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People of Different Cul- Tures May Have Different Logics) in Logical Terms Logique ck Analyse 161-162-163 (1998), 57-65 THE LOGIC OF LOGICAL RELATIVISM Gert-Jan C. LOKHORST Abstract We explicate the thesis of logical relativism (people of different cul- tures may have different logics) in logical terms. Our illustrations come from the field of paraconsistent logic. 1. Introduction Logical relativism is the claim that People of different cultures may have specifically different logics (for example, [there may bel a peculiarly Chinese logic distinct from West- ern logics). [16] Logical relativism was rather popular in anthropological circles before the Second World War [11, 14, 151. It then went out of fashion 171. The thesis is currently undergoing a revival in the so-called Strong Programme in the sociology of knowledge [2, I n Cultural anthropologists usually maintain that there is no evidence that there exist cultures which adhere to different logics than we do [23, 241. But I find this a strange claim. For one thing, even in my own country there is a subculture of people who try to adhere to intuistionistic logic rather than classical logic. For another, there are, as far as I know, no anthropolo- gists, logicians or sociologists of knowledge who have ever tried to formu- late the thesis of logical relativism in a truly precise way. As long as the latter has not been done, it does not make much sense to say that there is no empirical evidence for this thesis. In the following, I shall try to make the thesis more precise, and I will use some familiar logical tools to do so. All systems which I will present for illustrative purposes come from the field of paraconsistent logic. 58 GERT-JAN C. LOKHORST 2. Standard Doxastic Logic When talking about the logical properties of other peoples' opinions, it is natural to turn one's attention to doxastic logic, the logic of belief [12, 191. It is, however, impossible to make sense of the relativists' claim in terms of standard doxastic logic. Standard doxastic logic is based on modal system K. This system has, amongst others, the following rules. C stands for classical propositional logic. RI F -R2 F c çcR 3c 1 —F- › Reading D p as "the agent believes that ço," these rules imply (1) that all FcK agents believe all classical tautologies, (2) that agents' beliefs are closed tDp unde-r C-provable consequence, and (3) that if an agent believes that (p, he P also 1qbelieves all propositions which are C-provably equivalent with q). Thus(-; st an dard doxastic logic portrays all doxastic agents as followers of classD1ical logic. When one wants to deal with logical relativism, one has to adopP1t a less parochial outlook. Romane Clark has written that belief ascription is —1 q >mainly a matter of keeping the references and concepts of those of us Diwho are scribes, recording the occurrences of psychical happenings, P•distinct from those of the agents to whom we ascribe mental events. [6] ; The logical relativist seems committed to the claim that belief ascription involves keeping the agents' and our own logics distinct as well. Just as x y —) (0Fx [ i F y ) is not an acceptable principle of doxastic logic, so 1-c- tli o t p is, from the relativist's perspective, not acceptable either. 3. Adhering to a Particular Logical System What does it mean to say that somebody adheres to a particular logical system? In the literature about logical relativism, this expression is used in at least two quite different senses. First, it is sometimes said that people may adhere to different logics in the sense of embracing different sets of logical truths (see, e.g., [241). Second, it is sometimes said that people may adhere to different logics in the sense of following different logical rules THE LOGIC OF LOGICAL RELATIVISM 59 (see, e.g., [13]). The first conception of adherence to a logic seems related to R 1 above, whereas the second seems closer to R2 and R3. We accordingly propose the following three definitions: Def 1 T h e agent adheresi to logical system X according to logical system iff Fx h y 0 c p . Def 2 T h e agent adheres2 to logical system X according to logical system iff I - Dexf 3 T h e agent adheres3 to logical system X according to logical system tiff t t o ' ) P1 y1The1s e s4ens es of "adherence to a logic" should be carefully distinguished from each other, as the following example makes vivid. -D O .Let CC be the complement of the propositional calculus, i.e., the set of all- non-theorems of C. This system has been axiomatized as follows ([4] see- also [3, 25, 26]). ' D A l p (p atomic) P A2 — p —› p (p atomic) • R la (pip --> çc (p atomic, p does not occur in t Rib i —› ( p atomic, p does not occur in tp) R2 o(P )t i l l (io —› (cP —) R3 till(X ( P ) R4 cP —) 0 R5 1 0 R6 —( c› tPy —11 ( I ) —› tp R7 cP ——› (0 —( › x)/ ( c P —› R8 cp 4S , P— > —1(cp — › S, where S is of the form S = Si or SX S i —) (S2 —* (S n -1 S n )— ), with S, = pi or Si = —1 t Pk for i k , and p E lPi, P2, P n ) for all p which occur in Pi) (P —› qi• , —P i CC is perfectly› unsound and completely antitautological in the sense that kcc (PM' 10c (p141. It will be clear that CC is paraconsistent (it contains all logical falsehoods but no logical truths) and non-monotonic. Now define C-i-ECC as follows. • Axioms: All classical tautologies. • Rules: I. hC-FOCC DP; 60 GERT-JAN C. LOKHORST 2 .3. Mo d u s Ponens. I It will "be clear that an agent whose beliefs are described by this system does noCt adhere to classical logic in the sense of Def. I. Nor does he adhere to clascsi cal logic in the sense of Def. 2, for we have Fc I - 4h C A-LTIC4C Elb -I- -u4 Dt I . Then agoent tdoes however adhere to classical logic in the sense of Def. 3. For suppose that I-c (1, I l l . It follows that Fc (p iff hc 0 0, whence I-cc cp iff Fcc 0. So either (1) Fcc tp and Foc 0 or (2) l F a,nd k c c 0. In the first case, we have Fc+L cc El cp and F c C +'w hecncce 1 -( c + EcP+asiec, cwe haEve Fc -i-Dtcc p- - , lAEE c-kcp M a n d 1aLn y- tcaspce, QED. C0A It+ is cO n otC difCficu lt to contrive systems in which an agent adheres to a -aparticularI I —I lo gic in the sense of one of the other definitions. One may for example use the logic of awareness discussed in 1121, §9.5. Int i 1 l Dda nDo , P 4whd. Adhh erencee to Differentn Logical Systems ceh e; l W-n hat cdoes it mean to say that two agents adhere to different logics? We pCceropos e the following three definitions, corresponding to Defs. 1-3 above. +Oe1A° m-EeansI thatC agenCt i believes that tp. Ic -D+el 4 Agents i and k adhere C1tEe mt Xoc i ff dit isi fnof t ether ceasne tht a t Hx Eitp <=> F x Okcp. +lc o g i c s LaDef c5 Acgenots i randd k adi here2 to different logics according to logical sys- intWem Xg iff it is not the case that Ex LAI) < = > x Mop -) Ea /. ct) o Def 6 Agents i and k adhere3 to different logics according to logical sys- cl o g i c t-em X iff it is not the case that Ex Elicp D I P i=> Ex Lktp Da- l TPs-he followyi ng systesms illustrate the notion of several agents each follow- =-iEng his own lo gic in the senses of all three definitions. They are simplified I0versions of the systems discussed in 1171. l; Each system Ti + (Ecek)o,k,„, 0 i < to, is defined as follows. li qn • Axioms : A ll axioms of ( it e ik,n oiw.ne p.ar,ac onsistent hierarchy [8], plus Lk(tp —› — › (Elkg) —› for all k, 0 k < co. .h t h e i - Se t h os s y s t e m we i n ec D a ho C o s t an a ' s vd w e l e l - F c + D c C 1 1 1 ( 1 ) ' L i t ' i n THE LOGIC OF LOGICAL RELATIVISM 61 • Rules: Modus Ponens and 1-(C k F - D o p , It is of bvioous thrat in each system T i + (Ere, 0 0adheare to differentl l logics in the senses of Defs. 4-6. < Actually,k th, e agen ts adhere to different logics in an even stronger sense of the word. Following [21], one may call a theory of X iff F contains all k < 0 axioms of X and is closed under conjunction and Modus Ponens. One may , , ka l l a g e n t s kthen ob,serve th at eaich agennt k ad, heres to c-Ck in the sense that for all theo- ries <F of Ti + (1 7 1 Tk)k k i n , <Xilop E 11 is a theory of ( ,Cs,tr,ko n. ga sAenngsde ein nstaatndslar dl d oxastic logic. sa Wede tqhhinuke thate r nthee c just-described systems capture the thesis o f logical etrelastivi som in a p articularly clear way.
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