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Niki Pfeifer*Experimental probabilistic pragmatics beyond Bayes’ theorem DOI 10.1515/zfs-2016-0006 1 Introduction Quite recently, probabilistic approaches have become popular in formal e

Niki Pfeifer*

Experimental probabilistic pragmatics

beyond Bayes’ theorem

DOI 10.1515/zfs-2016-0006

1 Introduction

Quite recently, probabilistic approaches have become popular in formal episte-mology 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 experimental-psychological 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 posi-tion paper by Franke and Jäger (this volume), henceforth F&J, is a strong exam-ple 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 probabilis-tic 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

atten-tion 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

investi-gates (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

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 de-grees of belief Many probabilistic approaches define conditional probability p(B|A) by the fraction.1

, where it is assumed that p(A) > 0. (1)

p(A ∧ B) p(A) Assuming p(A) > 0 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) > 0 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

well-known fraction xy/z.2The 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 phenom-ena, 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 infer-ence 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:

1 A ∧ B denotes the conjunction of A and B as defined in classical logic.

2 For the sake of simplicity we suppose throughout this comment that the premise

probabil-ities (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.

– How should the uncertainty be transmitted in a coherent (or rational) way?

– How do people transmit the uncertainty? What probability constraints

lis-teners do actually infer from the speaker’s premises?

I think that it is plausible to assume that the normative and experimental re-sults 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

2 Sample theorems beyond Bayes’ theorem

This section sketches the proposed probability-logical approach to experimen-tal pragmatics by two examples The first example illustrates a pragmatic phe-nomenon 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 coher-ence 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

any-thing 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

Sanfilip-po 2016):

Theorem 1 From p(B|A) = x and p(C|B) = y infer 0 ≤ p(C|A) ≤ 1.

But why is Transitivity as an inference rule intuitively compelling to many people? By conversational implicature, so Adams’ argument goes, people

inter-pret the second premise by p(C|A ∧ B) = y Then, the following probabilistically

informative argument form results (Gilio 2002):

Theorem 2 From p(B|A) = x and p(C|A ∧ B) = y infer xy ≤ p(C|A) ≤ xy + 1 − x.

Recent work suggests that conditional probability is formally and descriptively useful to represent (uncertain) conditionals (see, e.g., Baratgin, Over, and Polit-zer 2014; Fugard, Pfeifer, and Mayerhofer 2011; Pfeifer 2013) If we interpret Theorem 2 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 prem-ise, which results into the following Cut rule (also called “Cumulative Transitiv-ity 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

corroborat-ed by the following psychological experiment In a between-group design, Pfei-fer and Kleiter (2006) used for the first group of participants tasks of the follow-ing 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 so-called “paradoxes of the material conditional.” There is nothing paradoxical

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:

It is easy to generate instantiations in natural language which form counterex-amples 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 explana-tion in purely semantical terms If the condiexplana-tional in (3) is interpreted by a conditional probability, we obtain the following probabilistically non-informa-tive inference rule (Pfeifer 2014):

Theorem 3 From p(C) = x infer 0 ≤ p(C|A) ≤ 1.

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),3the probabilis-tic non-informativeness of Theorem 3 holds within the framework of coherence

even if p(C) = 1 Moreover, it has been proven that not only Theorem 3 but also

Theorem 1 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

3 Traditional approaches to probability lead to the following counterintuitive consequence If

P(C) = 1, then by the fraction in Equation (1) we obtain P(C|A) = P(A∧C)=P(A) Thus, P(C|A)

= 1 if P(A) > 0, otherwise P(C|A) is undefined P(C|A) = 1 is counterintuitive and the other case, where p(C|A) is undefined because of a fraction over zero, is an undesirable result as

well.

Interestingly, experimental evidence shows that people seem to understand the probabilistic non-informativeness described in Theorem 3 Pfeifer and

Kleit-er (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 3

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.

3 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 rele-vant for formal (probabilistic) experimental pragmatics

Finally, I stress that the choice of an appropriate probability theory is im-portant not only for the basis of a probability logic but also for any other prob-abilistic 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 1 and in Theorem 3 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)

This is counterintuitive and does not match the experimental data The coher-ence approach to probability, however, avoids such problems with zero

anteced-ent probabilities and has already been successfully applied to many fields in-cluding human reasoning (Pfeifer and Kleiter 2009; Pfeifer 2013) Thus, I am

convinced that coherence based probability logic and experimental

investiga-tions beyond Bayes’ theorem fruitfully extend the probabilistic approach to pragmatics outlined by F&J

Acknowledgment: The author thanks two anonymous referees for helpful

com-ments Niki Pfeifer is supported by the DFG grant PF 740/2-2 (within the DFG Priority Programme SPP1516 “New Frameworks of Rationality”)

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