Non-Classical Operations Hidden in Classical Logic

Non-Classical Operations Hidden in Classical Logic Vladimir Sotirov* * Institute of Mathematics and Informatics,

Bulgarian Academy of Sciences, 8, Acad. G. Bonchev Str., 1113 Sofia, Bulgaria [email protected], http://www.math.bas.bg/˜vlsot









ABSTRACT. Objects of consideration are various non-classical connectives “hidden” in the classical logic in the form of with — a classical connective, and — a propositional variable. One of them is negation, which is defined as ; another is necessity, which is defined as . The new operations are axiomatized and it is shown that they belong to the 4–valued logic of Łukasiewicz.A 2–point Kripke semantics is built leading directly to the 4-valued logical tables.

 

KEYWORDS:



negation, necessity, 4–valued logic, Kripke semantics

Dedication In the issue of this Journal, which was dedicated to the memory of George Gargov, Johan van Benthem mentioned what he called “Sofia school of modal logic”. Indeed, many Bulgarian logicians have been successfully working in the field of modal and non-classical logics for 40 years until now. Prof. Dimiter Vakarelov is at the foundations of this school. He taught and tutored dozens of Masters and Ph.D. students of mathematical logic. Some of his colleagues — as, for instance, the late George Gargov and myself, worked or have worked in close collaboration with him. Prof. Vakarelov is still generating logical ideas and is pleased to watch these ideas being realized by his disciples. At the same time, I would like to note his capability to make mathematics of everything, be it philosophical ideas or puzzles. During the most dogmatic period of Bulgaria, he published a logical theory of dogma. One of his books was devoted to the large variety of permutation toys like Rubik’s cube; and the complete solution gave rise to new theorems of the theory of groups. Another aspect of his work that has always impressed me is the clarity and surprising naturalness of his ideas, even Journal of Applied Non-Classical Logics — Special issue..., th submission.

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Journal of Applied Non-Classical Logics — Special issue..., th submission.

when emerging from the most sophisticated demonstrations. Leaving aside the technicalities of the proofs, every logical invention of Prof. Vakarelov could be explained to the non-specialist. A popular presentation of his achievements in logic, algebra, topology, geometry, and philosophy, would be interesting and fascinating reading. Long life to you, Mitko, and to your logical school!

1. Preliminary and Historical Notes “Hidden” are logical operations, which do not occur explicitly in a certain logical system but can be defined within it. For example, the disjunction is “hidden” in implicational calculus because it can be expressed in it by    thepure

 . We will denote the basic systems by listing the main elements of their signature. So   is the system of the classical implication,    is that of implication plus conjunction, etc. Maybe Ch. Peirce in the 1880-s was the first to observe that implication involves many properties of negation if any subformula of the form  , where  is a distinguished propositional letter, is interpreted as ! . Let us denote the negation  "   (read so obtained by “negation produced by implication and a propositional variable  ”). When it is clear from the only  the sign will be used. A few ex #context,   

$  &$ #&  produces amples: one of the transitivity laws    ' # % one of the laws of contraposition '( ) "  after replacing & with  .  ( In the same manner, the law of commutativity of the antecedents '

  *  !&+  gives another law of contraposition  !"!   *"&%    .   *-&+ +   ,. ++  ,.&  Frege’s law of self-distrubutivity ,  +"   -/ +""  . gives a form of reductio ad absurdum: - .  Modus ponens presented by the formula (

(&   &+  gives  0the&+ weak law of double negation 0

"   *

. The law of reduction   0.&+ gives a variant of the Clavius law,  *",  * . Further on, the  Peirce law (&+ 1 gives the other Clavius law, or consequentia ( 1 .   Sometimes different laws can be derived: the taumirabilis   (  tology  2  2  gives both the law of alternative   '( + &  2

'( )(  and the law  '(&++" .



It is reasonable to ask: which are the axioms of    ? In other words, what is the volume of the negation “hidden” within the classical implication? It includes a part  (

 . of the intuitionistic negation, but not all because it does not satisfy  ' On the other hand, the new negation is not part of intuitionistic negation because  ! !3 +3 is not an intuitionistic law. If disjunction were present, even the law of excluded middle #! would be on the list. Where is the exact place of    on the scale of known negations? Of course, we have to define precisely the possible occurrences of the letter  in a tautology. Obviously we shall not interpret 4( as ' . Perhaps it would be not  %best to touch such subformulas at all. But, what to do with the tautology  3  ?

Hidden Non-Classical Operations

3

One way is to restrict the consideration to formulas not containing the letter 5 as an antecedent (except the case of 56-5 ). A second way is to “incorporate” 526 into the paradigm of 65 . This means to reduce any expression of the first kind to an expression of the second kind. Both variants will be regarded. Let us shortly retrace the history of the “hidden” negation and the main related results. According to [Prior 62] (Part 1, Ch. 3, §1), the idea to use “maximum inference” and “minimum something else” for building up the whole propositional calculus was promoted by Peirce in 1885 and implemented by M. Wajsberg in 1937. The latter introduced the constant 7 (“falsehood”), defined 89 by 906(7 , added a new axiom 76: to the axioms of implication, and obtained the full classical propositional calculus in the form ; 6"< 716(:= . The alternative way for introducing negation was the axiomatic one. Then the full classical propositional calculus was obtained again but in the form ; 6< 8+= . The two systems are not equivalent because the language of the first one is richer: it contains a constant while the second language contains variables only. Nevertheless they are equipollent. A. Church devoted to this question three pages of his eminent book [Church 56] (§23). That is why we will not discuss this topic in detail. Given a formula > of ; 6< 8+= , after replacing all its subformulas of the kind 8*? with ?*6.7 , its representative >+@ in ; 6< 76:= will be obtained. Then it can be proved that > is a theorem of ; 6"< 8= iff >@ is a theorem of ; 6"< 716:= . In this sense we may say that the negation 8,A 6"< 7CB is axiomatizable by the axioms of ; 6< 8+= . Earlier on, A. Kolmogorov [Kolmogorov 25] and I. Johansson [Johansson 37] defined “minimal” negation in two ways. The first one was by adding a propositional constant 7 to the syntax of the positive implication 6D without special axioms for it. The negation 8#9 was an abbreviation of 9(6"D7 . We will denote this form of negation by 80A 6D$< 7CB . The second way was axiomatic and used the single axiom :4E : A :'6"8#>+B$63A >%68':B . Denote this form of negation by 8 A 6"D$< :4ECB . The two systems are equipollent again. H. Curry [Curry 63] (Ch. 6, Sec. C, §6) noticed that Johansson had introduced also a system denoted by F1G which was an extension of ; 6D$< :4EC= by the additional axiom A 8,:*6-:B6-: . Curry judged that “no applications are known for F1G , and the system has been little studied. Johansson [. . . ] suggested that it formed a natural system of strict implication, but this has not been worked out”. Adding the Peirce law to F1G , Johansson obtained the system F1H and proved that the axiom A 8':'6(:B$6(: is superfluous in it. As we see, F1H I%; 6"< :4EJ= and is equipollent with ; 6< 7C= .

S. Kanger in [Kanger 55] also considered the system ; 6"< :4EJ= and noted that it was “a weakened classical calculus in the same sense as the minimal calculus is a weakened intuitionistic calculus”. He obviously had in mind the axiom 8':#63A :#6>+B which, when added to ; 6< :4EC= , produces the classical calculus, and, when added to the minimal calculus, produces the intuitionistic one. Kanger proved that each formula

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of K L"M N4OJP has a representative NQ in K L"P obtained by replacing all subformulas of the form R'S with S#L(T ( T is a variable occurring neither in S nor in the axioms of K LM N4OCP U . Then, N is a theorem of K L"M N4OJPWV X Y Z []\ K L^$M _$`JP U iff NQ is a theorem of the classical (intuitionistic) implicational calculus K L"PV X Y Z []\ K L"^)P U .

It will be shown below that the answer of our question about RaV L"M T U is: its adequate axiom is N4O . This means that the two systems, K L"M N4OJP and K LP , are equipollent. However, this fact cannot be immediately derived from the equipollency of K LM N4OCP and K L"M bCP . It is not correct to transfer the operations with a constant to operations with a variable, even with an “arbitrary but fixed” one. The syntax of K L"P is weaker than the syntax of K LM bCP because the constant b is not definable in K L"P . Furthermore, the system K LM bCP contains theorems such as bcLR#N , which is obtained from S LaV N'LS+U but has no analogue in K LP . Therefore a special proof for R,V L"M T U is needed and it cannot be the proof of Kanger. He interprets K LM N4OCP into K L"P but we need an interpretation in the opposite direction. A short exposition of some results concerning “hidden” negation were presented in [Sotirov 01].

2. The “hidden” negation

In a two-part paper published in 1967–1968 [Vakarelov 65], [Vakarelov 66], D. Vakarelov investigated various aspects of some kinds of modalities and negations added to the classical propositional logic, some of them “hidden” according to our terminology. He axiomatically introduced two unary operators: d (“doubtful”) and e (“verisimilar”). The axiom schemes (beside those of the classical implication) were the following five:

eV eN'LNU ; V N'L(SU)LaV e$NL(e$SU ; V N'L(SU)LaV dS#Ld2NU ; V e$NL(SULd dS ; d2NL3V NLaV eWS#L(S+U U .

Let us denote this system by K L"M eWM d2P . Afterwards Vakarelov built a translation of formulas of K LM e4M dP into formulas of K L"P using a propositional letter T , which does not occur in the given formulas, by induction on the construction of formulas:

f)g

f)g V hcUi"h when h is a propositional letter; )f g V e$hcU)i'T4L f)g V hcU and f)g V d2hcU)i f)g V hcU)L(T ; )f g V h0LjUi f)g V hcUL f)g V jU .

Hidden Non-Classical Operations

5

A lemma follows: if k is a theorem of l mn o4n pq then r)s t ku is a theorem of l m"q . For the converse proposition Vakarelov defined a translation vs from l mq into l mn o4n pq by induction on the construction of formulas: vs t wcux"w for w — a propositional variable or the formula y4m(y ; vs t y4mwcux"o$vs t wcu and vs t w*m(y u)xp2vs t wcu for wx' z y; vs t w*m{ u)xvs t wcumvs t { u for w1n {*x' z y. A next lemma follows: if k is a theorem of l mq then vs t ku is a theorem of l mn o4n pq . Combining both lemmas and using vs t r)s t ku u2x*k , a theorem is obtained: a formula k is provable in l mn o4n pq iff r)s t ku is provable in l mq . (This theorem provides the decidability of l mn o4n pq .) Therefore l mn o4n pq and l m"q are equipollent.

One can count the result of Vakarelov as close to the answer of our question because his p is our |,t mn y u . Two peculiarities of his exposition however prevented the direct answer. The first one is the “direction” of the view: Vakarelov transforms the theorems of l m"n oWn p2q into l m"q while we want to transform the theorems of l mq into. . . And here is the second peculiarity: the system of Vakarelov includes o besides p0xc| . The two operations interact in axioms and it is impossible to separate them. To reverse the direction of the view an inverse equality is needed: r)s t vs t ku ux%k . Then the definition of r)s must be modified with the permission for y to occur in k . The demonstration is almost the same. As a result, the following theorem is obtained: T HEOREM 1. — A formula k is a theorem of l

mq

iff vs

t ku

is a theorem of l

mn o4n pq .

Roughly speaking, the system of o and p axiomatizes ym and my . However, the inconvenience with the appearance of o remains. Indeed, we could benefit from this inconvenience because it gives an answer to the question: what are the axioms characterizing the two operations generated by the two positions of the letter y in the implication? The axioms are those of o and p . However, if we insist on the unique operation generated by m(y , the second part of Vakarelov’s paper provides a solution. Following the spirit of the paper, we shall change some denotations and shorten the proof.

o$w can be represented as p }(mw with } — a fixed theorem. This is the mentioned above incorporating of the expression y m into an expression containing only may . In addition, replace p with | . Denote this transformation by ~ . It is easy to show that the axioms of l m"n oWn p2q so transformed can be deduced from k4 . On the other hand, commute | to p in l m"n k4Jq . Then k4 so transformed can be deduced in l mn o4n pq (actually Vakarelov worked with the equivalent pair of axioms for negation t k#m€+u$m3t |#€ m|'ku and k#m|+|k which are closer to his axioms of o and p ). Hence T HEOREM 2. — A formula l mn k4Cq .

k

is a theorem of

l mn o4n pq

iff

~]t ku

is a theorem of

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Combining the two theorems gives rise to the following C OROLLARY 3. — A formula ‚ ƒŠ 4‹C„ .



is a theorem of

‚ ƒ"„

iff

…]† ‡ˆ † ‰ ‰

is a theorem of

The corollary is an expression of the fact that although ‚ ƒ„ and ‚ ƒŠ 4‹C„ possess different syntactic capacity, they have equal semantic capacity having isomorphic sets of theorems. This is the precise sense in which we say that 4‹ axiomatizes the negation Œ † ƒ"Š  ‰ “hidden” in ‚ ƒ„ .

The translation … ‡ˆ transforms a formula containing both cƒ and ƒ into a formula containing Œ . As we mentioned, the class of transformed formulas might be restricted to formulas without “left  ”, i. e., without subformulas of the form ƒŽ (except  ƒ- ). The corollary and the axiom of Œ*† ƒŠ  ‰ would then be the same. However the proofs become much simpler.



L EMMA 4. — If  is a theorem of not containing left  .

‚ ƒ„

not containing left  , there exists a proof of

P ROOF . — If + Š    Š 4‘"’* is the proof of  , replace each left  in the inference with “*ƒ” (“ is a fixed theorem). • Š    Š ‘• ’*• is obtained. To be it an inference of • , it has to be filled out by the proofs of “ , † “*ƒ ‰2ƒ” , and ƒ”† “!ƒ. ‰ together with the members providing the substitutivity of equivalent formulas. In result an inference of • will be obtained not containing left  . But it will be an inference of  as well because •4’ . ■ We shall modify the transformation –)ˆ of

‚ ƒŠ 4‹C„

into

‚ ƒ„ :

–)ˆ † Žc‰’"Ž when Ž is a propositional letter; )– ˆ † ŒŽc‰)’'–)ˆ † Žc‰ƒ( ; )– ˆ † Ž0ƒ—‰’–)ˆ † Žc‰ƒ(–)ˆ † —‰ . To prove that –)ˆ † ‰ is a theorem of ‚ ƒ"„ when A is a theorem of ‚ ƒŠ 4‹C„ it is enough to prove –)ˆ † 4‹C‰’ † ƒa† ˜ ƒ( ‰ ‰ƒa† ˜#ƒa† 'ƒ( ‰ ‰ but it is obvious. Modify also the converse transformation for formulas not containing left  (except 4ƒ( ): ‡ˆ † Žc‰’"Ž for Ž — a propositional variable or the formula 4ƒ( ; ‡ˆ † Ž*ƒ( ‰’Œ‡ˆ † Žc‰ for Ž’' ™ ; ‡ˆ † Ž*ƒ— ‰)’‡ˆ † Žc‰ƒ‡ˆ † — ‰ for Ž1Š —*’' ™ . To prove that ‡ˆ † ‰ is a theorem of ‚ ƒ"Š 4‹J„ when  is a theorem of ‚ ƒ„ , it is enough to prove all “  -readings” of the implicative axioms. We can choose them to be, say, ƒš† ˜.ƒ›‰ , † -ƒ›˜‰ƒš† † ˜.ƒœ‰"ƒ/† -ƒœ‰ ‰ , and † † ƒ˜+‰2ƒ‰2ƒ” . Only the last two axioms have such readings and these readings are unique: † 'ƒ˜+‰ƒa† Œ'˜ ƒŒ‰ and † Œ'ƒ‰ƒ . It is not dif-

Hidden Non-Classical Operations

ficult to deduce them from ž4Ÿ . Again theorem is obtained:

 )¡ ¢ £¡ ¢ ž¤ ¤W¥'ž

7

. In such a way the following

T HEOREM 5. — A formula ž containing right ¦ only is a theorem of is a theorem of § ¨"ª ž4ŸJ© .

§ ¨©

iff £¡

¢ ž¤

Summarizing the results, we see that the weak law of contraposition characterizes axiomatically the full variety of negations “hidden” inside the implication, both the intuitionistic and the classical one. Rearranging the matter and changing the emphases, we find these results in the paper of Vakarelov. One can conclude that they are not surprising taking into account the constructions by Peirce, Johansson–Kolmogorov, Wajsberg, and Kanger. However, the next result of Vakarelov was really surprising.

3. 4-valued semantics of the “hidden” negation Vakarelov found that the logic of the “hidden” negation coincided with the 4valued modal logic of J. Łukasiewicz [Łukasiewicz 53] and therefore it possessed a truth-table semantics. But a new peculiarity appeared in the proof: it used the presence of the classical negation. To carry out the proof strictly in the system of the plain implication, the multiplication of truth-tables will be applied. I do not know who invented this method. Łukasiewicz used it in his main paper of 1953 without references. We meet it in an earlier paper of 1950 by J. Kalicki [Kalicki 50]. H. Rasiowa in 1955 applied the same method with a reference to Kalicki and to a paper of 1936 by S. Ja´skowski as well. Kalicki himself called the matrix multiplication “well-known” and referred to a paper of 1935 by Wajsberg. And so on. . . Obviously it was folklore of the Polish logicians. Anyway, the multiplication of two tables successfully works in a case when a new propositional operation has to combine the properties of two given operations. I shall explore it to build a 4-valued table for “hidden” negation. Suppose «¬ and «®­ are the matrices of two binary operations, ¯ ¬ and ¯ ­ are the matrices of two unary operations, and 1 is the designated truth-value. Denote by °)¬ and °]­ the sets of tautologies produced by the operations with matrices «"¬ , ¯ ¬ , and «­ , ¯ ­ respectively. L EMMA 6 (K ALICKI –Ł UKASIEWICZ ). — The Cartesian products «"¬±«®­ and define operations whose set of tautologies is the intersection of °)¬ and °]­ , (1,1) being the designated truth-value “true”.

¯ ¬±c¯ ­

In the system of ² , its origin (the string ¨›¦ ) suggests that the tautologies of § ¨ª ²+© coincide with those formulas which are tautologies when ² is treated both as a (classical) negation ³ (in the case ¦®¥´ ) and the constant “truth” µ (in the case ¦"¥¶ ). The same can be observed in the fact that the axiom of ² remains true when ² is replaced both with ³ and µ . It will be convenient for the further to denote by ·4¢ ž¤1¥3¢ ¸ ¢ ž¤ ª ¹ ¢ ž¤ ¤ the value of ž in the set of pairs of 0 and 1. Denote also by º and & the non-formal implication and conjunction expressed either by words or by numbers. Multiplying the corresponding 2-valued ma-

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trices of implication (obtaining the Cartesian square) as well as the matrices of » and ¼ , the 4-valued operations ½ and ¾ are defined by ¿À Á,½Â+ÃÄ*À Å À Á,ÆÂ+à , Ç À Á!ÆaÂ+à ÃWÄ!À Å À Á+ÃWÆaÅ À ÂÃ È Ç À ÁÃ4Æ Ç À Âà Ã4Ä!À ÉÊÅ ËcÌ"Å ËÅ ÍÈ ÉÊ Ç ËcÌ Ç Ë Ç Íà , where for short Å À Á+ÃÄ!Å Ë , etc. ¿4À ¾!Á+ÃÄ*À ÉÊ"Å Ë , 1). Numbering the pairs from (1,1) to (0, 0) with the figures from 1 to 4, the operations shape into Table 1:

½ 1 2 3 4

1234 ¾ 1234 3 1133 3 1212 1 1111 1 Table 1

Î 2 1 2 1

Ï

» 4 3 2 1

1 2 3 4

1234 Ð 1234 2 2244 2 3434 4 4444 4 Table 2

Ñ 3 4 3 4

R EMARK . — » is the classical negation introduced with ¿À Î is coming to be considered.

»)ÁÃÄ#À É$ÊÅ ËWÈ É$Ê Ç Ë$Ã ;

T HEOREM 7. — A formula Ò containing right Ó only is a theorem of is a theorem of Ô ½"È Ò4ØJÕ . T HEOREM 8. — The system

Ô ½"È Ò4ØJÕ

Ô ½Õ

iff Ö×

À ÒÃ

is characterized by the 4-valued matrix.

The proof is given by Lemma 6 and the construction of the matrices for ½

and ¾ .

We can notice after Łukasiewicz that the order of the coordinates of À ÉÊÅ Ë$È É Ã was of no importance and could be reversed. In that case a new operation would be obtained with the same grounds to be called “negation” as ¾ . We denote it by Î and its values are shown in the table. Łukasiewicz observed a curious fact: there are “twins” in his logic — operations, which have the same properties when regarded separately, but different when they appear together. In our setting ¾ and Î are “twins”. The symmetry between them can be derived also from the fact that both of them satisfy the axiom of the negation. As a result they take part in the same tautologies. However they are not equivalent. The “twins” help us construct a new intuitive semantics, other than that of Łukasiewicz. Suppose the world is divided in two parts according to some criterion. To give some names, let us call them “here” and “there”. Our sentences about the two “worlds” are classical, i. e., they are true or false but with an indication where they are true or false — “here”, or “there”. If, for example, it rains here, but not there, the situation is determined by the couple (1, 0), shortly denoted by 2. If it is cold there but not here, the situation is À Ù È É Ã+Ä*Ú . In such a case, is the sentence “If it rains, it’s cold” true? It depends: the implication is false here but not there. Therefore its truth-value is À Ù È É ÃÄ,Ú . In such a way the whole matrix of the implication can be filled up. It is natural to introduce a “global” negation » inverting the truth-values by coordinates: the negation of “It rains here but not there” (1, 0) will be “On the contrary, it doesn’t rain here but there” (0, 1). Imagine further that in the two halves of the world are two extremely dogmatic sects each of them denying every statement about “that” world but confirming as a truth everything concerning their own world.

Hidden Non-Classical Operations

9

So two “local” negations are obtained, Û and Ü . As we see, the 4-valued logic can serve in this manner the “bi-polar” thinking.

4. The “hidden” necessity Now we turn to the system Ý Þ"ß à)á , axiomatized as usual. This time the occurrence of a propositional letter â inside a conjunction will be read as a necessity ã . For example, ä å1à+æ+ç$Þå gives ã4å'Þå after replacing æ with â . It is not difficult to realize that well-known tautologies produce some of the most popular laws of necessity, e. g., ãåà1ãè Þãä å"àèç together with the converse implication, ã4å#Þ(ã+ã4å , ã ä å Þ3è+ç4Þ-ä ã4å Þ3ã+è+ç and the stronger law ä å%Þ3è+ç4Þä ãå Þ3ã+è+ç , and so on. Especially the standard representation of ãå as åcàãé (é is a fixed theorem) will be used. The most important law missing in our list is ãé with é — a theorem. Respectively, the rule of necessitation inferring ãå from å is unsound, too. Because the conjunction is symmetric, we have only one possibility to define the necessity by â . At the same time we have to take care of some bad instances like â1Þä â4à®â ç , ä å"àâ çÞ(â or even åàcä â$àâ çÞåàâ .

It will be shown that the axioms of the “hidden” necessity are ä åaÞêèçcÞ and ã4åÞå . Denote this system by Ý Þß à$ß ã+á . It is known as the Ł-modal logic of Łukasiewicz although his original axiom includes a variable functor ë and therefore has wider expressive capability. Łukasiewicz refers to an unpublished paper by Wajsberg about the completeness theorem with regard to the 4valued truth tables. However the first published proof is that of T. Smiley [Smiley 61]. The adequacy of the 4-valued matrices for the plain ã -axioms together with the â interpretation of ã can be extracted from the second paper by Vakarelov because his ì and í are inter-definable with ã (using î ). His proofs are purely syntactic. Ten years later the paper [Porte 79] by J. Porte appeared. It contained the adequacy of the â -interpretation for the 4-valued matrices (in fact Porte used a constant instead of the variable â ). His proof applied multiplication of matrices and obviously was independent of Vakarelov’s papers.

ä ãåÞã+è+ç

The procedures of the previous sections will not be given in detail but only the main points will be marked. Of course, it would be elementary to reduce the modality to the “hidden” negation using the classical negation, but we prefer not to introduce additional operations. Define a translation ï)ð of a formula of Ý Þß ã+á into letter â , according to the construction of the formulas:

ï)ð )ï ð )ï ð )ï ð

Ý Þ"ß à)á

ä ñcçò"ñ when ñ is a propositional letter; ä ãñcç)ò'ï)ð ä ñcçCàâ for ñaò" ó é (é is a fixed theorem); ä ãé4ç)ò'â for é — the fixed theorem; ä ñ!à2ôçò'ï)ð ä ñcçCàï)ð ä ôç ;

using a propositional

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õ)ö ÷ ø0ùúûüõ)ö ÷ øcûù(õ)ö ÷ úû . In such a way any theorem of ý ù"þ ÿ
is translated into a theorem of ý ù"þ  because this is true for the axioms of ý ùþ ÿ , as we saw in our examples. For the converse translation  ö we have two possibilities. If  occurs somewhere in  out of conjunctions or in    , then  ö ÷  û could be  ÷   ÿ
4û , — a fixed theorem. It is the simplest way but then theorems like ÷  ùa÷    û ûù3÷ ÷    ûù  û would lose their more informative image of the form ù”÷ ÿ  ù  û with an appropriate . That is why we prefer to obtain a ÿ -image which is maximally close to the original. The definition of  ö follows (the manipulation over is not specially described):  ö ÷ øcûü"ø for ø — a propositional variable or  ù  ;  ö ÷  2øcû)ü  ö ÷ ø  ûü'ÿ  ö ÷ øcû for ø ü  ;  ö ÷    ûü'ÿ ;  ö ÷ ø2úûü  ö ÷ øcû  ö ÷ ú û for ø1þ úü  ;  ö ÷  ùú û)ü'ÿ
"ù  ö ÷ úû for úü  ;  ö ÷ ø*ù  ûü  ö ÷ øcûù(ÿ for ø ü  ;  ö ÷ ø*ùú û)ü  ö ÷ øcûù  ö ÷ ú û for ø1þ úü  . It has to be proven that if  is a theorem of ý ù"þ  , then  ö ÷  û is a theorem of ý ùþ $þ ÿ . Consider firstly the axioms. The axioms of implication do not contain  and come under the last case of the definition. The result is obvious. If the axiom is ÷  ûù , the possible translations are ÷ ÿ
"ÿ
4ûù›ÿ , ÷ ÿ
  ö ÷ û ûcù ÿ
Wþ ÿ   ö ÷  ûcù ÿ , and ÷ ÿ  ö ÷  ûÿ  ö ÷ û ûcù ÿ  ö ÷  û for  þ ü  (for the third formula use the standard representation). All three formulas are theorems of ý ù"þ ÿ
. The case of ÷  +û+ù is the same. The axiom  ù-÷ !ù÷  +û produces ÿ #ù-÷ ÿ #ù3ÿ û , ÿ #ù-÷  ö ÷ +û4ùaÿ  ö ÷ û û ,  ö ÷  û2ù÷ ÿ
*ùÿ  ö ÷  û û , and  ö ÷  û2ù÷  ö ÷ +û2ù÷  ö ÷  û  ö ÷ +û û where  þ ü  ; the standard representation is needed again. To check ù that modus po cannot be nens preserves deducibility is not difficult because  and in  ö ÷ +  û ö ÷  û ù ö ÷   û  . Therefore a theorem   is obtained from   and  ö ÷  û which ) õ ö ÷ ö ÷ û û   is equivalent to  because the are theorems by assumption. This time strict coincidence of both formulas would require too complicated rules for õ)ö and  ö . Therefore the system ý ù"þ þ ÿ
may be considered as an axiomatization of the necessity “hidden” in the system ý ù"þ  : T HEOREM 9. — A formula  is a theorem of ý ùþ  iff  ö ÷  û is a theorem of ý ùþ $þ ÿ . 5. 4-valued semantics of the “hidden” necessity To obtain a 4-valued matrix adequate for the necessity, almost the same observation can be made as in the case of negation. Following Łukasiewicz, note that the

Hidden Non-Classical Operations

11

axioms of  remain true when  is replaced with both the constant “falsehood” and the operator of identity. The same conclusion comes from the nature of  : because 
is generated from !" , #$ is obtained in the case of "
%$ , and 
# in the case of "%& . We find a third reason for such a treatment in the fundamental paper by S. Kripke [Kripke 65]. He deduces '() *,+-
*/. as a theorem (in the system with ' ) where and * may be assumed not to have variables in common. Then the theorems of 0 +1 !1 
2 are exactly those formulas, which are theorems simultaneously of the two systems obtained by adding respectively ' and *,+* as a new axiom. The first additional axiom reduces  to the operator “falsehood” and the second one reduces 
to . In other words, the system 0 +1 !1 
2 is the intersection of the Falsum and the Trivial systems. The last fact was first noticed by A. Prior [Prior 57] (Ch. 1). He appreciated it as an additional motivation for the two-component necessity. “Necessarily 3 , Prior proclaims, on no account asserts less than that 3 is actually true, and never asserts more than that 3 is at once true and false (for this last is a kind of upper limit to all assertions — if you’d believe that you’d believe anything)”. His conclusion is that Łukasiewicz’s logic is the logic of modalities each of them covering the maximal part of the natural spectrum of truthfulness. Anyway the multiplication of the two 46574 matrices for ! gives 8) *9!;:
.?/= @1 A >?
A @.7%) = >= @1 A >A @. . The matrix for  is obtained by 8) *.7% ) = >1 $ . . The inverse disposition ) $ 1 A >. gives B , the “twin” of  . The results of

renaming the pairs of 0 and 1 are presented in Table 2. T HEOREM 10. — The system

0 +1 !1 
2

is characterized by the 4-valued matrix.

The proof follows from Lemma 6 and the construction of the matrices for and  .

+1 !

,

6. Kripke semantics of the “hidden” operations What is the Kripke semantics for 0 +C1 !1 2 ? Does any ready to use result exist? Since