How to Improve the Convergent 
Argument Calculation 

ALEXANDERV. TVAGLO National University of 
Internal Affairs (Ukraine) 

Abstract: Quantitative apparatus for as-
sessing strength of an argument was im-
proved in the article. A probative generali-
sation and verification of the R.YanaJ's algo-
rithm to calculate convergent argument were 
proposed. The generalization obtained deals 
with both quantity (or number) and quality 
(or truth-values) of the argument's reasons. 

Resume: L'article presente l'approche 
quantitative a I'estimation de la force de 
l'argument. II a ete procede a la 
generalisation prouvee et a la verification 
de l' algoritmc du cal cuI de l' argument con-
vergent, propose par R.Yanal. Cette 
generalisation concerne la quantite et la 
qualite des raisons de 1 'argument. 

Keywords: informal logic, theory of argument, convergent argument, evaluation, 
Robert Yanal 

1. Introduction 

This article will pay attention to the contemporary tendency to elaborate a quanti-
tative apparatus for assessing the degree of support that reasons lend to a conclu-
sion, and hence the strength of an argument. This tendency or even program was 
well presented, for instance, by John Black. See Black (1991, 21). I will try to 
support this one tor the sake of both research and pedagogic use. 

There is no problem of finding argument diagrams, even in Beardsley's old 
book Practical Logic. The traditional approach to argument diagramming is quite 
friendly to intuition and simple in use. Perhaps owing to these properties, it seems 
hard to find principal changes in the essence of the approach or in diagramming 
techniques (see, e.g., Beardsley 1950, 18-26; Fisher 1988, 18-19; Moore and 
Parker 1998,261-264; Browne 1998,29-30; Thomas 1997,49-63; Walton, C. 
2000, TS 29-31). However, a few years ago Walton, D. (1996) published a mono-
graph that provides relatively stricter grounds -directed graphs (digraphs) theory-
for the techniques. This innovation permits us to describe argument structure in a 
more sophisticated manner than does the traditional approach, and because it is 
able to solve a lot of widespread tasks, I will use this variant of argument diagram-
ming further. 

©Informal Logic Vol. 22, No.1 (2002): pp.61-71. 



62 Alexander V. Tyaglo 

2. A short survey of traditional argument diagramming and evaluation 

In the simplest case, an argument consists of one reason (premise) and one con-
clusion. If we designate the reason by R, the conclusion by C, and the logical 
connection between them by a solid arrow directed from R to C, then the simplest 
argument's structure will be represented by the diagram in Figure 1. 

Figure 1 

a b 
Figure 2 

In more complex cases when the conclusion is supported by more than one 
reason, the arguments (and their diagrams) are divided into two main classes: 
those with mutually dependent reasons (such arguments are known as linked) 
(Figure 2a) and those with independent reasons (convergent arguments) (Figure 
2b). 

There exist other logical divisions of arguments, for example "convergentl 
divergent' (Figure 3a) and "one-steplmany-steps (serial)". When the conclusion 
of the first argument is a premise of the second one, etc., we have a serial argu-
ment (Figure 3b). 



How to Improve the Convergent Argument Calculation 63 

a 

Figure 3 b 

Diagramming techniques permit us not only to visualize an argument's struc-
ture, but also to reflect the strength of connection between reason(s) and conclu-
sion. This approach provides a way to calculate an essentially quantitative charac-
teristic of an argument-the probability of a conclusion being true if its reasons 
are acceptable. 

About 10 years ago an American researcher, Robert Yanal, proposed a relevant 
algorithm to calculate the strength of a convergent argument. (Yanal 1988,39-55; 
1991,137-144). He suggested dividing all arguments into strong, medium, weak 
and deductively valid depending on the degree of support that the reasons of an 
argument lend to their conclusion ("degree of validity"). I The difference between 
such sorts of arguments is described quantitatively by a set of p(ClRx) - the 
probability that an independent Rx lends to C or, in other words, the strength of 
connection between Rx and C. 

In addition, Yanal pointed out that "we can evaluate the degree of validity of an 
argument without concerning ourselves with the truth of its premise(s)" (1988, 
75). It becomes apparent that the degree of validity is indifferent to the truth-
values of the reasons or the conclusion and, therefore, does not characterize argu-
ment exhaustively. 

For instance, the argument "My dog has fleas, therefore all dogs have fleas" 
was named weak and designated as W. The argument "All dogs have fleas, there-
fore my dog has fleas" was named deductively valid and denominated as nv, etc. 
(1988,83). In accordance with Yanal's proposition, the traditional diagrams are to 
be completed by signs S, M, Wand nv. 



64 Alexander V. Tyaglo 

w DV 

a b 

Figure 4 

Yanal's algorithm to calculate the probability that the conclusion is true was 
substantiated by intuitive reasoning for a convergent argument with reasons Rl 
and ~. Suppose a reason, RI' is brought forward to support a conclusion, C; and 
suppose that Rl lends 0.3 probability to C. Analogously, independent R2 lends 0.4 
probability to C. Yanal writes: 

Obviously, C is supported by two reasons whose probability together is 
greater than the probability of either one individually. But how much greater? 
It would be an error to simply add the probability of each reason (0.3 + 0.4), 
thereby coming to the conclusion C is now supported with 0.7 probability. 
Here's why. lfwe know R

1
, then we have 0.3 certainty that C is true. Certainty 

can be thought as 1.0 probability. In other words, knowing that C is true on 
the basis of knowing that R j is true is the same as knowing C with probability 
0.3 times 1.0 (= 0.3). Therefore, merely knowing that RI is true leaves us with 
0.7 (= 1.0 minus 0.3) of uncertainty. Now, when R. ... is brought forward, we 
have decreased our uncertainty by 0.4. In total, we know the conclusion with 
0.3 (the probability ofRj) plus 0.28 (the probability ofRz times the remaining 
"unknown" left over from 0.3), which equals 0.58 ... We shall call this the 
"ordinary" way of summing probabilities. (Yanal 1988,54.) 

On the basis of this algorithm, Yanal proposed a criterion to distinguish argu-
ments with independent and dependent reasons (1991, 140). But this criterion 
was substantively questioned (Conway 1991, 145-158). 

It is necessary to point out that this algorithm de Jacto was supported by 
John Black (1991), but without reference to Yanal. Black simply postulated the 
rule without further test or even any illustration. Therefore for the sake oftheoreti-
cal perfection and practical reliance, it is necessary to verify Yanal's algorithm in 
probative fashion. How it is possible to reach this goal? 

3. Generalization and verification ofYanal's algorithm 

To start with, let us define the next quantitative frames to differentiate connections 
between premise and conclusion: p(ClRx) (see Black 1991,27). 



How to Improve the Convergent Argument Calculation 65 

Weak connection 

Medium connection 

Strong connection 

Deductively valid 

Argument's connection 

o S; p(ClRx) S; 112 
112 < p( ClRx) S; 2/3 

2/3 < p(ClRx) < 1 

Evaluation of the strength of argument with independent reasons is deter-
mined by two factors: the reasons' truth-values and the strength of connections 
between the reasons and conclusion reflected by p(ClRx)' 

Next, let us take into consideration the diagrams with true reasons only and 
denominate these reasons as T x' 2 In cases with true reasons, the convergent argu-
ment's strength, i.e. the conclusion's probability of being true P( ..• T X",), should 
be equal to the "ordinary sum" of p(ClT x), How can we explore Yanal's algorithm 
in a more probative way? 

The questioned algorithm is clearly represented by formula (IV 

It is reasonable to generalize (1) for an unlimited number of reasons (or 
premises) T x' I S X S 00. In this way we will arrive at (2). 

p(ClT1,Tz) = p(ClT1) + {1- p(ClT1)} p(ClTz) 

p(ClT1,Tz,TJ = p(ClT 1) + {1- p(ClT1)} p(ClTz) + {1- p(ClT,) - [l-p(ClT,») x 
x p(ClT

2
)} x p(ClT

J
) = 

= p(ClT1,T)+{1- p(ClT"T
2
)} p(ClT

J
) 

In order to verify (1) and each of its generalizations, let us introduce a set of 
natural essential conditions Cl - C4. 

The first condition Cl presupposes that the correctly calculated probability 
p(CI ... R x"') is to lie between 0 and 1.

4 C2 demands that this probability does not 
depend on a calculating process (in other words, it is possible to include any 
p(ClRx) into the "ordinary sum" in an arbitrary order). C3 demands: if a conclu-
sion has at least one sufficient reason Rx and, therefore, there is a probability 
p(ClR) =1, then all others are not able to change anything in p(CI ... R x"') =1. 
And condition C4 demands: if a conclusion has irrelevant reason(s), it does not 
influence one's probability. 

In the special case with true premises, CI-C4 will be modified to next Cl '-
C4'. 



66 Alexander V. Tyaglo 

O:S; p(CI ... Tx"'):S; 1, Cl' 

p(CI ... T x' TX+l"') = 
= p(CI ... Tx+I' T x"') etc., C2' 
if p(ClT x) = 1, 1 < X :s; co, 
then p(ClT1 ... T x''') = 1, C3' 
if p(ClTx) = 0,1 < X:S; co, 
then p(CI ... T x.I ' Tx' T x+ I "') = 
= p(CI ... T x.I ' T x+ I ''') C4' 

Now, let us test if (1) meets Cl'-C4'. 
If the initial conditions were stated correctly, then 0 s p(ClT

1
) s 1 and 0 s 

p(ClT
2

) s 1, therefore p(ClT1,T2) 2 O. On the other hand: 1 - p(ClT 1) 2 [1 -
p(ClT1)] p(ClT2). Hence, p(ClT1) + [1 - p{ClT 1)] x p(ClT2) s 1. Thus, Cl' is met. 

C2' is met because (I) includes p{ClT
1
) and p{ClT) symmetrically. 

Under condition p{ClT I) = 1, C3' is met because multiplier near p(ClT 2) equals 
o and this component's share is nullified. An analogous conclusion for p{ClT 2) = 1 
will be drawn if we take into account C2'. 

Ifp{ClT1) = 0, then C4' is obviously met; owing to C2' we arrive at analogous 
conclusion in the case with p{ClT

1
) = O. 

There is no problem in proving that generalization (2) meets Cl'-C4' as well, 
but in analytical form it is a task for the most curious people. I will simply verify 
(2) by a concrete example. 

For the diagram on Figure 5, let us suppose: p{ClT1) = 112, p(ClT2) = 2/3, 
p{ClT

3
) = 0, p(ClT

4
) = 1. 

w 

Figure 5 

Step by step from (2), we will figure out p(ClT
1
,T

2
,T

J
,TJ 

p(ClT1,T2) = 112 + (l 112) x 2/3 5/6, 
p(ClT

1
, T

2
,T

J
) = 5/6 + (l - 5/6) x 0 5/6, 

p(ClT
1
,T

2
,T

J
,T

4
) = 5/6 + (1 5/6) x 1 = 1. 



How to Improve the Convergent Argument Calculation 67 

This result meets Cl', C3' and C4' clearly. Repeat the exercise for, e.g., 
p(ClT

2
,T

4
,T1,TJ. 

p (ClT
1
,T

4
) = 2/3 + (1 2/3) x 1 = 1, 

p (ClT
2
,T

4
,T1) = 1 + (l 1) x 112 = 1, 

p(ClT
2
,T

4
,T1,T3) 1 + (1 - 1) x 0 1. 

Hence, p(ClT1,T2,T3,T4) p(ClT2,T4,T1,T3) in accordance with C2¢! 

Further generalization of (1) and (2) permits us to take into consideration an 
argument with unknown true-values of reasons. More exactly, in this case we 
know accurately only P(Hx)--the probability that a reason Hx is true. In terms of 
Black's paper this is "the assigned probability" (see Black 1991,27, 30). Let us 
name such a set of reasons and arguments as hypothetical. 

It is impossible to identify the hypothetical argument's strength 
P(ClH1& ... H x&"')' i.e. the probability of the reasons and conclusion being true 
at one time, with the "ordinary sum" p(CI ... R x"') as it was possible in the situa-
tion with a true premises argument. Indeed, in general, the argument's strength 
depends both on reason-conclusion connections p(ClHx) and the probability of a 
reason being true P(Hx)' 

In accordance with the elementary theory of probability that we have for the 
conditional probability p(ClH) = P(C&H) 1 P(H), we arrive at (3): 

P(C&H) = P(H) p(ClH) 
For instance, if P(H

1
) 0.7 and p(ClH1) 0.7, then the probability of the 

argument's reason and conclusion being true onetime will be 0.49. Suppose p(ClH
1
) 

1, but P(H) 0.1. Then P(C&H
1
) 0.1, and we get a deductively valid but 

practically unsound argument. 

On the basis of the last generalization, it is possible to calculate a serial argu-
ment that might include hypothetical premises. For instance, if p(HIT) 0.7, p( ClH) 
= 0.7, then we will get the result stated above: P(C&H&T) = peT) p(HIT) x 
p(ClH) = 1 x p(HIT) p(ClH) 0.49 (Figure 6). 

M 

M 

Figure 6 



68 Alexander V. Tyaglo 

It is not too difficult to introduce a fonnula (4) for a convergent argument with 
two hypothetical reasons. 

P(C&HI&H
1

) = P(H) p(ClH I) + [1 - P(HI) p(ClHI)] P(H2) p(ClH) 
Let us check whether (4) meets the principal conditions C1-C2 by a particular 

example relevant to Figure 7. 

Figure 7 

The probability that the statement "The brake in a repaired car is out of order" 
is true, equals P(H) = 0.0001. The probability that statement "The car's motor is 
out of order" is true, equals P(H) = 0.01. When the car has malfunctioning brakes 
then the probability of a quick fender-bender is p(CjH) =: 0.9; with malfunctioning 
motor it equals p(CjHz) =: 0.25. How much is the probability that the forecast about 
the car's imminent crash will be true? 

P(C&HI&H
1

) = P(HI) p(ClHI) + [1 - P(HI) p(ClH I)] P(H2) p(ClH) ::= 
0.0001 x 0.9 x (1 - 0.00009) x 0.01 x 0.25 0.002589775, 

P(C&Hz&H I) =: 0.01 x 0.25 + (1 - 0.0025) x 0.0001 x 0.9 = 0.002589775. 

So, C1 and C2are met. 

Another case is possible in which probability of a motor defect is excluded, i.e. 
P(.H

2
) 1 and p(CI.Hz) =: 0: in this case we calculate the possibility of a crash, 

caused by a malfunctioning brake only P(C&H I&.H2). 

P(C&HI&.H2) = 0.0001 x 0.9 + (l - 0.00009) x 1 x 0 =: 0.00009. 
Thus, (4) meets C4 as well. Checking the C3 condition for this case and 

generalizing from (4) does not create any significant problems. However, because 
of the technical complexity I will avoid these exercises now. 

Let us instead take into consideration a case with zero probability that a reason 
is true. The meaning of this case is quite clear: a reason Fx isfalse; i.e., it does not 
correspond to reality. 

Examine, e.g., the next diagram with P(H) = 0.5, p(ClH) 0.6, P(F) 0, 
p(ClF) 1. 



How to Improve the Convergent Argument Calculation 69 

Figure 8 

P(C&H) = P(H) p(ClH) = 0.3. 

P(C&H&F)= P(H) p(ClH) + (1- P(H) (ClH)] P(F) p(ClF) P(H) p(ClH) = 
0.3. 

Hence, P(C&H&F) = P(C&H). Even though spite of the connection between 
F and C is DV, the false reason must not lend any real support to the conclusion. 
This naturally presupposed result verifies the questioned algorithm once more. 

4. Brief conclusion 

In this article a probative generalization and verification of Yanal' s algorithm to 
calculate a convergent argument have both been proposed. The results obtained 
deal with both quantity (or number) and quality (or truth-values) of the argument's 
reason(s). Generally speaking, the quantitative apparatus for assessing the strength 
of the arguments was improved. 

It is particularly noteworthy that the generalization proposed has the same 
limits as Y anal's basic algorithm. First, the initial assignment of accurate numerical 
values both to reason-conclusion connections p(ClHx) and the probability of rea-
son being true P(Hx) in many practical cases is intuitive. Therefore it might be 
subjective and controversial (c/, Black 1991,29; and Walton 1996, 128-129). 
Second, as Professor Wayne Grennan has hinted to me, the formula (2) and (4) 
are only approximations to accurate formulae. Mathematically correct formulae 
might be derived based on a relevant 1. M. Keynes' equation. But in a regular case 
this equation includes additional unknown variables and so isn't solvable. s In order 
to get any numerical solution we must return to approximate Yanal's algorithm and 
its different modifications. Correspondence between the approximations and rel-
evant accurate solutions constitutes an open research problem, and I intend to 
study it further. 

The results obtained do not exhaust all the possibilities for improving traditional 
argument evaluation. It seems reasonable, for example, to reflect in an argument 
diagram and calculation not only pro but contra reasons as welJ.6 The issue de-
mands further investigations both scientifically and pedagogically. 



70 Alexander V. Tyaglo 

Acknowledgements 

The core idea of the paper arose through discussions with Dr. Edward F. McClennen, Dr. 
M. Neil Brown and Dr. StuartM. Keeley during my research visit to the Bowling Green 
State University in 1998. This visit was supported by the International Research and 
Exchange Board (lREX), with funds provided by the United States Informational Agency. 
I would like to thank the referee of an earlier version of this paper, Dr. Wayne Grennan, 
for his constructive criticism and remarks. 

Notes 

, An analogous degree of support scale was introduced in Thomas (13 7-138, etc.). Specifically 
important in this case is the explicit connection between degrees of validity and the decision 
making based on quantitative evaluation. 
2 Let us denote false reasons as Fx and reasons with unknown truth values as Hx' A set of all 
possible reasons will be R = TuHuF. 
3 An equivalent formula was introduced in Black (24). 
4 This most obvious condition appeared in the Black's article. 
; Grennan refers to the Keynes' formula for a two-premises convergent argument: 
P(C/T,&T)= P(C&T,IT2) 1 P(C&TlIT2) + P(.C&T,IT2), or P(C/T,&T2)= I/1+B where B = 
= P(.C&T,IT

2
)1 P(C&T,IT

2
) (see Kcynes 1973, 165-166). In addition to the sought value 

P(C/T,&T
2

) this equation includes generally unknown P(C&T,IT
2

) and P(-,C&T,IT
2
). Yet an 

equation with two or more independent unknown variables is unsolvable. 
6 See, e.g., the intention in Moore and Parker (264 ), Thomas (386-388) and Black (25). 

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How to Improve the Convergent Argument Calculation 71 

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Alexander V. Tyaglo 
Department of Philosophy and Political Science 

National University of Internal Affairs 
27 Prosp. 50-richya SRSR 

Kharkiv, 61 080 
Ukraine 

alext@adm. univd kharkov. ua