CONi)I'I'IONA1,S AN 1) COUNTI~,I~.I~ACI'UAI,S I N I'ROI.OG 
.1. t'h. I Ioepchnan (IB M- IW B S,1 Icidelberg, F RG) 
A_I. M. van I toof (FhG-IAO,Stuttgart,FRG) 
INTRODUCTION. 
In our contribution for COLING 88 (1III '88), we 
introduced the concept of dialogical failure in the 
framework of dialogue games, by defining rules for 
the operator F, where FA is to be interpreted as 
"A is not winnable in this stage of the dialogue". 
We showed that F can be employed in the formu- 
lation of provisional implications. Provisional im- 
plications can be considered as the dialogical 
interpretation of defaults.It was shown that our 
solution works well for a range of examples. We 
concluded our paper with the announcement of a 
treatment of counteffactuals in terms of dialogical 
failure and of an implemented theorernprover for 
conditionals and counteffactuals based on our the- 
oretical developments.In this note we will briefly 
describe the considerations: leading to our treatment 
of counteffactuals and introduce the implementa- 
tion of a theoremprover for conditionals as a sub- 
system of PROLOG. For a more detailed 
description of the formal properties of the system 
we refer to (ttli.91). 
I. A DOUBLE CIIANGE OF 
LOGICAL ROLES 
1. If it were the case that A, 
then it would be the case that C 
is usually called a counteffactual, because its 
antecedent is pretended not to have tile truth-value 
it "rcally" has. Tichy (T.84) shows that none of the 
prominent theories on counteffactuals is successfull 
on all members of a set of mostly very plausible 
testeases, and wonders why it is, that in settling the 
validity of a counterctual, nobody ever refers to 
such matters as world similarity and other well 
known paraphernalia of traditional counteffactual 
theory. We take seriously Tichy's suggestion to 
look for the use of counteffactuals and formulate 
a semantics for counteffactuals not in terms of 
"troth" and "falsehood", but rather in terms of what 
is done and not done in a dialogue in which a 
counterfactual appears. One thing which simply is 
qmt done" when discussing a eounterfactual is the 
following: Suppose you forward the thesis "if 
kangaroos had no tails, they would topple over" 
(I.73) to somebody who has already admitted that 
kangaroos do have tails. Suppose your adversary 
accepts the invitation to discuss, and takes the 
antecedent of the counterfactual as a temporary 
additional concession. Then it would be com- 
pletely out of order Ibr you to claim to have won 
the the discussion on the ground that your adver- 
sary has now "contradicted" himself.tlowever, if 
contradicting oneself - as an opponent - in a dis- 
cussion is no longer a reason for losing that dis- 
cussion, then we seem to be playing according to 
roles which are similar to those for minimal calcu- 
lus. Define negation as implication of some absurd 
statement, eg. a contradiction, "%": 
2.-'A= A-> % 
In intuitionistic and in classical logic the proponent 
who utters the absurd statement loses against any 
thesis whatsoever. Not so in minimal calculus: here 
the opponent, having uttered the absurd, loses only 
if he, in a later move, attacks the absurd brought 
forward by the proponent. However, count- 
effaetuals cannot simply be treated as implications 
in minimal calculus: Suppose Jones steps on the 
brake (B), and is "alive (A), and that is all we assume 
or admit. We will not accept the counteffactual "If 
Jones would not step on the brake, then he would 
not be alive" as winnable (i.e. holding) under these 
circumstances. But in minimal calculus we have 
3. A,B? --B-> -~A = yes 
min 
Suppose we add to tile concessions A and B a 
concession to the effect that stepping on tile brake 
is the only reason for Jones' being alive (--, B -> 
-, A). Now we will not want the counteffactual "If 
Jones would not step on tile brake, then he would 
be alive" ( -1 B - > A ) to be whmable. But it is, 
since in minimal calculus we have 
4. Q,C? A-> C = yes 
rain 
368 1 
\]B,~\] INTI:RNA L U,YE O/VL Y 
In lhe usual minimal games the Ol)pOl)cIll , having 
admitted (2 has no opportunity to briug at U addi-+ 
tional reasons into play which would allow him to 
retract C afler accepting the anleccdcnt A. Wc 
therelbre need two things: a treatment of negation 
which is even weaker than that of minimal 
calculus,and the introduction of an opportuuity for 
the opponent to make us(" of his own conccssiot~s 
as exception rules+ The second of these is easily 
etlk'ctuated: by inlroducing flK+ fail- operator twice 
we cause a first change of rolcs which gives the op- 
ponent, tmw as a ternporary proponent, the op+ 
portunity to britlg additional concessions into play, 
The second fail operator then restores the initial 
order of roles. What we get is 
5.1;(A ~. > I;((;)) 
()ur counterlhctual becomes,intuitively, "You,the 
opponent,will fail in showing that C fails, aftcr A 
has been added to the concessions". 
2. WI:,AKliNING Nt//)ATION 
We obtain a system with negation which is weaker 
than minimal calculus negation, by assuming that 
there is not just one absurd statement, but possibly 
infinitely many. l)efinition 2. implicitly considers 
the absurd as a function taldng formulae A to fo> 
mulac %(A) under the assumplion that %(A) - 
%(B) for any A and 13. If wc drop this assump- 
tion, which is actually a very strong one, we get a 
family of logics, for which the only axiom governing 
negation i'; 
6. (A -.'> ~A) o> -~A 
Valcrius (V.90) calls this thmily "most minimal cal- 
culus '\] and shows that adding the assumplion that 
%(A) = %(B) for any A arm B is equivalent to 
7.(A <-> 11) <+:> (%(A) <-> %(I3)) 
and brings us back to minimal calculus. Our final 
analysis of counteffactual implications "if it were 
the case that A, then it would be the case that B" 
1lOW is 
8. F (A-> I r 13) 
kin kin 
where the subscript "kir(' refers to the fact that the 
checking dialogues induced by the fail operator, are 
conducted according to the/'ules of classical games, 
but for the fhct that negation is handled by the rules 
R)r most minimal calculus. We will demonstrate 
that lhis :\[brmalization leads to satisfactory results 
on all of the examples presented in (T.84). 
3. IMIU,IiMI:,N'I'ATION 1N Pl~.()l,O(~ 
The prover is implemelded as a I>ROI ,()(} subsys- 
tem. One distinguishes between lhe syntax of the 
data in the dalabasc, and the s) lHax of qttcries +is in 
PR()I,()(}. As for data, the propram acceF, ts \[acts 
and rttles. Apar| from the, usual operators ",\] and 
";", lhere ;.lle "lieS" t'()r ne~,ation, " .: +~ " !br pro@. 
sional, non-monotonic implication, "< +" i0r ordi+ 
nary PP, OI,OG implication and "= >" ff~r the 
cotmterthctual.For atomic statements the program 
accepts standard Iq<OL()(} syntax. 'I'hey can also 
be built ins, which have to be declared i,~ order to 
bc accessible to the recta-interpreter. 'lhe recta- 
interpreter is called by %ucccss/l" and %uccess/2". 
success/l takes a query, success/2 lal<es a query as 
first argument and a list of additional facts which 
can be used in the proof in addition to the facts in 
the database. The implementation of the recta-. 
inlerpreter makes use of the PI{OI,OG internal da-. 
labasc facility. Interpreted date arc stored in six 
internal databases: literal/pos, literal/neg, if/pos, 
iffneg, pmviffpos, provif/neg. 
Irurther code is stored it~ the normal PROI.OG 
code space. In contradistiction to the PROIOG 
interpreter, the recta-interpreter performs loop 
checking+ A further additional feature is a consist- 
ency check. If" the goal is not a built_in,recta- 
interpretation searches through the internal 
databases in a PROI,O(Mike head_matching 
search.lt determines whether it has to match a pos- 
itive or a negative head. 'lhen a database search is 
started in the following order: facts,monotonic 
rules, non-monotonic rules. Apart from the results 
on counterfactuals mentioned above, the prover 
works well on a range of cases of default reasoning, 
including "double diamonds", hierarchies of predi- 
cates and relevant implication. 
CtIC\] 

References

It11.88 llocpelman,J.lq~.,van lloof, A..l.M.: 
The Success of l'ailure. The Concept 
of Failure in l)Mogue l,ogics and its 
P, elevance for Natural l~anguage Semantics. 
Proceedings of Coling '88. 
Budapest,pp.250o255 

11Ii+91 tloepeJman,.l.Ph.,van lloof,A.J.M.: 
'l'wo-l>, ole,Two-l~arty Semantics: 
Knowledge Representation, 
Conditionals and Non-Monotonicity. 
Oxford Urdversity Press,to appear. 

1,73 l,cwis,I).: 
Countcrtactuals, Oxford, 1973 

T.84 'l'iclly,P.: 
Subjunctive Conditionals: 
Two parameters vs Three. 
l'hik~sophical Studies 45,1984,pp. 147+ 174. 

V.9() Valerius, R.: 
Die t;ogik yon Rahmen- und Stoprcgeln 
ill l,orcnzcn Spielen. 
I)iss+ \[Jniversity of Stut|gart,1990. 
