A BEGRIFFSSCHRIFT FOR SENTENTIAL LOGIC John EVENDEN 1. INTRODUCTION The title of this essay is probably a misnomer, but the aim is to achieve f o r sentential lo g ic something v e ry simila r t o Frege's achievement for general logic. A good starting point is the thought that sentential logic is truth functional in the n a rro w sense that each o f its propositions is sufficiently specified b y a mere permutation o f truth values. This suggests constructing the logic in a script entirely comprised b y tru th tables, though it mig h t be supposed that the re su lt wo u ld b e in to le ra b ly cumbersome. Fortunately, however, a type o f diagram discussed b y Hubbeling is su f- ficiently compact t o p a rt ia lly overcome t h is d if f icu lt y a n d moreover, ma n y wo rth wh ile insights ca n b e obtained i n a discussion confined to two variables, wh ich further simplifies the tables. Th e resulting scrip t displays features o f sentential logic that are concealed b y more fa milia r symbolism. 2. BASIC SYMBOLISM The following are examples o f the diagrams described b y Hubbeling. - - + +1 P In each case t h e corresponding Peano-Russel expression has been writte n after the diagram. I t w i l l be seen that the arguments o f each table have been entered on the table and combined i n a manner v e ry simila r t o th e use o f Euler o r -
414 J O H N EVENDEN Venn type diagrams. The upper half of the square is reserved for p, the lo we r half fo r p, the left hand half fo r q and the right hand half for q. Consequently a «+» in the left upper half denotes p q, i n the rig h t upper h a lf p q and so on. Finally, when there is more than one plus the table denotes the disjunction o f the combination; f o r example, pluses f illing both the upper half and the left hand half denotes p V q. The tables can be extended to three or more arguments, but the present discussion will be confined to two arguments. As i s we ll known, t o each function specified b y a tru th table there corresponds a connective, though o n ly fo u r connectives are in regular use. When interpreted as a connective the table will be placed in square brackets and the four connectives in regular use are as follows. " [ - To avoid a ll punctuation, the syntax o f Polish notation will be adopted: eg. we shall write E+71+- 1 The connective tables can be interpreted as a n operation assignment for the reduction of the expression to simpler form. The fo llo win g are a ke y t o positions in th e connective, a n expression and its reduced fo rm. [ 1 2 3 4 [ + + + + a n d not ++-_!-+ + + + I I + + 1 The p lu s i n p o sitio n 1 o f t h e connective ta b le means «wherever both the first and second argument tables have a plus, place a plus in the resultant table» and this occurs in position 3 o f the two argument tables. The minus in position 2 o f the connective table means «wherever a plus occurs in the first argument table and a minus in the second argument
A BEGRIFFSSCHRIFT FOR SENTENTIAL LOGIC 4 1 5 table, place a minus i n the resultant table» a n d th is occurs in position 1 of the two argument tables. The plus in position 3 of the connective means «wherever a min u s occurs i n th e first argument and a plus in the second, place a plus in the resultant» and simila rly fo r the last position. The operations ju st described w i l l be called «contraction» and any composite expression can be contracted to a single table b y f irst contracting the sub-formulae and wo rkin g u p to the main connective. The single truth table resulting fro m these operations will be called the «fully contracted form» o f the expression. Expressions that are not f u lly contracted will be called «composite». + Consider [ + the second argument is re a lly o n ly a dummy, i t can be anything whatever without affecting the f u lly contracted fo rm o f the expression. For the connective is a degenerate binary connective: it is one of the two forms of negation, and these are unary. Such expressions can be writte n [ + _ + _ V+IL' -Ft + with p D q and V q + + - +, etc., je. in the first example one can add a dummy second argument i f one like s and in the second e x- ample a dummy first argument, if one likes. From time to time comparisons will be made with Principia Mathematica. In many cases the point made is va lid fo r most or a l l o th e r presentations o f sentential lo g ic, P.M. me re ly being taken as th e example. Alternative presentations o f a Boolian function in the present scrip t d o n o t correspond wit h alternative presentations in P.M. Eg. compare: ++ 1111 and[++-1-117 +11++-1111
416 J O H N EVENDEN 3. FORMA L INFERENCE The following is an example of a proof: E+ ++ + + + + 2. 3. 4. Clearly, b y such proofs a ll theorems can be proved and the method is a decision procedure. Also, i t is easy to see that every line in the proof is equivalent to ve ry other line (even if the function is not a tautology). No w in the sentential part of P.M. e ve ry line of every proof is equivalent to every other line o f e ve ry proof but the point is concealed b y the symbolism. Indeed, equivalence is in an important sense the basis of proof in P.M., fo r a ll theorems are so many different presentations of tautology. In the present script this point is made evident. The fa ct that a ll lines o f the above p ro o f are equivalent leads one to ask whether there is not a va lid proof running from step 4 to step 1. Consider lines 3 and 4. The connective is analogous to say «X» o r «-F-» in arithmetic. In seeking an inference fro m step 4 to step 3, f o r the given connective ([ + T ), it is as i f one were seeking a p a ir o f factors f o r lin e 4 (c.p. «X») o r t wo members whose sum is lin e 4 (c.p. «±» ). O r in seeking the second argument f o r the g ive n connective and the first a r- + I + gument a n d ' 1 ), i t i s a s i f o n e we re a skin g whether a number (the f irst argument) is a factor o f lin e 4.
A BEGRIFFSSCHRIFT FOR SENTENTIAL LOGIC 4 1 7 Also, u n like the questions usually asked in arithmetic there is the further type o f enquiry in which, g ive n both the a r- guments i n lin e 3, one asks whether there is a connective that will contract them into line 4. On any of these alternatives a problem o f inverse operation is posed that should be ca - pable o f solution, though it certainly has some unusual features. In the present paper we shall o n ly investigate the procedure in wh ich an inference is made fro m a given connective and the procedure will be called «expansion b y a given connective», o r mo re b rie fly, «expansion». Th e function t o b e expanded w ill be called «the contraction» and the t wo functions fo rmin g the members o f the expansion w i l l b e called its «first and second arguments». In what follows the diagram [ 3 4 It is easy to see that the interpretation of the connective is as follows. Fo r a sign a t position I, «wherever this sign occurs in the contraction, the sign «±» ma y be placed in the corresponding position i n both arguments o f the expansion». For a sign at 2, «wherever this sign occurs in the contraction the sign «-I-» ma y be placed in the corresponding position in the first argument of the expansion and the sign «--» in the corresponding position o f the second». Simila rly f o r signs in the other two positions. In general, this yields several expansions of a given contraction. Notice that the position of a given pair of signs in the two arguments corresponds to a g ive n position in the contraction but not a given position in the connective. The positions in the connective correspond t o pairs o f values i n th e arguments, e.g. position 1 to a «-E» in both arguments in any position. It is interesting to study the expansion of various functions by various connectives, b u t in the present essay discussion will be confined to the expansion o f tautology b y the 16 possible connectives. This is, of course, the domain of two-argument theorems and is the basis o f theorems having a greater number of arguments.
418 J O H N EVENDEN 4. THE EXPANSION OF TAUTOLOGY The expansion o f tautology b y 12 o f the 16 possible connectives yie ld s me re ly t riv ia l theorems. Expansion b y t h e remaining f o u r connectives i s mo re interesting. He re, t h e trivia l cases will be considered first. Expansion b y [ + + ] U s i n g the ru le described in the last section it can easily be seen that the first and second arguments can be anything, ie. th e «tautology connective» fo rms a tautology fro m any two arguments. Expansion b y [ - -. I t can easily be seen that the «contradictory connective» w i l l n o t expand tautology. Expansion b y [ + ], [----- + ] [ + 1 or [ + each o f these tru th tables contains one and o n ly one plus, each connective expands tautology in t o one a n d o n ly one pair o f arguments. These p a irs are, taut/taut., taut./contr., contr./taut., contr./contr. Expansion b y [ + 1 - + tautology into two identical arguments, that may be anything, je. t h e «equivalence connective» f o rms t h e equivalence o f any function to itself. The second of these connectives expands tautology into two arguments, one o f which can be anything and the other of which is the negation of the first. Expansion b y [ + + + all unary connectives, one argument can be anything, the ex- pansions a re 1+ - - + I] I + + I [ I + and [ 1 The above expansions have points o f interest but scarcely
A BEGRIFFSSCHRIFT FOR SENTENTIAL LOGIC 4 1 9 enumerate an exciting o r rich domain o f theorems. The fo u r remaining connectives a re mo re interesting. Expansion b y or. The first o f these connectives expands tautology in to a ll combinations o f arguments except those having a plus in some position in the f irst a r- gument wit h a min u s i n th e corresponding position i n th e second. Th e second o f these connectives expands tautology into all combinations except those having a plus in some position in the second argument wit h a minus in the corresponding position in the first. Therefore the two connectives f o rm a pair whose expansions differ only in the order of the arguments. O n ly the first connective (implication) w i l l b e studied here. Clearly, the three pluses in the connective give three possible pairs of values fo r each position in the arguments o f the expansion and as there are fo u r positions, the total number of expansions is 3 the fa milia r Boolian lattice, b y taking advantage o f the fa ct that the connective is reflexive and transitive (as can easily be shown). The nodes of the lattice are the 16 fu lly contracted functions. Then 16 pairs o f arguments are the reflexive co n - nections o f the nodes wit h themselves, a further 32 pairs are formed b y the lines o f the lattice and the remaining 33 pairs are formed b y the transitive connections o f the nodes joined by t wo o r more intervening lines. Expansion b y or [ + 1. These fo rm a pair in that the pairs o f arguments o f the expansions o f the one are the negations o f the pairs o f the other. The first connective (disjunction) will be studied here. The total number of expansions is once again 81 a n d one mig h t expect that th e y could b e grouped into a lattice, but this is not so. I t is, indeed, easy to show that the connective is o n ly reflexive f o r tautology and that it is not transitive. On the other hand, the connective is symmetric, so that i f the expansions can be presented as a node-and-line diagram, th e lines can be read either wa y. A
420 JOHN EVENDEN + + - + + + +
A BEGRIFFSSCHRIFT FOR SENTENTIAL LOGIC 4 2 1 suggested diagram is presented above and bears n o obvious relation t o a lattice. O n t h e d ia g ra m tautology h a s been omitted as it disjuncts with every function, but its immagined presence, as it were, is indicated as i f it were in the centre, by the short, inward directed line from each function. Contradiction is also omitted; it o n ly disjuncts wit h tautology. Including the lines t o tautology there are 39 lines o n the diagram and these wit h the line fro m tautology to contradiction make 40 lines, each of which can be taken in either direction, making 80 pairs o f arguments. The reflexive disjunction of tautology with itse lf brings the total to 81. In conclusion, the richest realm of theorem are those based on disjunction and o n implication and th e ir pairs. Also, th e system o f theorems based on disjunction differs mo re ra d ically fro m the system based o n implication than one wo u ld expect f ro m the simila rity o f th e ir tru th tables o r f ro m the disjunctive equivalent o f implication i n th e system o f P.M. 5. MATERIAL TRUTH AND FALSITY Consider a f u lly contracted function, eg. a n d it s equivalent in P.M., vis. p.q. Whitehead and Russell endeavoured to distinguish between the positing o f th e function and th e assertion o f it s tru th, writing «pi.» for the former and «1 p.» for the latter, but as, in any case, every line of the relevant part of P.M. is a theorem, th e distinction wa s n u ll. M y impression is th a t i n any case i t is bound t o remain obscure i n th a t symbolism. We take up the point they were trying to make and endeavour to cla rify it, using fresh terminology as i t is debateable h o w close i s th e correspondence between P.M. a n d t h e present system. A function such as LI- TT is a mere permutation o f truth and f a lsit y possibilities a n d a s such i t is n e ith e r tru e n o r
422 J O H N EVENDEN false. Mo re o ve r, because i t contains b o th p lu s a n d min u s signs it does not specify either a true o r a false sentence: f o r this reason i t w i l l be said not t o be «actualised» (c.p. P.M. «merely posited)>). A function analogous to this may, however, specify a sentence that is tru e as a matter o f contingent fact and in that event the function will be said to be «actualised» (c.p. P.M. «materially true)> and «asserted»). Also, the positions in the truth table that wo u ld have been occupied b y minus signs, had the function n o t been actualised, a re n o w replaced b y dots, thus [ + +, because the function specifies a true sentence b y virtue of one o f its truth possibilities but has no falsity possibilities. The tautology, 1 + + h a s in any case no falsity possibilities and as such may be regarded as vacuously actualised (c.p. all theorems being asserted in P.M.). Th e contradiction has, correspondingly, n o tru th possibilities and so cannot be a c- tualised. When a composite function is actualised the minus sign need only be replaced b y dots on the main connective, fo r a little thought w i l l show that i t is a category mistake t o suppose that one p a rt o f a composite expression ma y be actualised and not another. I f such a function is contracted, o r further expanded, the dots can then be preserved on the ma in connective. Three modes o f inference f ro m premisses, that have some unusual features, c a n b e liste d i n t h e present symbolism. T hese are 1. Inference b y equivalence. A n y premiss can be expanded o r contracted. A s re ma rke d e a rlie r, t h is s imp ly presents different forms o f the one function. 2. Inference b y implication. Let one o r more dots in a premiss be replaced b y pluses, th u s f o rmin g a fu n ctio n h a vin g a wid e r ra n g e o f truth possibilities. Th e n a rro we r range o f tru th possibilities was actualised, therefore the wid e r range must be actualised, je. the inference is valid. 3. Inference b y adjunction o f pre-
A BEGRIFFSSCHRIFT FOR SENTENTIAL LOGIC 4 2 3 misses. Fu lly contract the premisses, fo rm a table b y placing a plus in every position where every premiss has a plus and placing a dot in e ve ry other position. Then because a ll the premisses a re actualised, t h e ir co mmo n t ru t h possibilities must be, je. the inference is valid. Notice that after a set o f premisses have been adjuncted, the original premisses can be restored b y implication. Indeed, if one wishes t o develop a ll th e consequences o f a se t o f premisses, one can first adjunct them a ll in to a single f u lly contracted premiss a n d then systematically a p p ly inference by implication and b y equivalence. Then any conclusion that can va lid ly be infered from the original premisses can va lid ly be inferred fro m the single f u lly contracted premiss. 6. S OME META-THEOREMS The present section refers to the system o f non-actualised (and vacuously actualised) functions. B y means o f re la tive ly informal meta-proofs o f w e l l kn o wn theorems t h e re la tio n between implication, equivalence, proof and theoremhood will be explored. Besides its intrinsic interest this w i l l imp licit ly display something o f t h e relationship beween t h e present system and mo re fa milia r systems. The turnstile, «1» will be used for «is a theorem», ie. «contracts to tautology» and positions in the tabled w i l l be identified b y the ke y 13 4 1 6.1 I f t and a r e steps in the same proof, then 1 ito F o r ex hypothesis i s a contraction of do o r vice versa. Suppose the former and that the fu lly contracted form of 1 then t h e fo llo win g p ro o f ca n b e constructed. A. c. L r (I) B. + T i T '11 all steps o f a proof are equivalent.
424 J O H N EVENDEN Corollary. 6.2. I f a nad ll steps a r e in a nthe y functions proof of a such theorem a t are + 0Il theorems. 0 I ll + 1 [+++ then (11W a n d [ A. + + 11) + B. + 4 3 ' IF' C. + + 1 IP' are the f u lly contracted forms o f 0 and T. No w the connective + 7_1 i s n o t used i n the p ro o f u n t ill the la st step and inspection shows th a t o n ly positions 1 a n d 4 a re then used. Consequently, t h e connective is e q u a lly va lid. Moreover, inspection o f th is la st step shows th a t 0 ' therefore [ + IP 0 a n d, [++ T 0. He n ce equivalent functions mutually entail each other. 6.3. I f 0 and T a re a n y functions such that [ + ] 0 tit and [ + + ill 0, th e n [ + I l l (and [ I F 0 ). F o r there must be proofs ending A [4+ - + ] 0 ' I W' B. I + + I I and A. L + r + + + F r o m the first o f these no p a ir of signs in 0 T ' can be +,, and fro m the second no pair in T ' 0 ' can be +,, and therefore no p a ir in 0 ' can be, +, Therefore o n ly positions 1 a n d 4 o f the connective can be used, etc. Hence functions that mutually entail each other are equivalent. 6.4. I f 0 is a n y function [ H + + F o r there must be a proof A. [+, 1 + + I + + + + I B. 1 F + + I + + I
A BEGRIFFSSCHRIFT FOR SENTENTIAL LOGIC 4 2 5 and this last line cannot contain a + wit h a, je. position 2 is not used in further contraction and therefore one ma y add C. + I He n c e a ll functions entail tautology. I + + I 6.5. I f (I) is any function such that H. [ ++ ] 1 H. O. Proof b y 6.3 and 6.4. 6.6. I f H [ + + ( I ) II' then (I) and 1 steps in the same proof. Fo r there must be a proof A. [ + T ' B. ++ ++ land from this last step (11' = 7. Therefore there must be a proof A B. ( = T ' ) C. T. These various theorems ca n o n ly b e presented a s me ta - theorems i n the present scrip t a n d th e y are, indeed, me ta - logical, a point that is obscured in P.M., where they are carried down into the object system. Relating this to earlier sections of this paper it will be seen that the Begriffsschrift displays and distinguishes three aspects o f sentential logic, th e lo g ic of the manipulation o f functions (and notably, tautology), th e logic o f inference fro m premisses and meta-logic. I n P.M. a ll three aspects are confused. John Evenden