A C T A U N I V E R S I T A T I S L O D Z I E N S I S
FOLIA PH1LOSOPHICA 9, 1993
Uwe Scheffler
C O N D IT IO N A L S, BA SED ON STRICT ENT AILM ENT
C o nditionals can be o b tain d by several ways: as a result o f em pirical investigations;
from o th er co nditional o r n o n o co n d itio n al p ropositions by logical rules; from definitions o r o th er term inological statem ents;
- postulated;
- from sentences ab o u t logical entailm ent.
T he last way o f o b tain in g co nditional p ro p o sitio n s is the m ain topic o f the p aper which deals only with first degree conditionals (conditionals, containing only one occurence o f the conditional operato r). Such conditionals are im p o rtan t: they are logically tru e and are used to draw conclusions from facts to get facts.
Every true entailm ent A |— В co rresponds to a tru e co nditional A -» B. W hat kind o f co nditionals we get depends obviously on the system o f logical entailm ent, which rules the entailm ents, an d on the logical rules, governing conditionals. T he basic system is in this case the system F s o f strict entailm ent constructed by Wessel. T he proposed in tro d u ctio n -ru le fo r conditionals allowes to use two im plicative structures: entailm ents and conditionals, with different properties. U sing som e co nditional principles the class o f co nditionals can change while the class o f entailm ents rem ains unchanged.
T he alph ab et o f F s consists o f
1) co u n tab le m any p ropositional variables p. q, r, Pj,...;
2) truth -fu n ctio n al connectives л (conjunction), V (disjunction), ~ (ne-gation);
3) the predicate o f entailm ent |— ; 4) parentheses.
D l. A form ula is a tru th -fu n c tio n a l form ula, if it is constructed by the usual rules with tru th -fu n ctio n al connectives only.
D2. A form ula is a form ula o f logical entailm ent, if it has the stru ctu re A I— B, an d A and В are tru th -fu n c tio n a l form ulas.
The postulates for F s are all form ulas o f logical entailm ent having the form o f one o f the following schem ata an d m eeting the conditions El and E2: E l . K A i B, then В co n tain s only such p ropositional variables, which are also in A.
E2. If A i— B, then A is not a co n tra d ictio n an d В is not a tautology.
A 1. A I— ~ ~ A A2. ~ ~ A I— A A3. А л в I— A A4. A A B i— В A A A5. ~ (A A B) i--- A V ~ В A6. ~ A V ~ В I— ~ (A A B) A7. (A V В) A С I— (А Л С) V В A8. (А A С) V (В Л С) (A A B ) A C A9. A I— А Л (В V ~ B) The rules o f F s are:
R l. If A i— В and В |— C , then A |— C. R2. If A I— В an d A |— C, then A |— В A C .
R3. If А э В an d В э A are tautologies, then С |— C[A/B], where C |A /B ] m eans that in С all o r som e (including no one) occurenccs o f A are to be replaced by B, and С is not a co n trad ictio n and C[A/B] is not a tautology.
Wessel proved: A form ula o f logical entailm ent A |— В is a theorem in F s if and only if: А э B is a tautology, В co n tain s only such variables, which are also in A, A is not a co n trad ictio n an d В is not a tau to lo g y 1.
To get a conditio n al system we introduce a n o n -tru th -fu n ctio n al connec-tive -*• (conditional o perator: if then) into the language:
D3. A form ula is a sentence, if the following conditions arc satisfied: 1. T ru th -fu n ctio n al form ulas are sentences. 2. If A and В are sentences, ~ A, (А A В), (A V В), (A -» B) are sentences.
T he constru ctio n is com pleted by the conditional axiom and the co n -ditional rule:
A10. I— A -> A
R4. If A I— В and |— В -> C, then |— A -» C. F o r the resulting system F SK it is easy to show:
51. If I— A in F SK, then A is a conditional.
52. A I— В is theorem in F s if and only if A [— В is theorem in F SK. 53. A |— В is theorem in F s if an d only if |— A -+ В is theorem in F SK. 54. If i A -» С an d |— В -> С are theorem s in F SK, then I— (A V В) -» С is theorem in F SK\
55. It I ( Aa B) -> С is theorem in F SK, and В contains no variables, occuring in A or in C, then | A -> С is theorem in F SK.
■II' А л B i С is theorem , A ist not a co n tra d ictio n and С is not a tautology. U nder this co n d itio n and because o f the restriction on В in С occur only such variables, which are also in A. Let W be a valuation, which prescribes A the value T an d С the value F. In any case W can be extended to a valuation W including the variables o f B. that prescribes the value Г to A В too. Because o f S3 and W essel's result m entioned above the sentence is proved. ■
56. II I A -* В is theorem in F SK. then | (А л С ) —*• В is theorem in F SK, where А Л С is not a contrad ictio n .
■Use A3 and R l. ■
O bviously A -» В is no t a theorem in F SK, if A is a con trad ictio n o r if В is a tautology. In a direct sense this system is a paraconsistent logic: the ap pearance o f co n tra d icto ry d a ta does not force the system to be explosive, to derive any form ula. T he unusual restriction not to conclude from co n tra d ic-tions is a difference between relevant and p araconsistent logics and F SK and has to be explained. In relevant logic from p Л ~ p does not follow q, but it follow s p an d also ~ p. Even if one stipulates th a t there are true co n trad ictio n s probably no t all co n tra d ictio n s are true, therefore in som e cases from a co n trad ictio n does no t follow all nonsense you w ant (as in classical logic), but a little nonsense anyw ay. In o rd e r to avoid any nonsense the restriction on the antecedents is m ade. O n the o th er h an d the restriction on the consequents is u n d erstan d ab le at once: why we should conclude tautologies, if we already know that they are tautologies? Such im plications are often funny, so there is an old germ an rule: If the cock crow s on the dunghill, the w eather is changing o r it rem ains unchanged.
T here are good reasons for the restrictions, bu t som etim es they seem to be very hard. System atically violating them we co n stru c t w eaker systems.
We start to build up several systems o f co n d itio n al logic by adding co nditional rules. In all system s the set o f entailm ents rem ains unchanged, it is the set o f theorem s o f F \ T he concrete choice o f rules, which we w ant to use, depends o f course on practical purposes. So it m ay, for exam ple, be useful to have the n on-m onotonic relation o f entailm ent together with a m onotonic co nditional o p erato r. Such things can be done, as we w ant to show.
A disadvantage o f F SK is the absence o f the substitu tio n rule. So it is necessary to distinguish logically between (p A q ) -> p (w hat is valid on the base o f A3) and (p л ~ p) -> p (w hat is invalid because o f E2), though the letter is derivable from the form er by substitution. Logicians w orking in relevant logic would argue, th a t su b stitu tio n is a logical rule and therefore the set o f conditionals, obtained from sentences ab o u t entailm ents, should consist
o f n o t only the corresponding cond itio n als, but also o f all substitutions in such conditionals.
We get F SK5 by ad d in g the following rule to F SK:
R5. If I A -> B, then | С D, where С -*■ D is the result o f substituting p ropositional variables o f A -» В by truth -fu n ctio n al form ulas.
In this system we can prove co nditionals which do not m eet the condition E2: it is possible to derive conditionals with co n tra d icto ry antecedents and tautological consequents.
O ne o f W esscl's system s allow s to prove entailm ents, fulfilling the condition E l but failing to m eet restriction E2. His system o f logical entailm ent Ss can be o btained from F s simply by rejecting E2, a system SSK can be constructed adding A 10 an d R4. O bviously F SK^ is a system between F SK and SSK: all theorem s o f F SKS are provable in SSK, but p -*■ p V ~ p is theorem in the latter and not in the form er system.
In F SK5 theorem s are all co nditionals, co rresponding to F s -entailm ents, and all co nditionals being su b stitu tio n s in such „in n o ce n t” form ulas. Such a construction is useful, if we w ant to introduce coun terfactu als with logically false antecedents into the system.
A dding rule R6 to F SK we get FSKf>: R6. If |— A -» B, then |- ~ В -* ~ A.
It is easy to see th a t som e form ulas being provable with R6 arc violating E l. So form ulas like ~ A -» ~ (A A B) are theorem s, but n o t A -> (A V B) (because there is no ordin ary transitivity-rule). In som e connections it m akes sense to distinguish between these form ulas. O ne may argue, th a t A - * (A V B) means: on the base o f A it is possible to introduce into the discourse w hat you w ant (If roses are red, then roses are red o r the m oon is a green piece o f cheese); bu t ~ A -» ~ (А Л B) m eans only som ething like the „m onotonicity o f negative in fo rm atio n ” (If som ething is not the case, then it is not the case w hatever happens).
Systems like the m entioned one m ay be used in deontic logic. T he well know n principle:
F rom A |- В follows O(A ) O(B)
produces paradoxical situations in classical, relevant and m ost o f m odal logics. T he reason is not only the R oss-paradox:
If the secretary has to mail the letter, she has to mail or to burn the letter;
but also the possible occurence o f co n tra d icto ry A. O f course, there are co n trad icto ry false norm ative contexts, but then it is necessary to decide, which norm s one has to meet. In no case it is in a ra tio n a l sense possible to oblige som eone to generate a co n tra d icto ry situation. T his is, by the way, the sense of a im p o rtan t philosophical principle in political and social philosophy: All,
w hat is orderd, is possible. R em em bering political practice it should be added: but be careful in ordering.
C oncerning the Ross- p arad o x confer the m entioned sentence with
II the secretary has to mail the letter, she has to mail the letter or to go to dinner.
Because the secretary m ay first go to have a din n er and then mail the letter o r vice versa, there is nothing parad o x ical at all. T he parad o x in the fam ous secretary-exam ple raises up from the fact, th a t burn the letter m eans not to mail it, and mail it m eans not to burn the letter. T herefore it is a ter- tiu m -n o n -d atu r-co n stru ctio n in the conclusion o f A | B, w hat m akes the m entioned deontic principle leading to parad o x . Such co nstructions are explicitely excluded by E2.
W ith sim ilar result it is possible to ad d R6 to F SK5 an d SSK. The following rule
R7. If |- A ->B, then |- (A A C) -* В
added to F SK allows to prove in the resulting system F SK7 conditionals with co n tra d icto ry antecedents. It is a system betw een F SK and SSK, different from |, | n p S K 7
one may
use j|ie m o n o tonic co nditional o r the non-m o n o to n ic entailm ent and also both together. This m ay be interesting in d ata systems, where the d a ta arc arriving from different sources: conclusions w ithin the different pools should be draw n with the help o f the m on o to n ic conditional, conclusions with d ata from different sources w hould be draw n on the base o f the entailm ents.The co nditional in F SK67, co n stru cted by ad d in g R7 to F SK6, is also a m onotonic one. In this system conditionals with tautological consequents are provable, it is an o th er system between F SK an d SSK, different from F SK5.
T ogether with F SK the follow ing rule constitutes F SK8:
R8. If |- A —► B, then |- A —► (В V C); where В V С contains only such variables, which occur in A.
By R8 co nditionals with tautological consequents are derivable, the system is not equivalent to one o f the form er m entioned.
Let Fs r be the system , constructed by adding R5 R8 to F SK. The conditional o p erato r, occuring in p ro b ab le conditional sentences o f this system, is not the m aterial im plication. T his is show n by an easy sentence:
57. If |- A -» B, then there is a prop o sitio n al variable, occuring in A and in B.
■ I se induction: the postulates have the prop erty , the rules hand it dow n. ■
58. By adding the transitivity-rule for conditionals (If | A -» В and I В -> C, then I A -» C) the co nditional o p e ra to r becomes m aterial im plication.
■ I. (В Л ~ В) V ~ A I ~ A 2. |— A —* (А Л (B v ~ B)) 3 . I - (А Л (В V - В)) (В V - B) 4. |~ A -» (В V -v B) ( F s ) (F SK, R6, T rans.) ( F SR) (T rans.) ■
In o rd e r to get the last system o f the p ap e r we have to accept two ad d itio n al rules:
R9. If A h В and В | A and | C -> D, then |- С -> D[A/B], RIO. I l | (A V В) -+ C, then I A -> C, if A and С are sharing a com m on p ropositional variable.
These rules together with F SR co n stitu te the system F SR. contain in g all m eans to construct norm al form s.
The rule RIO w ithout restriction is one o f the often discussed rules in co nditional logic. T here are som e counterexam ples against this rule, for instance:
From „1Г the secretary has to write a letter or to go home, she would go home" follow s by unrestricted RIO „ If the secretary has to write a letter, she would go home".
The restriction on RIO prevents the appearan ce o f sUch exam ples, for-m ally it prevents the validity o f (p A ~ p) -> q. T herefore S7 holds also for F s r .
S9. If A -+ B, there is a form ula С such, th at |- A -> С a n d | С -» В, and in С are only these variables, which occur also in A and in B.
■In are all m eans to co n stru c t for any form ula the corresponding to rm u la in extended disjunctive norm al form (a disjunctive n orm al form such, th at for all occuring variables holds: they occur with or w ithout negation in all elem entary conjunctions). Because o f R9 it is sufficient to show S9 for form ules in extended n orm al form .
Let A and В be form ulas in extanded disjunctive norm al form and | A -> B. Let С be the result o f erasing in A all p ropositional variables, which do no t occur in В. С exists because o f S7.
F o r all elem entary conjunctions Aj o f A there is an elem entary conjunc-tion Cj o f С such, that for the sets o f occuring atom ic form ulas {Aj} and [Cj[ holds {Cj} £ [Aj). F o r these A, an d Cj the co nditional Aj -> Cj is provable because o f A3, and so is |- Aj -» С for all Aj. By S4 follows |- A -+ C.
Since |- A -> B, for all A j because o f RIO is valid |- A j -» B. A n y A j is a co njunction C j Л D j, and D j does not share variables with C j an d B; therefore (by S5) follows |- Cj —» B. Since this holds for all C j, | С —>B is valid because o f S4. ■
If I— A -* В and A is not a contradiction, then there is a formula С such, that A |— С and | - С -* В. and С contains only variables, occuring in B.
F SR is no t equivalent to the relevant system F D E o f first degree entailm ent. In all m entioned system s |- ( ( ~ p q) p) —*• q (the у-principle) is valid, in F D E not. In F D E we have unrestricted transitivity, S8 show s, th at Fs r together with transitivity collapses to a system o f m aterial im plication.
F SR is not equivalent to the first degree fragm ent o f the system SI o f strict im plication. T he so called paradoxes o f strict im plication are no t prov ab le in F , but it is easy to see, that F SR is a subsystem o f SI.
Hum bold-University, Berlin Germany
Uwe Scheffler
O K R ESY W A R U N K O W E O PA R T E N A ŚCISŁYM „E N T A IL M E N T "
W artykule rozważa się okresy warunkowe oparte na systemie ścisłego entailm ent Fs skonstruowanym przez Wessela. Poprzez uzupełnienie aksjomatyki i reguł inferencji Fs otrzymuje się system F SK, posiadający dwie struktury implikacyjne, typu: entailm ent i okresu warunkowego. Konsekwencją dalszej modyfikacji systemu FSK poprzez wprowadzenie dodatkow ych reguł inferencji. systemy FSKS, F SK6 i FSK7, jest zmiana odpowiednich klas okresów warunkowych bez zmiany entailment.
