09/12/2016 07:31:28AM via University of Exeter 90 DE GRUYTER MOUTON Niki Pfeifer overview see, e.g., Coletti and Scozzafava 2002) Coherence means to avoid bets which lead to sure loss Historically, coherence goes back to Bruno de Finetti’s conception of subjective probability, where probabilities are conceived as degrees of belief Many probabilistic approaches define conditional probability p(B|A) by the fraction.1 p(A ∧ B) , where it is assumed that p(A) > (1) p(A) Assuming p(A) > is important for traditional approaches to avoid fractions over zero In the coherence approach, however, conditional probability is primitive This allows for assigning directly probability values to the conditional event B|A, without assuming p(A) > or knowledge about p(A ∧ B) and p(A) This has various practical advantages including the possibility to properly manage zero antecedent probabilities of conditionals This is important, for example, to deal with counterfactuals (i.e., conditionals in the subjunctive mood, where the antecedent is factually false) In the context of coherence-based probability logic, Bayes theorem can be understood as an uncertain argument form consisting of the conclusion p(A|B) = u and the three premises p(B|A) = x, p(A) = y, and p(B) = z Here, the probability of the conclusion is a point probability value given by the wellknown fraction xy/z The propagation of the probabilities from the premises to the conclusion is traditionally studied in a non-interactional (single agent) context For applying probability-logical theorems to study pragmatic phenomena, I suggest that the probability propagation in uncertain argument forms can also be studied in interactional contexts The premises could, for example, represent what the speaker says and the conclusion would represent what inference the hearer could draw from the speaker’s premises This, I argue, should be studied both, normative-formally and experimentally Consequently, for studying pragmatic phenomena in interactive settings, I propose the following research questions, respectively: A ∧ B denotes the conjunction of A and B as defined in classical logic For the sake of simplicity we suppose throughout this comment that the premise probabilities (x, y, and z) are understood as point probability values They could be imprecise in the sense of interval-valued probabilities (characterised by lower and upper probability bounds) Then, u would of course be imprecise as well - 10.1515/zfs-2016-0006 Downloaded from De Gruyter Online at 09/12/2016 07:31:28AM via University of Exeter DE GRUYTER MOUTON – – Experimental probabilistic pragmatics 91 How should the uncertainty be transmitted in a coherent (or rational) way? How people transmit the uncertainty? What probability constraints listeners actually infer from the speaker’s premises? I think that it is plausible to assume that the normative and experimental results obtained in single-agent contexts can be transferred to communicative contexts between speakers and listeners In fact, any reasoning task can be seen as an interactive multi-agent task on a meta-level: the experimenter can be conceived as a speaker and what he says is expressed in the premises which are presented in the instruction of the reasoning task Likewise, the participant can be seen as the hearer who draws inferences or evaluates conclusions in the light of the speaker’s premises Sample theorems beyond Bayes’ theorem This section sketches the proposed probability-logical approach to experimental pragmatics by two examples The first example illustrates a pragmatic phenomenon in the context of Transitivity The second example illustrates that not all phenomena, which are traditionally conceived as pragmatic ones, are in fact pragmatic: it explains a “paradox of the material conditional” in purely semantical terms The second example should not be seen as an argument against (formal) pragmatics Rather, it highlights the importance of the coherence approach for formal modeling 2.1 Example 1: Probabilistic informativeness of Transitivity Historically, pragmatic considerations in the context of probability logic go back at least to Adams (1975) In his seminal book on conditionals, Adams pointed out that conversational implicatures are at work when we talk about transitive inferences While Transitivity is logically valid, one cannot infer anything about p(C|A), if the premise set consists of p(B|A) = x and p(C|B) = y, i.e., Transitivity is probabilistically non-informative (Gilio, Pfeifer, and Sanfilippo 2016): Theorem From p(B|A) = x and p(C|B) = y infer ≤ p(C|A) ≤ But why is Transitivity as an inference rule intuitively compelling to many people? By conversational implicature, so Adams’ argument goes, people inter- - 10.1515/zfs-2016-0006 Downloaded from De Gruyter Online at 09/12/2016 07:31:28AM via University of Exeter 92 Niki Pfeifer DE GRUYTER MOUTON pret the second premise by p(C|A ∧ B) = y Then, the following probabilistically informative argument form results (Gilio 2002): Theorem From p(B|A) = x and p(C|A ∧ B) = y infer xy ≤ p(C|A) ≤ xy + − x Recent work suggests that conditional probability is formally and descriptively useful to represent (uncertain) conditionals (see, e.g., Baratgin, Over, and Politzer 2014; Fugard, Pfeifer, and Mayerhofer 2011; Pfeifer 2013) If we interpret Theorem in terms of conditionals (⇒), it is easy to see that the antecedent of the first premise is added (or cumulated) to the antecedent of the second premise, which results into the following Cut rule (also called “Cumulative Transitivity rule”): From A ⇒ B and A ∧ B ⇒ C infer A ⇒ C (2) From an experimental pragmatics point of view, it is interesting to note that Adams’ conjecture that people interpret Transitivity as Cut has been corroborated by the following psychological experiment In a between-group design, Pfeifer and Kleiter (2006) used for the first group of participants tasks of the following kind to investigate Transitivity (Theorem 1): Exactly 80 % of the cars on a big parking lot are blue Exactly 90 % of blue cars have grey tyre-caps Imagine all the cars that are on the big parking lot How many of these cars have grey tyre-caps? The second group of participants received the same tasks, with the difference that the second premise was replaced by “Exactly 90 % of blue cars that are on the big parking lot have grey tyre-caps” to investigate the Cut rule (Theorem 2) Pfeifer and Kleiter (2006) observed that there were no statistically significant differences between participants who solved a number of Transitivity tasks and those who solved corresponding Cut tasks People did not infer probabilistically non-informative responses in the Transitivity condition but rather inferred probabilistically informative inferences as in the Cut condition (i.e., most of the responses were located within the normative correct probability bounds given in Theorem 2) This is strong evidence for the pragmatic hypothesis that − by conversational implicature − people interpret Transitivity as Cut 2.2 Example 2: A paradox of the material conditional Another example of probability-logical modeling is illustrated by one of the socalled “paradoxes of the material conditional.” There is nothing paradoxical - 10.1515/zfs-2016-0006 Downloaded from De Gruyter Online at 09/12/2016 07:31:28AM via University of Exeter DE GRUYTER MOUTON Experimental probabilistic pragmatics 93 about the material conditional per se, but the paradox arises if natural language conditionals are formalised by material conditionals For instance, consider the following inference rule: From C infer A ⇒ C (3) It is easy to generate instantiations in natural language which form counterexamples to (3) For example, it is counterintuitive to say that the conditional If there is life on Mars, then children like candy follows from the premise Children like candy Traditionally, pragmatic reasons were entertained to explain this fact that while (3) is logically valid under the material conditional interpretation, (3) appears counterintuitive Coherence-based probability logic, however, offers a compelling explanation in purely semantical terms If the conditional in (3) is interpreted by a conditional probability, we obtain the following probabilistically non-informative inference rule (Pfeifer 2014): Theorem From p(C) = x infer ≤ p(C|A) ≤ Here, the paradox is blocked: probabilistic information about C alone does not constrain the probability of the conclusion Thus, p(Children like candy | There is life on Mars) could be low even if p(Children like candy) is close (or even equal) to one We note that, contrary to traditional approaches to probability (which define conditional probability as in Equation (1) above),3 the probabilistic non-informativeness of Theorem holds within the framework of coherence even if p(C) = Moreover, it has been proven that not only Theorem but also Theorem is probabilistically non-informative even in the special case with probability one in the premises (Pfeifer 2014; Gilio, Pfeifer, and Sanfilippo 2016) This is one of many reasons why I think that the coherence approach to probability should be preferred to traditional approaches Traditional approaches to probability lead to the following counterintuitive consequence If P(A∧C) P(A) P(C) = 1, then by the fraction in Equation (1) we obtain P(C|A) = P(A) = P(A) Thus, P(C|A) = if P(A) > 0, otherwise P(C|A) is undefined P(C|A) = is counterintuitive and the other case, where p(C|A) is undefined because of a fraction over zero, is an undesirable result as well - 10.1515/zfs-2016-0006 Downloaded from De Gruyter Online at 09/12/2016 07:31:28AM via University of Exeter 94 Niki Pfeifer DE GRUYTER MOUTON Interestingly, experimental evidence shows that people seem to understand the probabilistic non-informativeness described in Theorem Pfeifer and Kleiter (2011) used vignette stories which asked the participants to imagine a factory which produces playing cards They were told that there is a shape (triangle, square, etc.) of a certain color (green, blue, etc.) on each card The premise was of the following kind: 90 % certain that a card shows a square Then, the participants were asked to evaluate the corresponding conditional (i.e., the conclusion): If the card shows a red shape, then it shows a square Most of the participants inferred that − based on the premise − one cannot infer anything about the conditional This is just what is predicted according to Theorem This example shows that probability-logical analyses may also demonstrate that alleged pragmatic phenomena can actually be explained by (probabilistic) semantics alone, without the need of pragmatic ad hoc hypotheses Concluding remarks In my comment, I proposed coherence based probability logic as an important method to formalise probabilistic premises (i.e., what the speaker says) and for investigations on what kind of conclusions the hearer should draw or actually draws I illustrated why and how formal-normative and experimental results of the proposed approach, which originally stem from semantic contexts, are relevant for formal (probabilistic) experimental pragmatics Finally, I stress that the choice of an appropriate probability theory is important not only for the basis of a probability logic but also for any other probabilistic approaches including the one proposed by F&J Traditional approaches to probability which, for example, define conditional probability (p(C|A)) by the fraction of joint (p(A ∧ C)) and marginal (p(A)) probabilities presuppose that the probability of the conditioning event is positive (p(A) > 0) However, as explained in Section 2, such a definition of conditional probability can lead to odd predictions In Theorem and in Theorem the probabilities of the conclusions would jump to one in the traditional framework when the premise probabilities are equal to one (Pfeifer 2014; Gilio, Pfeifer, and Sanfilippo 2016) - 10.1515/zfs-2016-0006 Downloaded from De Gruyter Online at 09/12/2016 07:31:28AM via University of Exeter DE GRUYTER MOUTON Experimental probabilistic pragmatics 95 This is counterintuitive and does not match the experimental data The coherence approach to probability, however, avoids such problems with zero antecedent probabilities and has already been successfully applied to many fields including human reasoning (Pfeifer and Kleiter 2009; Pfeifer 2013) Thus, I am convinced that coherence based probability logic and experimental investigations beyond Bayes’ theorem fruitfully extend the probabilistic approach to pragmatics outlined by F&J Acknowledgment: The author thanks two anonymous referees for helpful comments Niki Pfeifer is supported by the DFG grant PF 740/2-2 (within the DFG Priority Programme SPP1516 “New Frameworks of Rationality”) References Adams, Ernest W 1975 The logic of conditionals An application of probability to deduction Dordrecht: Reidel Baratgin, Jean, David E Over & Guy Politzer 2014 New psychological paradigm for conditionals and general de Finetti tables Mind & Language 29(1) 73–84 Coletti, Giulianella & Romano Scozzafava 2002 Probabilistic logic in a coherent setting Dordrecht: Kluwer Franke, Michael & Gerhard Jäger This volume Probabilistic pragmatics, or why Bayes’ rule is probably important for pragmatics Zeitschrift für Sprachwissenschaft 35(1) 3–44 Fugard, Andrew J B., Niki Pfeifer & Bastian Mayerhofer 2011 Probabilistic theories of reasoning need pragmatics too: Modulating relevance in uncertain conditionals Journal of Pragmatics 43(7) 2034–2042 Gilio, Angelo 2002 Probabilistic reasoning under coherence in System P Annals of Mathematics and Artificial Intelligence 34 5–34 Gilio, Angelo, Niki Pfeifer & Giuseppe Sanfilippo 2016 Transitivity in coherence-based probability logic Journal of Applied Logic 14 46–64 Hailperin, Theodore 1996 Sentential probability logic Origins, development, current status, and technical applications Bethlehem: Lehigh University Press Pfeifer, Niki 2013 The new psychology of reasoning: A mental probability logical perspective Thinking & Reasoning 19(3–4) 329–345 Pfeifer, Niki 2014 Reasoning about uncertain conditionals Studia Logica 102(4) 849–866 Pfeifer, Niki & Igor Douven 2014 Formal epistemology and the new paradigm psychology of reasoning The Review of Philosophy and Psychology 5(2) 199–221 Pfeifer, Niki & Gernot D Kleiter 2006 Is human reasoning about nonmonotonic conditionals probabilistically coherent? In Proceedings of the th Workshop on Uncertainty Processing (WUPES’06), Mikulov, Czech Republik, September 16–20, 2006 138–150 Mikulov: UTIA, Czech Academy of Science Pfeifer, Niki & Gernot D Kleiter 2009 Framing human inference by coherence based probability logic Journal of Applied Logic 7(2) 206–217 - 10.1515/zfs-2016-0006 Downloaded from De Gruyter Online at 09/12/2016 07:31:28AM via University of Exeter 96 Niki Pfeifer DE GRUYTER MOUTON Pfeifer, Niki & Gernot D Kleiter 2011 Uncertain deductive reasoning In Ken Manktelow, David E Over & Shira Elqayam (eds.), The science of reason: A festschrift for Jonathan St B.T Evans 145–166 Hove: Psychology Press - 10.1515/zfs-2016-0006 Downloaded from De Gruyter Online at 09/12/2016 07:31:28AM via University of Exeter ... conclusion is a point probability value given by the wellknown fraction xy/z The propagation of the probabilities from the premises to the conclusion is traditionally studied in a non-interactional... GRUYTER MOUTON Experimental probabilistic pragmatics 93 about the material conditional per se, but the paradox arises if natural language conditionals are formalised by material conditionals For... conception of subjective probability, where probabilities are conceived as degrees of belief Many probabilistic approaches define conditional probability p (B| A) by the fraction.1 p(A ∧ B) , where it