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Zeitschrift für Sprachwissenschaft 2016; 35(1): 89–96
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 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: [email protected]
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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) > 0.
(1)
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 wellknown fraction xy/z. 2 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:
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 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.
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–
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How should the uncertainty be transmitted in a coherent (or rational) way?
How do people transmit the uncertainty? What probability constraints listeners do 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.
2 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 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-
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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 Politzer 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 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
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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 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),3 the probabilistic 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(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)
= 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.
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Interestingly, experimental evidence shows that people seem to understand
the probabilistic non-informativeness described in Theorem 3. 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 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 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 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).
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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
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Pfeifer, Niki & Gernot D. Kleiter. 2011. Uncertain deductive reasoning. In Ken Manktelow,
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