HANS REICHENBACH
Elements
of SI^boIic Logic
fHE FREE PRESS, New york coLLIER-MAcMTLLAN LTMITED, London
II. The Calculusof Propositions
38
'c'must be analyzedfor each of the 23:8 possiblecombinations of truth-values of 'a', 'b' ,' c' . We dassify the formulas with respect to the number of propositional variables or to the kind of operations. The first group referring to only one propositional variable contains fordr,rlaspresentedin traditional logic as the laws of thought, such as the law of identity or the law of contradiction; we see that we are con@rned here only with some special fomrulas out of a long list of other equally important laws of ttrought. The explicit fonnulation of these other formulas, few of which were known in traditional logic, is due to the work of the first logisticians such as de Morgan, Boole, Schrtider, Peirce, Russell, and Whitehead. We add some remarks concerning the equivalence relation. Though not all tautologies are equivalences, equivalences play an important role because their function in logic corresponds to the function of equations in mathematics. Most of our formulas are therefore equivalences. Becauseof formula 7a we obtain from every formula containing an equivalence another formula which states, instead, an implication; thus from rb we get ova)o
G)
(at b) t (6> a)
(6)
from 6c we get We do not include such formulas in our list becausethey can easily be obtainedl we rather follow the practice of writing an equivalence sign instead of an implication wherever it is possible. In group 8 we collect ttrose formulas for which an equivalence sign would have been false, calling them 'one-sidedimplications'. [Ex.] TAUTOLOGIES IN THE CALCI'LUS OF PROPOSITIONS Concerning oneproposition: n. o:o l rb. ovo-ol tc. rd. te. rf. r.B ,
tL o=a d=o ov ii aZ o )A=A
I
rule of identity nrle of double negation tertium non datur rule of contradiction reductio ad absurdum
Sum: 2a. ov b = bv a zb, ov (bv c) = (ovD) v c = ovbv c
commutativityof 'or' associativity of 'or'
$ 8. Surveyof PossibleOperations
39
Product: commutativity of 'and' 3a. o.b = b.o associativityof 'and' 3b. a.(D.c): (o.b).c= o,ba Sum and product: 4a. a.(bv c) = a.bv o.c rst distributive rule 4b. ov b.c= (a v D).(ov c) znd distributive rule 4c, (o v b).(cy il) = s.c,v b,cv a.il v b.d I twofold distribution 4d. o.rv c.il,: (av c).(bvc).(ov Q.(bv d) | = a.(ov = b) av a.b o 4e. redundanceof a term Negation,product, sum: Sa . o tr = av 6\ hsaLing of negationline 5 b . ZvT : d. 6J dropping of an always true factor 5c. o.(bvb) = a dropping of an always false term 5 d . o vb. b= a redundanceof a negation 5 e . o v d. b= ov b Implication, negation,product, sum: 6 a . a ) b = A v b) dissolutionof implication 6 b . o )b= o. - b J = 6 c. o )b bt a contraposition 6 d . o ) ( b) c ) = b) ( o) c) = a .b )c symmetry of premises 6 e . (a ) b) . ( a) c ) : a) b .c l 6 f. (a) c ) . ( b)c ) = ay [1 t I merging of implications 6 9 . (ot b) v ( a) c ) : o > b v c I 6 h . (ot c ) v ( b> c ) : o .b tc ) Equivalence,implication,negation,product, sum: ?a.. (a : b) = (o: D).(6:a) \ dissolution of equivalence lb. (a = b) = a.bv d.lb I T€ i= ( o= 6) negationof equivalence f. negation of equivalent terms Zd . (a : b) = 1a = $' 1 One-sidedimplications: 8 a . a ) ov b addition of an arbitrary term 8 b . a .b>a implication from both to auy 8 c. o l (blo) l arbitrary addition of an implica, 8 d . d l ( at DJ tion 8e. a.(o>b) > b inferential implication 8f. (a addition of a term in the implicate 8g. (a > b) ) (a.c) b) addition of a factor in the implicans 8 h . (a v c t b) t ( at b) dropping of a term in the impli8i. (o ) b.c): (a : D) 8j. 8 k. 81. 8m.
(a> b).(c)d) t (o.c>b.d) \ (o ) b) . ( c td) t ( av c ) b v A I (ot b) . ( bt c ) > ( at c ) (o = b).(b= c) t (o = c)
CADS
droppiag of a factor in the implicate derivation of a merged implication transitivity of implication transitivity of equivalence
