Aristotle, Rhetoric and Probability

7/27/2019 Aristotle, Rhetoric and Probability

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Aristotle, Rhetoric and Probability

Katarzyna Budzynska1, Magdalena Kacprzak2

1Institute of Philosophy, Cardinal Stefan Wyszynski University in Warsaw, Poland

[email protected]

2Faculty of Computer Science, Bialystok University of Technology, Poland

[email protected]

Abstract. The aim of the paper is to explore and link three concepts togetherthe role ofprobability in rhetoric, the meaning of this notion according to Aristotles theory ofpersuasion and the formal tools for representing the probability in important in rhetoric

interpretations. We focus, especially, on applications of objective and subjectiveprobability. For their formalization we adapt logics introduced by J. Y. Halpern.

Introduction

In the paper we are interested in exploring the following research questions: What role playsa notion of probability in rhetorical studies? Exists only one way of referring to this notion

during the persuasion? For example, does an expression Probably a person who commits acrime leaves trace evidence use the notion of probability in the same meaning as anexpression John is probably guilty? What Aristotle states about the probability? Whatformal tools should we use to represent the interpretation that is important for rhetoric?

We examine two levels of applying the notion of the probability in rhetoric. The first

level pertains to a frequency of satisfaction of an argumentations premise that expresses astatistical relationship. It uses the objective interpretation of the probability like e.g. in the

statement: Probably a person who commits a crime leaves trace evidence. It suggests astatistical relationship between committing a crime and leaving trace evidence, i.e., when thefirst phenomenon takes place then probably (almost always, usually) the other will take place

as well. The second level of application uses the subjective interpretation of probability whichis related to the power of arguments or the strength of belief in a thesis. In other words, it is

the measure ofthe uncertainty of persuasions participants. For example, the probability usedin the expression John is probably guilty refers to the uncertainty about Johns guilt. Thekey difference between these two types of probabilities is the fact that statements of objective

probability refers to real, statistical state of the world, while statements of subjective

probability refers to someones beliefs about the world.It is hard to not notice that the notion of probability plays an important role in

Aristotles rhetoric. Originally he uses the objective interpretation of probability referring to

the fact that people more often persuade each other about what happens usually than aboutwhat happens always. However, the subjective interpretation of probability can be also found


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in his theory. It is connected to the notion which is especially important in his rhetoricwithsuccess of convincing and persuasiveness. Obviously, the aim of a speech is to make the

audience believe our thesis in the highest degree. When we want to express that some

arguments are stronger and the other are weaker, we need to refer to the (greater and less)

grades of audiences uncertainty.

Moreover, we consider what formal tools can be adapted to use the notion ofprobability in the rhetorical studies efficiently. We show that the aspects of rhetoric

mentioned above can be nicely expressed by means of two deductive systems proposed by

Joseph Y. Halpern. They are the first-order logics of probability. One of them puts theprobability on the domain and the other on possible worlds. Therefore, the former issuitable to reason about statistical information that may be conveyed by premises of

argumentation and the other to reason about degrees of audiences beliefs. Let us define the key words that we use throughout the paper, namely: rhetoric,

persuasion and probability. We understand rhetoric as the theory of persuasion. The

persuasion is a type of action which aim is to influence the change of someones beliefs. Atleast two individuals take part in argumentation: a persuader (proponent, speaker) a personwho gives arguments to make someone believe his thesis, and an audience a person towhom the argumentation is addressed. According to Aristotle, rhetoric may be defined as thefaculty of observing in any given case the available means of persuasion (Rhet. I.2 1355b).Finally, the probability of an event is a measure of the possibility of that event to occur as a

result of an experiment. In probability spaces different probabilitys measures can beassumed. Thereby there is not one formal probability system. As a result various

interpretations of the probability are discussed (see e.g. (Kyburg 1970) or (Skyrms 2000)).

We focus on objective and subjective interpretations expressed in terms of the logics of

Halpern. Both of them satisfy classical Kolmogorovs axiomatization.The rest of the paper is organized as follows. In the first section we discuss where we

can find probability in rhetoric and where the analysis of this notion can be fruitful for

development of this discipline. In Section 2 we examine how Aristotle understands the

probability and its role in persuasion. In the last section we present the formal tools that canbe used to investigate the probability in rhetoric.

1. Probability in rhetoric

In this section we present two fields of rhetoric where the notion of probability may be

applied to describe some phenomena characteristic for rhetorical actions (see also (Budzynska

2004)). The first field is related to the objective probability. It is understood as statistical

measures that represent various assertions about the objective statistical state of the world. Inargumentation, this probability can be assigned to the statistical relationship described by its

premise. It plays an important role when we want to consider to what degree we can be

mistaken when reasoning with the use of such a premise. That is, it allows expressing the

degree of validity of a given persuasion.The second field of application is associated with the subjective probability. It is

understood as uncertainty measures or degrees of belief that represent various assertions

about the subjective state of someones beliefs. The subjective probability can be assigned toany sentence in argumentation according to the judgment of its participant (proponent or

audience). It indicates in what degree the participant believes in the truthfulness of a sentence

(a thesis or an argument). It plays an important role in rhetorical studies when we want to

express the successor more generallythe effects of a particular persuasion.


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1.1 Objective probability

There are two main types of relationships that argumentation may refer to: universal and

statistical ones. The statistical relationships are characteristic for those fragments of the

reality where phenomena are shaped by various random factors. In consequence, therelationships are rarely exceptionless, i.e. they are not universal. Since people tend to discussdifficult and complex topics rather than simple ones, they often refer to statistical

relationships. In this section, first we describe what we mean with an objective statisticalprobability. Then, we demonstrate how it influences the validity of an argumentation.

Let us start with an example. Consider the following conditionals: (1) If someonecommits a crime using weapon [A], then he is a criminal [B], (2) If someone commits acrime using weapon [A], then he leaves trace evidence [C]. Say that the relationshipexpressed in the first statement is universal. This means that every person who commits a

crime using weapon is a criminal (i.e. it always happens). The second relationship can be

viewed as statistical in this sense that for the most part a person who commits a crime using

weapon leaves trace evidence (i.e. it happens usually but not always).We may represent the relationships in terms of relations between sets (see Figure 1).

We can understand the statement (1) in the following way: the antecedent of the conditionaldescribes the set of people who committed a crime using weapon (the set symbolized as A)

and its consequent describes the set of criminals (the set B). Then, the conditional If A thenB expresses that the set A is a subset of B, i.e. for every person x, if x commits a crime usingweapon, then x is a criminal.

The statistical relationship represents a different kind of relations between sets. In the

example (2), the antecedent describes A and its consequent describes C - the set of people

who leave trace evidence on the scene of the crime. The conditional If A then C expressesthat the most part of the set A is a subset of C, however a part of the set A is not a subset of

C. In other words, the most part of A is in their intersection, however there are still some A

which are not C, i.e. a person who commits a crime using weapon usually leaves traceevidence, however there are cases when someone commits a crime using weapon and leaves

no trace evidence.

Fig.1 Argumentation refers to different types of relationships: (1) universal, (2) statistical.

The objective probability can be viewed as a ratio of the number of the elements of the

sets intersection to the number of all elements of one of these sets. In the example, the fact

that the most of people who commit a crime using weapon leaves trace evidence means thatthe ratio of the number of people who commit a crime using weapon and leave trace evidence

A

B

Universal relationship:If A, then B

A

C(1) (2)

Statistical relationship: If A, then C


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(A and C) to the number of all people who commit a crime using weapon (A) is close to 1.

Furthermore, each universal relationship has the probability 1. Notice that the probability

specified in this manner is an objective property of the reality itself. Once the premises used

in a persuasion refer to such statistical relationships, the probability will become the property

of the sentences as well.

Now we are ready to demonstrate how the objective probability influences a validityof rhetorical arguments (see also (Budzynska 2006)). Deduction employs schemes which

express the universal relationships of probability 1. This means that a conclusion always

inherits truthfulness from premises. We say that this type of reasoning is valid, i.e. if itspremises are true then there is 100% guarantee that its conclusion will also be true. Recall the

example of the universal relationship If someone commits a crime using weapon [A], thenhe is a criminal [B]. Say that a prosecutor wants to prove that John is a criminal. If he is ableto show that John really committed a crime using weapon [A], then on a basis of this

relationship he will deduce true conclusion John is a criminal [B]. Observe that theprosecutor achieves 100% guarantee that the conclusion is true in condition that a person who

commits a crime using weapon is always a criminal (with no exception). We say that the

conclusion: John is a criminal inherits truthfulness from its premise: John commits a crimeusing weapon.

Unfortunately, argumentations are rarely deductions. Instead, they are often built onthe statistical relationships. Once the reasoning is founded on such a scheme then it is highly

probable (guaranteed) that we will obtain true conclusion from true premises, i.e. thetruthfulness will be almost always inherited by the conclusion from the premises. We may say

that this type of reasoning is valid in some degree (close but not equal to 1). Consider the

example of the statistical relationship: If someone commits a crime using weapon [A], thenhe leaves trace evidence [C]. Say that John committed a crime using weapon [A]. Does thismean that he left trace evidence? Probably, but not necessarily. If we apply this scheme to

conclude about different people then most of the times our reasoning will be correct. But

there is still a chance that it may lead us to a false conclusion.

In the case of reasoning based on probable scheme, the degree of guarantee that

truthfulness will be inherited is the same as the schemes probability is. If the probabilityequals 0.9, then it is warranted that in 9 argumentations out of 10 true conclusion will bededuced from true premises. Obviously, the schemes used in argumentation may have also

the probability different than 1 or close to 1. In those cases they refer neither to universal nor

to statistical relationships. Such schemes are commonly used in sophistic types of persuasion

where low probable but appealing schemes are applied.

The probability of a relationship determines the probability of a guarantee that we will

obtain true conclusion if we assume true premises. Observe that it does not determine the

probability of obtaining true conclusion for true premises. Say that the prosecutor argues:John is guilty, therefore 2+2=4. The probability of the relationship If someone is guilty,then the sum of two numbers equals other number is 0. However, the conclusion of thisargumentation is 1. The true conclusion is accidental here, i.e. its truthfulness is not inheritedfrom the true premise. The objective probability 0 shows that there is no guarantee to obtain

true conclusion from true premises, but it does not excludes obtaining true conclusion.

1.2 Subjective probability

The aim of persuasion is to change beliefs of individuals and to reach particular values of

certainty concerning the truthfulness of a thesis (see also (Budzynska-Kacprzak 2008)). Itmeans that a proponent forces an audience to act or pass judgments which satisfy him as well


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as to identify with the proposed thesis. A typical example we can be found in a legal

argumentation where a lawyer tries to convince the judge (or jury) that the accused is

innocent.

In the court hardly ever the audience changes its views after short and fast

argumentation. Assume that the audience does not believe a given thesis, what we call as a

belief with a degree 0 (with a probability 0) about the thesis. Let Pr[i,T] stand for the measureof uncertainty of a person i about a thesis T. Thus Pr[audience,T]=0. Observe that in such a

situation only in few cases after giving an argument (arguments) the audience changes its

degree of uncertainty and moves it to 1, i.e. Pr[audience,T]=1. Similar situation occurs whenthe audience believes in the thesis with probability 1 and the goal of the prosecutor is to

convince it to believe the thesis with probability 0. It is very hard to do it immediately. More

often such a change is a very slow process and the proponent has to make a big effort to reach

a desirable result. What is more, before persuasion the audience may or may not have

established attitude to the thesis defended by the proponent. It may take place in a courtroom

for example. At the beginning of a trial, a judge (as a person who gives a verdict) is forbidden

to have established attitude to the accused or the case. Otherwise, the principles of fairness or

objectivity could be broken. Thereby, at the beginning the degree of judges belief about thethesis T = The accused is guilty should be neutral, what we signify by Pr[judge,T]=1/2.Specifically, it means that the judge neither accepts nor reject this thesis. Next, in the middleof the trial the initial degree of the judges belief changes so that at the end it reaches thevalue 1 or 0.

From the rhetorical point of view it is very important not only to make an assessment,

whether or not the audience is convinced about the thesis, but also to trace the change which

persuasion induced. Going back to the courtroom. Here the degrees of belief about the

accused persons fault can slowly rise or decrease or oscillate around the value 1/2. The lastcase is typical for the situations when the lawyer and the prosecutor exchange evidences. The

change of the judges beliefdegree is crucial for both the prosecutor and the lawyer. Thejudges subjective opinion about the case is the most important, since it is him to give theverdict. Therefore we focus on very subjective attitudes and beliefs which sometimes arecontrary to reality.

Fig.2 The change of degrees of subjective beliefs induced by argumentation during the trail.

The next question is how to model the subjective beliefs. One of the methods is to use

tools of probabilistic logics. Then, degrees of uncertainty of the audience (or other parties)

could be identified by the probability that a given sentence is true according to the audience.

For describing degrees of belief we can assume semantics of possible worlds. Suppose a

situation in which the judge considers 5 possible scenarios of a crime (see Figure 2). In 3 of


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them the accused takes a part in the dramatic events while in two of them not. In

consequence, we can say that the judge believes that the accused is guilty with the probability

3/5, i.e. Pr[judge,T]=3/5. Now, assume that during the trial one of the scenarios is refuted

such that now in 3 of 4 possible scenarios the accused is guilty. Thus, the degree of

uncertainty rises to 3/4, i.e. Pr[judge,T]=3/4.

Observe that we cannot express that the probability of the sentence John committed acrime is e.g. 3/4 in terms of the objective interpretation since John committed this crime orhe did not. Thus the objective probability of this sentence is 1 or 0. There is no other option if

we think about what happened in the reality. The probability of this statement can be onlyexpressed in terms of subjective interpretation. That is, we can consider in how many visions

of reality allowed by the judge the accused committed a crime. When we come to one real

world such sentences cannot be treated as probable unless one want to express that the person

committed a crime more (e.g. 3/4) or less (e.g. 1/4) instead he did it (1) or did not (0).

As we mentioned above, the subjective opinion ofpersuasions parties is a keyelement influencing the success of the process of convincing. In the court, the higher degree

of the judges certainty about innocence of the accused is, the higher success of the proponent(in this case the lawyer) is. By applying subjective degrees of beliefs we can also describewhich arguments are apt and affect the judge the most. For example the judge believes that

the accused is innocent with probability 2/5, the degree rises to 4/5 after the argument arg1was given. It means that the argument was well-chosen. At the same time, after giving the

argument arg2 the degree could change from 2/5 to 3/5 only. In this way we can analyze andcompare power and results of arguments as well as the degrees of uncertainty of the audience.

The interesting question is: does this two types of probability influence each other?Observe that whenever I use the objectively probable premise in my argumentation its

conclusion can be allowed only with some probability, namely, with some degree ofuncertainty. Thus the probability of conclusion is to be understood in the subjective manner.

Assume that the statistical probability of the relationship: If someone commits a crime usingweapon, then he leaves trace evidence is 0.9. Say that the prosecutor uses it to reason aboutJohn. When the prosecutor is absolutely certain that John committed a crime using weapon

(i.e. Pr[prosecutor, John_crime_weapon]=1), than he will be almost sure that John has left

trace evidence, i.e. Pr[prosecutor, John_leaves_evidence]=0.9.

2. Probability according to Aristotle

In this section we demonstrate how Aristotle understands the notion of probability and whatrole it plays in his rhetoric. We want to show that Aristotle originally refers to the objective

interpretation of probability, i.e. he associates the notion of probability with the nature of thereality. However, the subjective interpretation can also be found in his theory. Aristotle

emphasizes that rhetoric deals with (audiences) beliefs about the world not with the worlditself. Therefore, the probability plays a crucial role for persuasion as long as it influences the

uncertainty of the beliefs.


In this section, first we discuss why should we interpret probability in Aristotles rhetoric in

the objective manner. Then, we describe why he considers this notion as especially importantin the field of persuasion.


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Aristotle originally uses the objective interpretation of the probability, i.e., he refers to

statistical (not-universal) relationships which take place in the reality. Co mpare: Aristotleseikos (probability) and the knowledge that comes from it is rooted in the real order and it is

this existential aspect of it which makes it a legitimate source for further knowledge(Grimaldi 1998: 119).

What are the arguments for treating the probability objectively in Aristotles rhetoric?There are at least three of them: (1) the probability is compared with the truth, (2) the

probability is mentioned as the property of events, not beliefs, (3) the probability is caused by

complexity of the reality. First, Aristotle compares the notion of probability with the objective

truth, not with subjective beliefs about the truth: The true and the approximately true areapprehended by the same faculty; it may also be noted that men have a sufficient natural

instinct for what is true, and usually do arrive at the truth. Hence the man who makes a good

guess at truth is likely to make a good guess at probabilities. (Rhet.I.1 1355a). Next,Aristotle refers to the probability as to a property of events. That is, he defines the probability

as a thing that happens usually, not what we think (believe) that happens usually: AProbability is a thing that usually happens (Rhet.I.2 1357a), Probability is that which

happens usually but not always () Any argument based upon what usually happens isalways open to objection: otherwise it would not be a probability but an invariable and

necessary truth. (Rhet.II.25 1402b). Finally, according to Aristotle the probability is a resultof the complexity of the realitys fields to which persuasion refers. For example, Most of thethings about which we make decisions, and into which therefore we inquire, present us withalternative possibilities. For it is about our actions that we deliberate and inquire, and all our

actions have a contingent character; hardly any of them are determined by necessity (Ret.I.21357a). Complexity means that a given phenomenon is influenced by various random factors.

In consequence, the sentences describing this type of phenomena cannot be exceptionless.

This complexity reflects in uncertainty of our beliefs about the reality. However, initially the

complexity is a property of the reality.

What role does the probability play in rhetoric according to Aristotle? First, it is a

basis for an enthymeme, i.e. for a rhetorical type of a syllogism: Enthymemes are basedupon one or other of four kinds of alleged fact: (1) Probabilities, (2) Examples, (3) Infallible

Signs, (4) Ordinary Signs (Rhet.II.25 1402b). Furthermore, the probability plays a crucialrole in persuasion since the sentences that are probable can be much more often encountered

than the necessary ones (i.e. the sentences that are always true): There are few facts of the"necessary" type that can form the basis of rhetorical syllogisms. () the propositionsforming the basis of enthymemes, though some of them may be "necessary" will most of

them be only usually true. (Rhet.I.2 1357a). The necessary and invariable facts rarely refer tothe typical subjects of rhetorical speeches. Therefore, they seldom build the rhetorical

syllogism.Moreover, according to Aristotle the probability of sentences used in enthymeme

influences the degree of the enthymemes validity. That is, it influences the guarantee ofobtaining true conclusion: when it is shown that, certain propositions being true, a furtherand quite distinct proposition must also be true in consequence, whether invariably or usually,

this is called () enthymeme in rhetoric (Rhet.I.2 1356b). We can understand it as follows: in the syllogism with probable premises the truthfulness will be usually inherited by the

conclusion from the premises. Therefore, the frequency of obtaining true conclusion is either

100% (invariably) when necessary premises are used or close to 100% (usually) whenprobable premises are used.


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2.2 Subjective probability

No doubt, Aristotle uses the objective probability. Can the subjective interpretation be also

found in his rhetoric? There are some arguments for understanding his ideas in this manner.

In Aristotles view, the studies of rhetoric refer only to beliefs (endoxa). A persuader has a

chance to be successful when he uses opinions of his audience as premises: We must not,therefore, start from any and every accepted opinion, but only from those we have defined --those accepted by our judges or by those whose authority they recognize (Rhet.II.22 1395b).Thus, what is important in rhetoric is the beliefs about the reality rather than the reality itself.As a result, in enthymeme the sentences express someones beliefs about the truth and

probability rather than the actual truth and probability. Aristotle emphasizes it when he

differentiates demonstrations from rhetorical syllogisms. A demonstration is a deduction in

which the premises are true and an enthymeme is a deduction in which the premises are

accepted (believed) by someone.

Although initially Aristotle treats the probability as some property of the reality, in

rhetoric this notion becomes important only as long as it influences the character of opinions

used in argumentation: the eikos (probability) is so substantially and obviously grounded inthe real order that the majority of men accept it as a totally acceptable representation of the

truth (Grimaldi 1998: 119). That is, the probability is important since it generates uncertaintyof our beliefs: We should also base our arguments upon probabilities as well as uponcertainties. (Rhet.II.22 1396a).

Aristotle finds the close relation between objective and subjective probability since in

his view people have a natural disposition to the true (Rhet .I.1 1355a). Thus, there is no

unbridgeable gap between the commonly held opinions and what is true. In fact, there is a

close affinity between the true and the persuasive.

3. Formalization of probability

In order to formally model notions discussed in the previous sections we use first-order logicsof probability by J. Y. Halpern (Halpern 1990). The syntax of the adopted languages is

similar while their semantics are different. Thereby we consider two systems: the first oneputs a probability on the domain and is appropriate for giving semantics to formulas

involving statistical information, the second approach puts a probability on possible worlds

and is appropriate for giving semantics to formulas describing degrees of belief.


We start with a language which allows to reason about statistical information in large

domains. To present some intuitions to basic constructions suppose that the predicate

Help(x,y) says that x helped y with the crime and consider three terms with the followingmeaning:

wx(Help(x,y))it describes the probability that a randomly chosen x helped ywith the crime,

wy(Help(x,y))it describes the probability that x helped a randomly chosen ywith the crime,

wx,y(Help(x,y))it describes the probability that a randomly chosen pair (x,y)will have the property that x helped y with the crime.


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Now the formula wx(Help(x,y)) 1/2 can be interpreted as the probability that a randomlychosen x helped y with a crime is greater or equal to 1/2. Analogous for other terms.

Formally we consider two-sorted first-order language. Assume that is a collection

of predicate symbols and function symbols of various arities. The first sort consists of the

elements of , together with a countable family of object variables and describe thoseelements of the domain we want to reason about. Terms of the second sort represent real

numbers, which we want to be able to add and multiply. Thereby it consists of the binary

function symbols + and , constant symbols 0 and 1, binary relation symbols > and =, and a

countable family of field variables which are intended to range over the real numbers. Object

terms, field terms, and formulas are defined simultaneously by induction:

if f is an n-ary function symbol in and t1,,tn are object terms, thenf(t1,,tn) is an object term,if is a formula and is a sequence of distinct object variables, then

)(w n1 x,...,x is a probability term; if t1, t2 are field terms, then l1+l2 and l1 l2are field terms,

if P is an n-ary predicate symbol in , and t1,,tn are object terms, thenP(t1,,tn) is an atomic formula; if t1, t2 are field terms, then t1= t2 and t1>t2 areatomic formulas,

if 1, 2 are formulas and x is a (field or object) variable, then 1, 1 2,x 1 are all formulas.

Other Boolean connectives and the quantifier are defined in the standard manner. We call

the resulting language L1( ). Semantics of L1( ) is defined in a type 1 probability structure

(D, , ), where

D is a domain,

assigns to the predicate and function symbols in predicates and functions

of the right arity over D,

is a discrete probability function on D, i.e. is a mapping from D to the real

interval [0,1] such that d D (d)=1. For any A D, we define (A)= d A (d).

Given a probability function , a discrete probability function n on the product domain Dn is

defined by taking n(d1,,dn)= (d1) (dn). Moreover we define a valuation to be afunction mapping each object variable into an element of D and each field variable into an

element of R (the reals). Given a type 1 probability structure M and a valuation v, weassociate with every object (resp. field) term t an element [t](M,v) of D (resp. R), and with

every formula a truth value, writing (M,v) if the value true is associated with by(M,v). The definitions follow the lines of first-order logic. Here we give only main of them:

(M,v)t1=t2 iff [t1](M,v)=[t2](M,v),(M,v) xo iff (M,v[xo/d]) for all d D, where v[xo/d] is the valuationwhich is identical to v except that it maps xo to d,

[n1 x,...,x

w ( )](M,v)=n({(d1,...,dn) : (M,v[x1/d1,, xn/dn]) }).

Consider the following example to illustrate how we can apply the aboveformalization to express objective probability. Suppose that the language has only one


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predicate Help, and M=({John,Peter,Simon}, , ) such that (Help) consists of one pair

(John, Peter), (John)= (Peter)= (Simon)=1/3. Intuitively, (John) means that the

probability of picking John form the domain is 1/3, similar for Peter and Simon. Assume alsothat v is a valuation such that v(x)=John, v(y)=Simon. Then we have

[wx(Help(x,y))](M,v)=0 if we pick an x at random from the domain and fix yto be Simon, the probability that x helped y with the crime is 0 (Simon did not

take the part in the crime),

[wy(Help(x,y))](M,v)=1/3 - if we fix x to be John and pick a y at random from

the domain, the probability that x helped y with the crime is 1/3 (in fact it is a

probability that y=Peter),

[w(Help(x,y))](M,v)=1/9 - if we pick pairs at random the probability of

picking a pair (x,y) such that x helped y with the crime is 1/9.

As we can observe the proposed language and its semantics allow to reason about facts when

the probability of their occurrence is based on statistical information. However it is not well

suited for modeling subjective probability, i.e. degrees of belief (of e.g. audience). Notice that

in the logic we can express that the probability that a randomly chosen accused is guiltyequals 0.9: wx(Guilty(x))=0.9. At the same time we have no chance to express that the

probability that a particular accused (e.g. John) is guilty equals 0.9. It follows from the fact

that considering term wx(Guilty(x)), the wx binds the free occurrences of x in Guilty(x).

Therefore replacing x with e.g. y in predicate Guilty we obtain a term for which it is true that

either wx(Guilty(y))=0 (non of x satisfies Guilty(y)) or wx(Guilty(y))=1 (any x satisfies

Guilty(y)).

3.2Subjective probabilityFor reasoning about degrees of belief we use the language which the syntax is the same like

in L1( ) except that instead of probability terms of the form )(wn1 x,...,x

only probability

terms w( ) are allowed. The intended reading of w( ) is the probability of , e.g.w(Guilty(John))=0.8 says that the probability that John is guilty equals 0.8. We call the

resulting language L2( ).

Semantics of L2( ) language is given by a type 2 probability structure which is a tuple

(D,S, , ), where

D is a domain,S is a set of states or possible worlds,

is a function such that for every state s S, (s) assigns to the predicate and

function symbols in predicates and functions of the right arity over D,

is a discrete probability function on S.

The main difference between type 1 and type 2 probability structure is that in type 1 the

probability is taken over the domain D, while in type 2 the probability is taken over the set of

states S.

Given a type 2 probability structure M, a state s, and a valuation v, we associate with

every object (resp. field) term t an element [t] (M,s,v) of D (resp. R), and with every formula a

truth value, writing (M,s,v) if the value true is associated with by (M,s,v). Now, the


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meanings of the predicate and function symbols might be distinct in different states. Again we

give only main definitions:

(M,s,v)P(x) iff v(x) (s)(P),(M,s,v)t1=t2 iff [t1](M,s,v)=[t2](M,s,v),

(M,s,v) xo iff (M,s,v[xo/d]) for all d D,[w( )](M,s,v)= ({s S(M,s,v) }).

In this approach the formula w(Guilty(John))=0.8 is true in a structure M if the set of

all sates in which Guilty(John) is true has probability 0.8. The value 0.8 depends on the

definition of the discrete probability function . In particular we can assume a structure M in

which (S)=|S|

|'S|for any S S. Then

|}S's{|

|)}John(Guilty|)v,'s,M(:S's{|))John(Guilty(w )v,s,M( .

Intuitively it means that the probability that John is guilty equals the ratio of all states inwhich John is guilty to all states of the structure M. It could be the case where a judge

considers 5 possible courses of a crime and in 4 of them John is regarded as the guilty party.

Following this the probability that John is guilty is 4/5=0.8.

Summarizing, on the one side this approach allows expressing subjective probability

about properties of selected individuals. On the other side there is no possibility to reason

about objective probability using L2( ). Notice that in some situations we need to join both

probabilities. For example it is possible that degrees of belief are derived form the statistical

information. Suppose we know that the probability that a randomly chosen accused person is

guilty equals 0.9 what can be expressed in L1( )({Guilty,Accused}) by the conditional

probability statement wx(Guilty(x)|Accused(x))=0.9 which is an abbreviation forwx(Guilty(x) Accused(x))=0.9 (Accused(x)). Next, if we know that John is accused, then we

might conclude that the probability that John is guilty is 0.9. Now assume that

w(Guilty(John)) means the probability that John is guilty. We can write the implication:

Accused(John) wx(Guilty(x)|Accused(x))=0.9 w(Guilty(John))=0.9. See (Bacchus 1991)

for more details about the relation between statistical facts and degrees of beliefs.

Conclusion

The aim of this paper is to identify the places where the notion of probability may become

useful for the inquires conducted within the framework of rhetoric. We focus especially ontwo interpretations of probability theory which allows modeling subjective opinions of

individuals and objective or statistical premises on which argumentation is built. We present

the logical representation of those interpretations. In (Budzynska-Kacprzak 2007) we

presented the discussion about other formalizations which can be used for reasoning about

beliefs of parties of persuasion.

Our paper does not exhaust all applications of probability theory in rhetoric. In future

work we are going to continue the issue. Among other things we plan to study (a) probability

understood as chances that an audience will believe a thesis after a specific argument has

been given, (b) the probability as a rat io of individuals which are convinced of a thesis to allindividuals in a group that is an audience of a given persuasion, (c) the probability as a degree

of credibility of a proponent.


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References

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Budzynska, K. & M. Kacprzak (2007). Logical model of graded beliefs for a persuasion

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Aristotle, Rhetoric and Probability

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