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32Von Wright's truth-logic and around
Alexander S. Karpenkq
Devoted to Georg Henrik von Wright
abstract. In this paper von Wright's truth-logic T" is considered. It seems that it is a De Morgan four-valued logic DM4 (or Belnap's four-valued logic) with endomorphism e2. In connection with this many other issues are discussed: twin truth operators, a truth-logic with endomorphism g (or logic Tr), the lattice of extensions of DM4, modal logic V2, Craig interpolation property, von Wright-Segerberg's tense logic W, and so on.
Keywords: Wright's truth-logic, De Morgan four-valued logic, twin truth operators, tetravalent modal logic TML, truth logic Tr, modal logic V2, von Wright-Segerberg's tense logic
1 Four-valued classical logic C4 and four-valued De Morgan logic DM4
Let M4 be a four-valued logical matrix
MC =< {1, b, n, 0}, D, V, A,{1} >
which is obtained from the direct product of the matrix M (for classical propositional logic C2) with itself, i.e. M^ = M44 x M44, where matrix operations D, V, A, — are the following:
D 1 b n 0 11 b n 0 b 11 n n n 1 b 1 b 0 1111
x —x
1 0
b n
n b
0 1
V 1 b n 0
1 1 1 1 1
b 1 b 1 b
n 1 1 n n
0 1 b n 0
A 1 b n 0
1 1 b n 0
b b b 0 0
n n 0 n 0
0 0 0 0 0
Note that the set of truth-values {1,b,n, 0} is partially-ordered in the form 0 < n, b < 1, i.e. n and b are incomparable. As usual
x V y =: —x D y, x A y =: — (—x V — y), x = y =: (x D y) A (y D x).
It is well known that matrix ffl^ is characteristic for calculus C2. The logic with the above operations is denoted as C4. As usual, we will denote connectives and the similar operations by the same symbols.
Then the logic with the operations V, A and ~ is called four-valued De Morgan logic DM4, where ~ is De Morgan negation: ~ 1 = 0, ~ b = b, ~ n = n, ~ 0 = 1 (see [5], [9]). In another terminology, DM4 is Belnap's four-valued logic [3].
2 Endomorphismus in the distributive lattices
In [6] the authors point out the fact that the modal and tense operations in a number of modal and tense logics and in corresponding algebras are expressed in terms of endomorphismus in the distributive lattices.
Let us consider one-place operations g, e\ and e2
x g(x) ei(x) e2 (x)
1 1 1 1
b n 0 1
n b 1 0
0 0 0 0
which are the endomorphismus in the distributive lattices:
f (x V y) = f (x) V f (y), f (x A y) = f (x) A f (y), f (-x) = -f (x), f (1) = 1, f (0) = 0, f (xs) = (f (x))s, 1
where f can be any operations from g, e1 and e2. 3 Von Wright's truth-logic T"
Now in the new terms introduced above we can define Wright's truth-logic. The expansion of DM4 by the endomorphism e2 leads to the logic which G.H. von Wright in 1985 denoted as T"LM and called a ' truth-logic' (see [28]). For the sake of brevity, we will denote it as T". Here a truth-operator T is the endomorphism e2. Note that the following important definitions hold:
(*) ei(x) =: ~ (e2(~ x)) and e2(x) =: ~ (ei(~ x)).2
It is easy to show that all four-valued J^(x)-operations are definable in T"LM, where
J<(x)^o, ifx=!(i=1'n'6'0)-
Thus, we have:
x Ji(x) Jb(x) Jn(x) Jo(x)
1 1 0 0 0
b 0 1 0 0
n 0 0 1 0
0 0 0 0 1
jx, if 1 —x, i
if 5 = 1
s
x
—ix, if 5 = 0.
2In [19] a four-valued 'logic of falsehood' FL4 is formalized. In our terms it is the expansion of the language of DM4 by the endomorphism ei. So, in virtue of (*) logics FL4 and T" are functionally equivalent.
i
One may easily verify that
J1 =: ei(x) A e2(x), Jb =: ~ ei(x) A e2(x), Jn =: ei(x) A ~ e2(x), Jo =: ~ e1(x) A ~ e2(x).
Note that e2(x) =: J\ V Jb. Then Wright's logic T" is De Morgan logic DM4 with all Jj(x)-operators (but, it is important, without classical negation -i). Note also that in many finite modal logics the operator J\ is the modal operator of necessity □. Then the well-known tetravalent modal logic TML is DM4 with the operator □ added to its language (see especially [9]3). So T" is an extension of TML.
Now we need some additional definitions. A finite-valued logic Ln with all Ji(x)-operators is called truth-complete logic, and a logic Ln is said to be C-extending iff in Ln one can functionally express the binary operations D, V, A, and the unary negation operation, whose restrictions to the subset {0,1} coincide with the classical logical operations of implication, disjunction, conjunction, and negation. In virtue of result of [2] every truth-complete and C-extending logic has Hilbert-style axiomatization extending the C2. It means that Wright's T" logic has such an axiomatization. Moreover, it follows from [1] that we have adequate first-order axiomatization for logic T" with quantifiers.
It is very interesting to generalize given four-valued von Wright's logic, i.e. to consider an arbitrary finite-valued De Morgan logic with all Ji(x)-operators. As a result, we obtain an entirely new class of many-valued logics which I suggest to call ' Wright's many-valued logics' and a new class algebras which I suggest to call 'Wright's algebras'. Then again it follows from [1] that for such logics we have adequate first-order axiomatization.
3However, see also [5].
4 Properties of a truth-operator T and the twin truth operators
The following two properties of a truth-operator T are useful: (I) T(~ x) =~ T(x)
(II) T(x) V Tx) — the law of excluded middle.
Note that these two conditions are required in the Tarski's axiomatic theory of truth with a predicate symbol True (see [12]).
None of these conditions is fulfilled in the logic T". However it is interesting to consider the operations ei and e2 as the twin truth operators Ti and T2 bearing in mind (*). Then
(I') Ti(~ x) =~ T2(x)
(II') Ti(x) V T2(~ x) — the law of excluded middle.
Here we must note that the main goal pursued by von Wright has been the construction of paraconsistent logic. So the choice of the operations ~ and T2 is such that the law of contradiction
~ (T2 (x) A T2(~ x))
is not valid in T". But it is interesting that this law is valid in the form
~ (Ti(x) A T2x)) or ~ (T2(x) A Ti(~ x)).
We want to stress that von Wright's truth logic with the twin truth operators Ti and T2 seems to us very interesting.
5 Logic Tr
Let us consider the expansion of DM4 by the endomorphism g. Now the conditions (I)—(II) are fulfilled. Note that operators ~ and g commute among themselves, i.e.
g(x) = g ~ (x).
Moreover, this allows to define the classical negation -: -(x) =: ~ g(x).
We denote a truth logic with the set of operations {V, A, ~,g} by Tr.
There is a very simple and nice axiomatization of this logic (see justification below), where the operation T is g:
(A0) Axioms of classical propositional logic C2. (A1) T(A D B) = (TA D TB). (A2) -TA = T-A. (A3) TTA = A.
The single rule of inference: modus ponens.4
It is worth to mention that there is a generalized truth-value space in kind of bilattice (see [11]). Indeed, smallest nontrivial bilattice is just the four-valued Belnap's logic. In [8] M. Fitting extends a firstorder language by notation for elementary arithmetic, and builds the theory of truth based on bilattice. This four-valued theory of truth is an alternative to Tarsky's approach.
Also in one case, Fitting extends this language by the operation ' conflation' (endomorphism g).
6 Interrelations between T" and Tr
Let P4 be Post's four-valued functionally complete logic (see [20]). The set operation R is called functionally precomplete in P4 if every enlargement {R, f} = R U {f} of the set R by an operation f such that f / R and f € P4 is functionally complete.
It is not difficult to prove, that the logic with the set of the operations {V, A, e2,g} is four-valued Lukasiewicz logic L4 which first appeared in [15]). According to Finn's result L4 is precomplete in P4 (see [4]). Note that in L4
4At the time of my report G. Sandu had asked about the logic Tr with the axiom (A4) TA = A. Let's denote this logic by Trc. If we take the operation T as identity operation of C2 then the logic Trc is a conservative extension of C2.
x V y = max(x, y) and x A y = min(x, y),
i.e. the truth-values in L4 are linearly-ordered5.
As a result, we have the following lattice of extensions of DM4:
L 4 A
T" < > Tr
DM4
7 Modal logic V2
In [25] Sobochinski presents the formula (S2):
Dp V D(p D q) V D(p D -q).
He establishes that it is not provable in S5, and S5 plus (fi2) is not classical calculus C2. In [26] this logic is denoted by V2. As a consequence of Scroggs' result about pretabularity of S56 logic V2 is finite-valued one. It was remarked that four-valued matrix of 'group III' from [14], i.e. matrix
< {1,b,n, 0}, D, D, {1} >,
is characteristic for V2 (see e.g. [5, p. 190]).
In [6] it has been shown that logics Tr and V2 are functionally equivalent:
5In details about different finite-valued logics see in [13, ch. 5].
6A logic L is said to be pretabular if it is not finite (tabular), but its proper extension is finite. Scroggs [22] has shown that S5 has no finite characteristic matrix but every proper normal extension does.
□p =: p A g(p),
Op =: -□-p,
g(p) =: □p V (-p A Op).7
Note that in [5] an algebraic semantics (named to MB-algebras) has been developed for logic Tr (V2). MB-algebra is an expansion of De Morgan algebra by Boolean negation In this case g(x) = ~ -(x) = - ~ (x). It is interesting that Pynko [21] introduces a similar algebraic structure called De Morgan boolean algebra. He also suggests Gentzen-style axiomatization of four-valued logic denoted by DMB4.
In [17] Maksimova considers all normal extensions of modal logic S4 with the Craig interpolation property. From this it follows that modal logic V2 is the single normal extension of modal logic S5 with the Craig interpolation property (between S5 and C2). Since the logics Tr and V2 are functionally equivalent then the following theorem can be proved:
Theorem 1. A logic Tr has the Craig interpolation property.
8 Von Wright—Segerberg's tense logic W
It is interesting that we can come to the logic Tr on the basis of an entirely different considerations. In [27] von Wright presents a tense logic 'And next' which deals with discrete time. In [23] Segerberg reformulates it under the name W and provides other proofs of completeness theorem, and decision procedure.8
A logic W is a very simple propositional logic in which a new unary operation S with the intuitive meaning of 'tomorrow' is added to the language of the classical propositional calculus. W is axiom-atized in the following way:
(A0) Axioms of classical propositional logic C2. (A1) S(A d B) = (SA D SB). (A2) -SA = S-A.
7However, see [23, p. 49].
8For detailed overview of von Wright's tense logic see Segerberg's paper [24].
The rules of inference: R1. Modus ponens, R2. From A follows SA.
Segerberg suggests the following Kripke-style semantics for W (this semantics in a simplified way is presented in [7, p. 288]). Let N = 0,1,2,... be the set of possible worlds. Valuation v(pi, w) = 1,0 ('truth', 'falsehood') for propositional variables pi and w € N. For d and — as usual, and for SA : v(SA,w) = v(A,w + 1). Pay attention that W is the logic that defines the set formulas valid in N.
Concerning the logic W there are the following meta-logical results:
1) There is no finite axiomatization of W with modus ponens as sole inference rule [23].
2) Logic W is pretabular [7].
It is worth emphasizing that in [6] Muchnik has devised algebraic semantics for W, named 5g-algebras, and has proved Stone's representation theorem for them. Here it is noted that 5g-algebra with involution, where gg(x) = x, corresponds to the logic V2. Thus we again have come to the logic Tr.
Note than in [18] Kripke frame, consisting two possible worlds, is represented for V2. Here we describe Kripke frame i =<T, R> for W and Tr, where T is the set of instants of time.
A Kripke frame i =< T, R > is a frame for W if the following conditions fulfill:
1. Vw € T 3v € TwRv
'from every point (instant) something is accessible'.
2. Vw € T Vvi € T Vv2 € T(wRvi & wRv2 ^ vi = v2) 'from every point no more than one point is accessible'.
And for Tr it is necessary to add:
3. Vw1 € T Vw2 € T Vw3 € T(w1Rw2 & w2Rw3 ^ w3 = w1)
'from every point in two steps we once again find ourse2ves in the same point'.
Theorem 2. Logic W + axiom (A3) SSA = A and logic Tr are the same as the sets of derivable formulas.
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