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5Логические исследования 2023. Т. 29. № 1. С. 101-113 УДК 510.642
Logical Investigations 2023, Vol. 29, No. 1, pp. 101-113 DOI: 10.21146/2074-1472-2023-29-1-101-113
Dmitrij Skvortsov
On Finite Domains Based Slices in the Structure of Superintuitionistic Predicate Logics, Preview
Dmitrij Skvortsov
FRC "Computer Science and Control" RAS, Vavilova str., 40, Moscow, 119333, Russian Federation. E-mail: skvortsovd@yandex.ru
Abstract: The article presents a series of closely related papers that together make up a single piece of work, devoted to the description of a hierarchy of slices in the structure of superintuitionistic predicate logics and making probably the first attempt to transfer to the predicate case Hosoi's hierarchy of (finite) slices for the structure of superintuitionistic propositional logics.
Keywords: superintuitionistic predicate logics, hierarchy of slices, Jankov-style formulas, splittings
For citation: Skvortsov D. "On Finite Domains Based Slices in the Structure of Superintuitionistic Predicate Logics, Preview", Logicheskie Issledovaniya / Logical Investigations, 2023, Vol. 29, No. 1, pp. 101-113. DOI: 10.21146/2074-1472-2023-29-1-101-113
Dedicated to the memory of Tsutomu Hosoi (12.03.1937-28.12.2020)
This text announces a series of closely related papers that together make up a single work presenting the results of our recent research (made in 2020).
Introduction
In this work we mainly deal with superintuitionistic predicate logics (see, e.g., [Gabbay et al., 2009, Sect.2.6, Definition 2.6.3]), but we also consider, when necessary, superintuitionistic propositional logics. However, we are interested not so much in individual logics per se, but would like to describe the entire structure they form.
Recall that the set of superintuitionistic logics forms a structure (i.e., a lattice) ordered by inclusion, with the operations of set-theoretic intersection LinL2 and logical sum [Li + L2]; the latter is the deductive closure of the union of logics L1UL2 (i.e., the least logic that extends both these logics). This structure has the bottom element — intuitionistic logic (QH in the predicate
© Skvortsov D., 2023
case or H in the propositional case). It also has the top element — the inconsistent logic [QH+±] or [H+±j. For now, we denote the inconsistent logic, predicate or propositional (depending on the context) by .1 It is well known that the structure of logics, both predicate and propositional, is complete: for
every family (Lg : d £ 0) of logics there exists the intersection f] Lg and the
gee
logical sum £ Lg, which is the deductive closure of (J Lg. gee dee
Our work is inspired by (and based on) the work of two outstanding scientists who studied the structure of superintuitionistic propositional logics — Vadim Jankov and Tsutomu Hosoi.
In the early 1960s Vadim Jankov constructed [Jankov, 1963; Jankov, 1969] characteristic formulas for finite Heyting algebras whose top element has a unique predecessor. It is known that such algebras are just finite subdirectly irreducible Heyting algebras. In Kripkean terms, such algebras correspond to finite rooted propositional Kripke frames (partially ordered sets). Nowadays, characteristic formulas came to be known as Jankov formulas. We believe that Jankov formulas are not simply formulas (or the logics axiomatized by the formulas) but rather an embodiment of a fundamental idea bringing to light the various aspects of the structure of the superintuitionistic propositional logics. We briefly recall the crucial property of Jankov formulas later, in Subsection 0.3; the inexperienced reader can find a description of Jankov formulas and their main properties e.g. in [Gabbay et al., 2009, Section 1.13].
Around the same time in the 1960s (slightly later), Tsutomu Hosoi described2 (see [Hosoi, 1967; Hosoi, 1969 ]) an important hierarchy in the structure of propositional superintuitionistic logics. Namely, he partitioned the structure of logics into an (w + 1)-sequence of 'slices', generated by the decreasing sequence of the logics of finite chains. Each finite slice consists of logics lying between the logic Gn of the (n+1)-element finite chain, regarded as Heyting algebra, (i.e., which is the same, the logic of the n-element linear Kripke frame),3 and the logic axiomatized by the Jankov formula of the (n+2)-element chain (i.e., the (n+1)-element linear Kripke frame).4 The final, w-slice consists of the logics lying between Dummett's logic LC (the intersection of all logics Gn)
1Later, when considering our predicate hierarchy of slices, to achieve the uniformity of notation, we denote the inconsistent predicate logic by QC0.
2as Julius Caesar: veni, vidi, and described.
3these logics Gn were introduced by Godel in [Godel, 1933].
4More precisely, Hosoi used not Jankov formulas as such, but their slightly modified deductively equivalent versions (called n-th Pierce laws, for n > 1, while the 0-th slice consists of the inconsistent logic G0). Hosoi did not know about Jankov's results and formulas until the 1970s, so he independently re-establish Jankov's splitting properties for a particular case he was interested in.
and intuitionistic propositional logic H. This fundamental hierarchy of (finite) slices gives a global view of the structure of propositional superintuitionistic logics, as well as useful information about the logics themselves. E.g., all logics from Hosoi's finite slices enjoy the finite model property, and so they are Kripke complete (see e.g. [Komori, 1975]). Hosoi's classification also gives essential information about the formation of the entire structure of logics. E.g., every finite slice is embeddable (as a poset, and hence, as a lattice) into all the further slices; moreover, onto segments in those slices. The inexperienced reader can find a more explicit description of Hosoi's hierarchy of slices (in Krip-kean terms), together with a more detailed formulation of some of its known properties, in [Gabbay et al., 2009, Section 1.15].5
By the way, note that there exist two other, almost equally fundamental, (w + 1)-hierarchies of slices in the structure of superintuitionistic propositional logics. They are also generated by decreasing w-sequences of finite Kripke frames (posets) leading to two pretabular logics other than Dummett's LC (see in [Maksimova, 1972]). One was described by H. Ono [Ono, 1972-1973]; the second hierarchy was noticed by the author in an unpublished note [Skvortsov, in preparation]. Neither hierarchy, however, is as elegant as Hosoi's; hence we do not discuss them here. We only point out that the three hierarchies together produce a three-dimensional picture of the structure of superintuition-istic propositional logics. In particular, a logic is tabular iff it belongs to finite slices of all three hierarchies; this simple observation sharpens and clarifies the mentioned result by Maksimova [Maksimova, 1972] on existence of exactly three pretabular superintuitionistic propositional logics.
The fundamental achievements by Jankov and Hosoi are closely related through the technically important lattice-theoretic notion of splitting (cf. e.g. [Galatos et al., 2007, Chapter 10]).
In the subsequent preambulus, we briefly present a general description of hierarchies of slices in the structure of logics. We hope that such a description could provide a framework that is more suitable for the predicate case studying in the announced work, as well as for possible future investigations. This perspective also helps us explain the role of splittings in Jankov's and Hosoi's considerations for the propositional case. Then, based on this description, as well as terminology and notions introduced in its course, we shall describe, at the end of our text, the contents of the whole work.6
5Although this presentation is somewhat hasty, and so rather sloppy and opaque in places.
6The numbering of Subsections 0.1, 0.2, etc. in the preamble aligns with that in the announced work, where the introductory section, Preamble, is Section 0, followed by Sections 1, 2, etc. (see Subsect. 0.7 in this text).
Preamble. Systems and hierarchies of slices: An overview
0.1. A subset L' of the structure L of (all) superintuitionistic logics (predicate or propositional) is called convex in L if it satisfies the following property:
VLi, L2 £L' VLo £L (Li CLo CL2 ^ Lo eL').
This condition, in other words, means that:
L' is downward closed in (L')- = {LeL | 3L'eL' (L'CL)} or equivalently, L' is upward closed in (L')- = {LeL | 3L'eL' (LCL')}.
We are only interested in non-empty convex sets. We now mention two examples of convex subsets of L.
A segment (a bounded convex set) is a complete sublattice of L of the form [L1, L2] = {L | Li C L C L2} (for logics L1,L2 such that L1 C L2). An (open) interval is a subset of L of the form (L1, L2) = {L | L1 cLcL2}. Notice that the condition L1 cL2 is necessary, but not sufficient, for the non-emptiness of (L1, L2). The notion of convexity is essentially broader than these particular cases; convex sets may not be sublattices of L (e.g., every antichain in L is convex).
Clearly, a convex set is a segment if and only if it contains the greatest and the smallest elements. On the other hand, a subfamily of the structure L of logics is convex if and only if, whenever it contains two logics L1 c L2, it also includes the segment [L1, L2].
By a system of slices in the structure L of logics we mean a partition of L into convex subsets. This means that these subsets are non-empty, disjoint, and their union is the whole structure L. Of course, a system of slices is worth studying only if either the corresponding equivalence relation itself or the induced equivalence classes (i.e., slices) are interesting or meaningful. We suspect that in most cases, slices will turn out to be segments, although not always. Moreover, we cannot rule out systems containing slices that are not sublattices of L.7
0.2. We now consider a more elegant and nicer type of systems of slices.
A hierarchy of slices in a structure L of logics is a system of slices equipped with an antimonotone map to ordinals. This means that its slices LM are parametrized by ordinals ^ < (for some ordinal > 0) so that
V^1, ^2 < VL1 e LmVL2 e L№ (L1 c L2 ^ > ^2).
7However we do not forsee that in interesting systems their slices will be nontrivial (i.e., non-singleton) antichains etc.
In other words, this means that: (L»)-ÇL<» for every ^< or equivalently, (L»)- ÇL>» for every ^ <
where, naturally, L<» = U L» and L>» = U L» •
o<»'<» »<»'<»*
Remark 1. In general, both these inclusions may be proper. Definitely, it is not hard to construct appropriate (artificial) examples; we do not, however, wish to be distracted by pathological cases.
A hierarchy (LM : / < /*) of this kind is called a /*-hierarchy of slices; this is a sequence of slices of length /* + 1. Recall that the slices LM, for / < /*, generate a partition of L (i.e., all the slices are non-empty, disjoint, and their union is L).
Notice that similar hierarchies of the form (LM : /</*) (of length /*) for limit ordinals /* are definitely impossible. Indeed, let LM eLM for all /</*, then the intersection f] LM does not belong to (J LM = L.
0.3. We now can consider the notion of splitting (in a structure of superintuitionistic logics). To begin with, notice that there exists a single 0-hierarchy of slices; this trivial hierarchy consists of one slice Lo = L. Next, we say that a pre-splitting is just a 1-hierarchy of slices. So this is a 2-partition (L+, L-) of L in which L+ is upward closed and L- is downward closed. Thus, every upward closed subset L+ cL gives rise to a unique pre-splitting (L+,L\L+) and, dually, every downward closed subset L- c L gives rise to a unique pre-splitting (L\L-, L-).
Obviously, the lower part (portion, component) L- of a pre-splitting has the least logic (namely, intuitionistic logic, the least in the structure L), and the upper part L+ has the greatest logic (the inconsistent logic, the greatest in L). Now, a pre-splitting (L+,L-) is called a splitting if L- contains the greatest logic (denoted by L-) and L+ contains the least logic (denoted by L+). In other words, a pre-splitting is a splitting iff both its parts are segments in L; i.e.,
L- = [(Q)H, L-] = {L | L C L-}, L+ = [L+, L±] = {L | L+ CL}.
Thus a pre-splitting is just a partition of L into two convex sets, while a splitting is a partition of L into two segments.
One can easily see that a pair of logics (L+, L-) generates a splitting in L (or is a splitting pair, cf. e.g. [Galatos et al., 2007, Chapter 10, Section 10.1]) iff it satisfies any of the following equivalent conditions:
(i) VLeL [ (L+ CL) + (LCL-) ]
(where + means the exclusive 'or', i.e., the logical sum modulo 2);
(i)' VLeL ( L+ CL ^ LCL- ); (i)'' VLeL ( LCL- ^ L+CL );
(ii) (L+C L-) and VL eL [(L+ C L) V(L C L-)].
V. Jankov [Jankov, 1963; 1969] proved that every Jankov formula generates a splitting in the structure of propositional superintuitionistic logics. In effect, he established that a logic axiomatized by a Jankov formula, together with the logic of the finite Heyting algebra (or the associated finite rooted pro-positional Kripke frame) to which that formula corresponds, form a splitting pair. Thereby, Jankov anticipated, for the structure of propositional superin-tuitionistic logics, the important construction of splitting, which was explicitly introduced about a decade later, in the 1970s, in a general lattice-theoretic setting, by McKenzie (cf. [Galatos et al., 2007, the beginning of Chapter 10]).
Moreover, it is known8 that all splittings in the structure of propositional superintuitionistic logics are generated, in the described way, by Jankov formulas. In fact, this result immediately follows from a theorem of Jankov's, 1960s (namely, from the Theorem on conjunctively indecomposable formulas, see [Jankov, 1969, the beginning of § 3]).9
Indeed, let (L+, L-) be a splitting pair. Clearly, the logic L+ cannot be
presented as a sum of its proper sublogics (since ^ Lg C L- if all Lg c L+).
6>e©
On the other hand, every logic is the sum of its finitely axiomatizable sublogics. Hence we conclude that the logic L+ is finitely axiomatizable; moreover, it is axiomatizable by a conjunctively indecomposable formula, i.e., by a Jankov formula (due to the mentioned theorem of Jankov's). ■
However, Jankov in the 1960s did not state such a result; perhaps, he never gave a thought to this important (from the modern standpoint) corollary of his theorem.10
Now, we say a few words on splittings in the structure of predicate (su-perintuitionistic) logics. Again, for any splitting pair (L+, L-) the logic L+ is finitely axiomatizable (since it cannot be presented as a sum of its proper sublogics), i.e., L+ = [QH+A+] for a formula A+ (this formula is unique up to the intuitionistic deductive equivalence, i.e., the mutual derivability). We call such formula A+ the characteristic formula or the Jankov-style formula for a splitting (or for its splitting pair). Say that a splitting pair is determined by a frame F- if L- is the logic of F-. In this case we may call the corresponding axiom A+ (for L+) the Jankov-style formula for the frame F-.
8E.g., this follows from a rather general algebraic McKenzie's theorem, 1970s.
9This rather straightforward argument has, probably, remained, for over half a century, unnoticed by specialists. Hence, we sketch it here.
10Perhaps, Jankov could not even conceive of such a question because at the time he was affiliated with the Moscow school of constructivism, hence such a non-constructive object as the entire structure of logics did not attract his attention.
The mentioned description of splittings for the propositional case shows that every splitting is determined by a finite rooted Kripke frame (i.e., a poset) there. On the other hand, we cannot guarantee that every splitting in the structure of predicate logics is determined by a frame of any kind. Although splittings considered in this work (as well as splittings which we envisage for our possible future investigations), are (or could be) determined by predicate Kripke frames, however we cannot see too far into the future; perhaps, generalized Kripke-type semantics, known for predicate logics, could be used by future researchers to identify new splittings or somewhat else (why not?).
By the way, in a future continuation of the presented work (On a 'perpendicular' system of slices in the structure of superintuitionistic predicate logics) we hope to show that all propositional Jankov formulas are Jankov-style formulas for predicate splittings determined by suitable predicate Kripke frames (over posets, to which Jankov formulas correspond).11
0.4. Now we consider applications of splittings to hierarchies of slices.
Let (LM : y<y*) be a y*-hierarchy of slices. For every y<y* it generates a pre-splitting (L<M, L>(M+i)) We call a hierarchy y-split if this pre-splitting is a splitting. This means that the downward closed subset L>(^+1) CL contains the greatest logic L^+1 and the upward closed subset L<M contains the least logic L^own. Then, obviously, is the greatest element in LM+1, while is the least element in LM. However, the existence of the greatest logic in LM+1 and the least logic in LM is not sufficient for a y-split hierarchy, since these logics in general need not be the greatest in L>(^+1) and, respectively, the least in L<M; cf. Remark 1 to the definition of hierarchies of slices in subsection 0.2.
Remark 2. A y*-hierarchy of slices gives rise to another kind of pre-split-tings. Namely, for 0 <y < y*, we obtain a pre-splitting (L<M, L>M) (where L<^ = U LM'). However, these pre-splittings seem useless, as they do not
lead to new splittings. Indeed, such a pre-splitting for y+1 is just the pre-splitting (L<M,L>(^+1)). And if y is a limit ordinal, then the pre-splitting (L<M, L>M) cannot be a splitting since L<M does not have the least element.
11Some authors call logics L+ (from splitting pairs) splitting logics, while other authors use this name for logics L_. The author is inclined to the first option (position), since otherwise finitely axiomatizable (and hence convenient) logics L+ turn out to play a secondary, subservient role. In any case, to avoid misunderstanding, we do not use this term, opting instead for a more invariant notion of a splitting pair. Sometimes, logics axiomatized by arbitrary (perhaps infinite) families of Jankov formulas (or, in our case, Jankov-style formulas) are called join-splitting logics or union splitting logics. Again, until we have enough experience working with splittings in the structure of predicate logics, we prefer to avoid such terminology.
Let (Ld : / < /*) be a /*-hierarchy of slices, and let 0 < /0 < /*; put /+ = /0 if /0 is a limit ordinal and /+ = /0 + 1 otherwise. The hierarchy (Ld : /</*) is called < /0-split if it is /-split for all 0 < / < /0 (here we exclude the trivial case /0 = 0; otherwise all hierarchies would be < 0-split).
One can easily see that a hierarchy is < /0-split iff it satisfies the following four conditions:
(1) every slice Ld for 0 < /</0 contains the greatest logic L'p and the least logic L^own, i.e., Ld = L°wn, Ljf] (clearly, here LUP is the inconsistent logic; it is the greatest in L and is in the 0-th slice L0 as well);
(2) if /0 is not a limit ordinal, then the downward closed subset L>d0 contains the greatest logic L'p, i.e., L>№ = [(Q)H, L'p] = {L | L CL'p} (clearly, the logic L'p is the greatest in the slice Ld0 as well);
(3) both V,/" </0 (/' </'' ^ Ld°Wn c L'0wn)
and V/, < /+ (/' < ^ L'P c L'P)
(in other words, the lower logics (L'own : /</0) form a decreasing /0-chain and the upper logics (L'p : /</+) form a decreasing /+-chain);
(4) if /0 is a limit ordinal, then V/</0 VL eL>d0 (L c L'p)
(actually, due to (3), it is sufficient to check this condition only for arbitrarily large /</0).
Note that if, in a /*-hierarchy, every slice Ld for /+ < / < /* has a top logic L'p, then we can replace
the condition in (2) with: V/>/0 [L^p c L'p],
and the condition in (4) with: V/ > /0 V/' </0[L'p c L'p].
A hierarchy of slices is called split (or totally split) if it is </*-split, i.e., if it is /-split for every /</* (i.e., for all its slices). Such a hierarchy is an assembled combination of splittings (coordinated in a suitable way).
0.5. Note that one can transform, in an obvious way, every <w-split /*-hierarchy of slices (with /* >w) into a <w-split w-hierarchy. Viz., one can replace all slices Ld with / > w by their union L>w. The resulting hierarchy remains <w-split, and so, formally speaking, it becomes totally split. Nevertheless, it should be intuitively clear that a new hierarchy becomes worse than the original one: every gluing of succesive slices makes a hierarchy less expressive. Thus, we are interested in obtaining subtler hierarchies of slices, even if some formal properties (e.g., those related to splittings) are affected.
Now we briefly sketch an attempt to explicate these somewhat vague informal considerations.
Say that an (arbitrary) y*-hierarchy is constructed along (or under) a sequence (L^p : y<yo) (for some y0 <y*) if every L^p is the greatest logic in the slice LM. As explained, every <y0-split hierarchy is constructed along its decreasing y+-sequence (Ljf : y<y+). In particular, a totally split hierarchy is constructed along the sequence (L^p : y< (y*)+); note that this sequence includes all ordinals y< y* for the case of a non-limit y* and can omit (in general) the last ordinal y* in the limit case. However, if this last slice LM* has the top logic, then our y*-hierarchy is again constructed along the 'total' sequence (L7 : y <y*).
A decreasing sequence (LM : y<y0) is weakly (or locally) uninterrupted in the lattice L if all the intervals (LM+1, LM) between adjacent logics in this sequence are empty (for all (y+1) <y0) and LM = f] L^/ for all limit ordinals y<y0. Also, a decreasing sequence (LM : y <y0) (with the least element LM0) is strongly (or globally) uninterrupted in L if {L | LM0 C L} = |LM | y < y0}. Obviously, every strongly uninterrupted sequence is weakly uninterrupted. Moreover, every strongly uninterrupted sequence obviosly begins with L0 = Lx (the inconsistent logic, i.e., the top element of L). By the way, one can notice that a weakly uninterrupted sequnce begins with L0 = Lx iff the condition LM = n LM' is extended to y = 0, besides (infinite) limit ordinals. We tend
to believe this requirement to be too restrictive — a weakly uninterrupted sequence could begin with an arbitrary L0.
A hierarchy of slices is uninterrupted (in the weak or the strong sense) up to a slice LM0 (or up to the corresponding ordinal y0) if it is constructed along an uninterrupted sequence (L^p : y < y0) of the greatest logics in the corresponding slices.
It seems preferrable to construct uninterrupted (and split), as far as possible, hierarchies of slices, because they could clarify the formation of the structure of (superintuitinistic) logics. And clearly, the requirement of uninterruptedness prevents attempts to glue together adjacent slices in such a hierarchy.
Now, the reader familiar (even slightly) with Hosoi's hierarchy of finite slices for propositional superintuitionistic logics can recognize a split w-hierarchy constructed along a strongly uninterrupted (w + 1)-sequence of logics — namely, Godel logics (i.e., the logics of finite chains) together with their intersection, Dummett's logic.
Remark 3 (Warning). The inexperienced reader could be tempted to try to apply the uninterruptedness condition to the sequence of lower logics
(L'own : /</0) in slices of a split hierarchy. However, this idea seems useless. E.g., for the Hosoi's hierarchy (and for similar ones), the sequence of lower logics is not even weakly uninterrupted; namely, for every /<w there exist many logics between Ld+1n and L'own. Perhaps, this obstacle is crucial; deep regions of the structure of superintuitionistic logics are too saturated and do not allow for too long sequences of empty intervals between adjacent logics.
Now let us briefly conclude the considerations and observations from subsections 0.2 to 0.5.
We see that pre-splittings and splittings are particular cases of hierarchies of slices. They are very small: only one step beyond the degenerate case /* =0, so at first sight, they appear simple, even simplistic, plain, scanty, underdone. Nevertheless, when viewed adequately, they can be quite useful and productive for the study of the structure of (superintuitionistic) logics. These primitive, basic constructions are rather frequent; this is known for the propositional case, and we can only hope that the situation in the predicate case is not much worse. (Pre-)splittings can be regarded as building blocks for hierarchies of slices — moreover, for rather extensively split hierarchies of slices (perhaps, almost as good as the beautiful and elegant hierarchy of slices of Hosoi's for the propositional case).
0.6. In the presented work, we make probably the first attempt to transfer Hosoi's hierarchy of slices (and, less so, Jankov's approach) from the structure of propositional superintuitionistic logics to a richer and more complicated structure of predicate superintuitionistic logics.
Namely, we describe a hierarchy of slices constructed along an w-sequence of logics of classical frames (i.e., predicate Kripke frames with a single world) with finite domains. More exactly, we use the decreasing (w + 1)-sequence of logics
QC0 D QC D ... D QCm D QCm+1 D ... D QCW , (QC)
where QCm (for m<w) is the classical logic of an m-element domain (in particular, QC0 is the inconsistent predicate logic, i.e., the logic of empty domain), and their intersection QCW = P| QCm is the classical logic of all finite do-
m<ui
mains, with classical predicate logic QC (i.e., the classical logic of any infinite domain) ending this sequence.
To obtain this hierarchy, we introduce predicate counterparts of Jankov formulas for finite classical frames and establish that these formulas generate
the corresponding splittings in the predicate structure; these coordinated splittings constitute a <w-split (w + 1)-hierarchy (L0, L1, L2,..., Lw, Lw+1) (a sequence, of length w+2, of slices). Here Lm = {LgL | (L CQCm, LC QCm+1)} for 0<m< w (where QCW+1 means QC).12
All slices apart from Lw are segments; i.e., all finite slices and the last, (w + 1)-th slice Lw+1 = [QH, QC] (the family of intermediate predicate logics, i.e., the segment between intuitionistic predicate logic QH and classical predicate logic QC). We conjecture that this hierarchy is not w-split (i.e., not totally split) and, moreover, it cannot be extended to an w-split hierarchy.
Our version of counterparts of Jankov formulas allows us to prove that the sequence (QC) of upper logics in the slices of this hierarchy is strongly uninterrupted; this means that the family of all proper extensions of the logic QCW is just the w-sequence {QCm | m<w} of all classical logics of finite domains.
This also yields, for every finite m, an exact description of formulas that can axiomatize the classical logic QCm over classical logic QC — these are just arbitrary formulas from QCm\QCm+1, i.e., those formulas that are classically valid on the m-element domain and non-valid on the (m + 1)-element domain.
0.7. We now briefly describe a plan of the whole our investigation that consists of three parts making up a single larger work with a single enumeration of sections, statements etc., with numerous cross-references, and with a single list of bibliographic references.
Part I (Sections 1-6) contains mainly non-semantical considerations; in the part we use only classical predicate models. Section 1 contains necessary lattice-theoretic (and some other) preliminaries, as well as agreements on terminology and notation. Section 2 introduces the hierarchy of slices studied in the work.
In Section 3 the counterparts of Jankov formulas for finite classical domains are presented and their main properties are established. Most of these properties are parallel the well-known properties of propositional Jankov formulas; those properties are successfully transferred to the predicate case. The corollaries for the structure of superclassical logics (i.e., extensions of QC), mentioned at the end of the previous subsection 0.6, are also obtained here. Finally, the constructed Jankov-style formulas allow us to describe (for every finite m) the family of logics whose negative fragment (i.e., the set of provable formulas of the form -A) coincides with the negative fragment of QCm. Recall that an analogous description of predicate logics with the classical negative fragment (i.e., those whose negative fragment is the same as that of QC) was obtained by D. Gabbay in the 1970s [Gabbay, 1972]. This is an application of our predicate Jankov-style formulas that has no analogue in propositional logic; recall
12In the announced work, for clarity, the slices of this hierarchy are occasionally called FD-slices (i.e., finite domains based slices).
that by Glivenko's theorem, all consistent (i.e., intermediate) superintuitionistic propositional logics have the classical negative fragment.
Section 4 contains proofs of two crucial theorems on Jankov-style formulas postponed from Section 3 due to their technical character. They require a more elaborate consideration and special notation used only in these proofs. The first of them is the predicate analogue of the crucial Jankov's result on his formulas. The second one is a semantical completeness theorem for the logics QCm; they are axiomatized by our Jankov-style formulas over classical logic QC.
Section 5 contains a predicate version of Hosoi's result mentioned in Introduction: every finite slice Lm is isomorphically embeddable in all the further slices Lm' (for m<m'<(w + 1)); again, onto corresponding segments of those slices.
In Section 6 we transfer the result on isomorphic embeddings to the infinite w-slice Lw. This consideration makes it necessary to introduce a subtler classification within this slice. Together with the hierarchy of finite slices, this gives us a non-hierarchical (i.e., not parametrized by ordinals) system of slices in the structure of predicate logics. The main obstacle in the way of the proof is the lack of the least logics in the new slices (these are called the 'subslices' of the w-slice). Nevertheless, this obstacle is easily overcome, so the embedding result is extended to these subslices in a natural way.
Part II (Sections 7-9) describes and studies Kripke complete logics from finite and infinite slices of our hierarchy. Section 7contains semantical preliminaries, an extensive overview of Kripke semantics for superintuitionistic predicate logics. Sections 8 and 9 are devoted to logics of Kripke frames with expanding and constant domains (respectively) in different slices of the hierarchy.
Part III (Sections 10-12) concludes the whole work. In Section 11 we apply the semantics of Kripke sheaves (an introduction to this semantics is given in Section 10) to the proofs of Kripke incompleteness of various logics from finite slices. The final Section 12 contains the proofs of main results on decidability, axiomatizability, and non-axiomatizability for some notable Kripke complete logics from finite and infinite slices. These theorems are stated in Sections 8 and 9 (from Part II), but their proofs are postponed because of special technical machinery required (as in Section 4, Part I).
We hope that this work could become a starting point for further investigations. First, some unsolved questions are formulated, and some suggestions and prospects for future research are mentioned throughout the text. Second, other, new systems of slices (in the structure of superintuitionistic predicate logics) — both hierarchical and non-hierarchical — may be found and explored. The experience obtained in Section 6 gives some hope for that. We are not yet
ready to discuss these options, but we hope that somebody finds in our work
inspiration for further studies in the field.
Acknowledgements. The author would like to express sincere gratitude to Dmitry
Shkatov for editing the English text of the paper.
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