A Galois connection between intuitionistic & classical logics. I: Syntax

A Galois connection between classical and intuitionistic logics. I: Syntax

Abstract.

In a 1985 commentary to his collected works [?], Kolmogorov remarked that his 1932 paper [?] ‘‘was written in hope that with time, the logic of solution of problems [i.e., intuitionistic logic] will become a permanent part of a [standard] course of logic. A unified logical apparatus was intended to be created, which would deal with objects of two types — propositions and problems.’’ We construct such a formal system QHC, which is a conservative extension of both the intuitionistic predicate calculus QH and the classical predicate calculus QC.

The only new connectives and of QHC induce a Galois connection between the Lindenbaum posets (i.e. the underlying posets of the Lindenbaum algebras) of QH and QC. Kolmogorov’s double negation translation of propositions into problems extends to a retraction of QHC onto QH; whereas Gödel’s provability translation of problems into modal propositions extends to a retraction of QHC onto its QC+() fragment, identified with the modal logic QS4. The QH+() fragment is an intuitionistic modal logic — whose modality is a strict lax modality in the sense of Aczel — and thus resembles the squash/bracket operation in intuitionistic type theories.

The axioms of QHC attempt to give a fuller formalization (with respect to the axioms of intuitionistic logic) to the two best known contentual interpretations of intiuitionistic logic: Kolmogorov’s problem interpretation (incorporating standard refinements by Heyting and Kreisel) and the proof interpretation by Orlov and Heyting (as clarified by Gödel). While these two interpretations are often conflated, from the viewpoint of the axioms of QHC neither of them reduces to the other one, although they do overlap.

Supported by The Fund for Math and Russian Foundation for Basic Research Grant No. 15-01-06302
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1. Introduction

1.1. Problems versus propositions

The present series of papers (the sequels being [?] and [?]) belongs firmly to the field of Logic, but is motivated primarily by considerations of mathematical practice rather than any internal developments in the field of Logic. Therefore it is addressed not only to logicians, but to other mathematicians as well. The reader who is not familiar with any of the terms used can consult the treatise [?] as need arises; one of its main goals is precisely to make the present series accessible to a general mathematical audience.

This paper introduces a logical apparatus that enables one to study in a formal setting basic interdependencies between what can be called (cf. §6.1) two modes of knowledge: knowledge-that (or knowledge of truths) and knowledge-how (or knowledge of methods). In mathematical practice, these have been traditionally represented by propositions (i.e., assertions, such as theorems and conjectures) and problems (such as geometric construction problems and initial value problems). The English word ‘‘problem’’ is, in fact, somewhat imprecise; we will use it in the narrow sense of a request (or desire) to find a construction meeting specified criteria on output and permitted means (as in ‘‘chess problem’’). This meaning is less ambiguously captured by the German Aufgabe (as opposed to the German Problem) and the Russian задача (as opposed to проблема). The closest English word is task (other words with related meanings include assignment, exercise, challenge, aim, mission), but as it is not normally used in mathematical contexts, we prefer to speak of problems.

To appreciate the difference between problems and propositions, let us note firstly that the problem requesting to find a proof of a proposition is closely related to both (i) the proposition asserting that is true; and (ii) the proposition asserting that is provable. These are not the same, of course, whenever ‘‘proofs’’ are taken to be in some formal theory and ‘‘truth’’ is taken according to some two-valued model of , with respect to which is not complete.2 There are other reasons why ‘‘true’’ should not be equated with ‘‘provable’’; for instance, they differ also in the modal logic S4, where it is a simple consequence of the axioms that consistency is provable.3 Conversely, the proposition asserting that two groups and are isomorphic is closely related to both (i) the problem requesting to prove that and are isomorphic; and (ii) the problem requesting to construct an isomorphism between and . These are generally not the same because one proof that an isomorphism exists might represent several distinct isomorphisms or no specific isomorphism.

The logical distinction between problems and theorems (as they appear, in particular, in Euclid’s Elements) has been articulated at length by a number of ancient Greek geometers in response to others who disputed it. A detailed review of what the ancients had to say on this matter is included in the third part of this paper [?]. In modern times, the distinction was emphasized by Kolmogorov [?]:

‘‘On a par with theoretical logic, which systematizes schemes of proofs of theoretical truths, one can systematize schemes of solutions of problems — for example, of geometric construction problems. For instance, similarly to the principle of syllogism we have the following principle here: If we can reduce solving to solving , and solving to solving , then we can also reduce solving to solving . Upon introducing appropriate notation, one can specify the rules of a formal calculus that yield a symbolic construction of a system of such problem solving schemes. Thus, in addition to theoretical logic, a certain new calculus of problems arises. In this setting there is no need for any special, e.g. intuitionistic, epistemic presuppositions. The following striking fact holds: The calculus of problems coincides in form with Brouwer’s intuitionistic logic, as recently formalized by Mr. Heyting. In the second section we undertake a critical analysis of intuitionistic logic, accepting general intuitionistic presuppositions; and observe that intuitionistic logic should be replaced with the calculus of problems, since its objects are in reality not theoretical propositions but rather problems.’’

A key difference between problems and propositions is that the notion of truth for propositions has no direct analogue for problems, so that problems cannot be asserted. For instance, let be the problem Divide any given angle into three equal parts with compass and (unmarked) ruler. Then reads, Divide any given angle into three equal parts with compass and ruler or prove that it is impossible to do so (cf. [?]LABEL:int:about-bhk and 3.10 below). This is not a trivial problem; indeed, its solution took a couple of millennia. Even now that a solution is well-known, the problem still makes perfect sense: the law of excluded middle would not help a student to solve this problem on an exam (in Galois theory). By citing the law of excluded middle she could solve another problem: Prove that either has a solution or has no solutions; in symbols,

where denotes the proposition There exists a solution of the problem , and denotes the problem Prove the proposition .4 In fact, this problem is strictly easier than

(in words, Prove or disprove that has a solution), which requires a justified explicit choice. But the latter problem, which can be written equivalently as , is still strictly easier than the original problem, , for it is generally easier to prove that some problem has a solution than to actually solve it.

1.2. A joint logic

The present paper is devoted to the study of the logical operators and in a formal setting. Like in the previous example, is meant to refer to non-constructive proofs, whereas is understood to signify explicit existence. We extract axioms and rules governing the use of and essentially from two sources:

  • The problem interpretation of intuitionistic logic. This is essentially Kolmogorov’s 1932 explanation of the intuitionistic connectives [?], which had some parallels with the independent writings of Heyting (1931), and was slightly refined by Heyting (1934). A disguised form of this explanation, often incorporating a further refinement by Kreisel, has come to be known as the BHK interpretation of intuitionistic logic (see [?]LABEL:int:intro for a detailed review and discussion). We include Kreisel’s addendum in the following form, also found in the ancient commentary by Proclus on Euclid’s Elements (see [?]): A solution of a problem must include not only a construction, but also the verification, i.e. a proof that the construction meets the requirements specified in the problem (see [?]LABEL:int:about-bhk for a discussion of this principle).

  • The proof interpretation of intuitionistic logic. This is essentially the meaning explanation of intuitionistic logic given independently by Orlov (1928) and Heyting (1930, 31) (see a detailed review in §6.2.1), which was partially formalized in Gödel’s 1933 translation of problems into modal propositions (see [?]LABEL:int:provability), and further clarified by Gödel’s proof-relevant analogue of S4 (see §2.4.2).5

It then comes as a little surprise that the resulting axioms and rules harbor a great deal of unintended symmetries, and are also compatible with Kolmogorov’s double negation translation of propositions into problems (reviewed briefly in [?]LABEL:int:negneg). (For a different connection between Kolmogorov’s and Gödel’s translations see [?].) What is most surprising, however, is that nobody seems to have studied the operators and before, apart from hints of an abandoned project aimed at a similar study, found in Kolmogorov’s own writings. In his 1931 letter to Heyting [?], Kolmogorov wrote:

Each ‘proposition’ in your framework belongs, in my view, to one of two sorts: expresses hope that in prescribed circumstances, a certain experiment will always produce a specified result. (For example, that an attempt to represent an even number as a sum of two primes will succeed upon exhausting all pairs , , .6) Of course, every ‘‘experiment’’ must be realizable by a finite number of deterministic operations. expresses intention to find a certain construction. […] I prefer to keep the name proposition (Aussage) only for propositions of type () and to call ‘‘propositions’’ of type () simply problems (Aufgaben). Associated to a proposition are the problems (to derive contradiction from ) and (to prove ).

Apart from this fragment and the quote in the abstract, there are only a few further hints at how Kolmogorov envisaged the connection between problems and propositions. Several problems consisting in proving a proposition are also mentioned in Kolmogorov’s paper [?]. There is also a bit more in Kolmogorov’s letters to Heyting, which will be thoroughly reviewed in §6.2.2. There we note, in particular, that while Kolmogorov’s propositions of type () seem to stand precisely for the objects of intuitionistic logic, his propositions of type () could not be intended to exhaust all objects of classical logic; in fact, it appears that they can be identified with the ‘‘stable propositions’’ of §4. Another apparent divergence between Kolmogorov’s remarks and our approach is noted in 3.10 and discussed more thoroughly in [?]LABEL:int:about-bhk.

The joint logic of problems and propositions that is constructed in the present paper is presumably very unnatural in the standard constructivist paradigm (of Brouwer and Heyting) that views intuitionistic logic as an alternative to classical logic that criminalizes some of its principles. We work in the other paradigm (of Kolmogorov), which views intuitionistic logic as an extension package that upgrades classical logic without removing it. For us, the main purpose of this upgrade is solution-relevance (=‘‘proof-relevance’’), or ‘‘categorification’’. Thus from the viewpoint of the BHK semantics, topological (Tarski) models are in fact models of a ‘‘squashed’’ copy of intuitionistic logic — whose existence is only revealed with the aid of the new connectives and (see §5.2 below); whereas ‘‘true’’ models of the genuine intuitionistic logic are the (solution-relevant) sheaf-valued models of [?] (a special case of ‘‘categorical models’’ — not to be confused with the usual ‘‘sheaf models’’ of intuitionistic logic). Models of the joint logic of problems and propositions will be discussed in [?].

1.3. Double negation translation

Speaking of ‘‘intuitionistic logic as an extension package that upgrades classical logic without removing it’’, we run into the natural question: ‘‘Wait, but what about the double negation translation?’’ Indeed, there is a version of the double negation translation that redefines classical connectives in terms of intuitionistic ones and introduces no other modifications to formulas (see [?]LABEL:int:negneg). However, this syntactic translation fails to reflect actual mathematical practice. There are several levels at which this failure occurs:

(i) In the words of Kreisel [?], ‘’there is a good reason why mathematicians neglect’’ the double negation translation, in the form of ‘’replacing by and by ’’, ‘’namely, this: For the sense in which mathematicians actually understand the propositions of mathematical practice, … the difference between and on the one hand and their translations on the other … is not significant’’. ‘’Put differently, they do not understand the intuitionistic meaning of and which makes the [double negation] translation significant.’’

This is not merely a matter of mathematicians’ conventions, psychology or ignorance. For mathematicians to be serious about the intuitionistic meaning of propositions, in the tradition of Brouwer and Heyting, they would have to sacrifice their understanding of mathematical objects as ideal entities existing independently of one’s knowledge about them. But most of them certainly do not want to be ‘‘expelled from the paradise that Cantor has created’’, and for a good reason: the customary mental aid of Platonism does simplify their job immensely.

(ii) Kolmogorov’s problem interpretation of intuitionistic logic entirely avoids the issue of sacrificing platonist thinking. But then the double negation translation makes no sense, because, when understood in these terms, it conflates problems with propositions; and when corrected so as to respect their distinction, it is no longer a translation into plain intuitionistic logic. This ‘‘corrected’’ double negation translation (see §5.4) is, actually, quite meaningful from the viewpoint of mathematical practice; for instance, , ‘’there exists an such that ’’ is translated as , ‘’it is impossible to derive a contradiction from a construction of along with a proof of ’’. The ‘‘corrected’’ double negation translation is essentially equivalent to Fitting’s translation of classical logic into the modal logic QS4.

(iii) Even though the ‘‘corrected’’ double negation translation is no longer a translation into plain intuitionistic logic, one might still ask if its effect is significant from the viewpoint of mathematical practice. The assertion that its effect is trivial is equivalent (see [?]*LABEL:g2:nabla-K) to the so-called K-principle, , an independent principle of the joint logic of problems and propositions. But the effect of the K-principle is drastic: it immediately rules out independent statements (see [?]LABEL:int:BHK-to and [?]LABEL:g2:H-K).

1.4. Related work

Modern literature contains a number of attempts to blend classical and intuitionistic logics. On the one hand, there are the Linear Logic and the logics of Japaridze [?], [?], [?], [?] and Liang–Miller [?], [?], which all have something classical and something intuitionistic in them — albeit fused in far more elaborate ways than Kolmogorov could have possibly meant in his words: ‘‘A unified logical apparatus was intended to be created, which would deal with objects of two types — propositions and problems’’ [?].7

On the other hand, there is Artëmov’s Logic of Proofs LP, which he actually meant to address these very words of Kolmogorov [?]*p. 2. It is clear, however, from Kolmogorov’s letter quoted above, that he did envisage the ‘‘two types’’ to be on equal footing, which is not the case in Artëmov’s LP. Also, for instance, the following expression found in Kolmogorov’s paper [?]: ‘’in the case where the problem consists in proving a proposition’’ does not seem to be compatible with Artëmov’s approach. In a future paper the author plans to discuss a proof-relevant extension of the joint logic of problems and propositions which includes a variation of Artëmov’s LP.

What is more obviously related to Kolmogorov’s research program is the ‘’propositions-as-some-types’’ paradigm, and indeed our composite operator on problems is very similar to the squash/bracket operator in intuitionistic type theories (see §3 and 3.18). There are also similarities between our approach and some ideas behind the Calculus of Constructions [?] (see also [?] and [?]).

A direct type-theoretic analogue of our due to Aczel and Gambino [?]*§1.3 is dissimilar to in that it satisfies a reversible analogue of our schema () (see §2.4). In contrast, the reversibility of our () would amount to allowing the BHK interpretation to represent arbitrary, and not just constructive functions (see [?]LABEL:int:confusion). But there is nothing surprising here, since Aczel and Gambino do not assume the principle of excluded middle on either the two sides.

A type-theoretic analogue of our due to Coquand [?]*§1 satisfies an analogue of our schema () (see §3.6), which Coquand argues to express ‘‘Heyting’s semantics of the universal quantification’’. This time we see a full agreement on the syntactic level; but it is remarkable that our formalization of the BHK clause for the universal quantification is not (), which is reversible just like its Coquand’s version, but () (see §2.4), which is irreversible for the same reasons as ().

Disclaimer

Most translations quoted in the present series of papers have been edited by the present author in order to improve syntactic and semantic fidelity. When emphasis is present in quoted text, it is always original.

Acknowledgements

I would like to thank L. Beklemishev, M. Bezem, G. Dowek and D. Shamkanov for valuable discussions and useful comments. A part of this work was carried out while enjoying the hospitality and stimulating atmosphere of the Institute for Advanced Study and its Univalent Foundations program.

2. QHC calculus

In the present series of papers we work in first-order logic, but with some deviations from standard terminology, notation and conventions. Namely, our basic syntactic setup is the meta-logic of [?]LABEL:int:formal, which is a slightly simplified and ‘‘mathematicized’’ version of the meta-logic used in the Isabelle proof-checker. The simplification is mostly concerned with omission of features that are not needed for dealing with first-order logics (without equality). To be precise, in the present series of papers we use the straightforward extension of the setup in [?]LABEL:int:formal to the case of many-sorted first-order logics.

The following includes a quick summary of [?]LABEL:int:formal which should suffice for the reader who is familiar with some conventional treatments of first-order logic as well as simply-typed -calculus and natural deduction.

2.1. Simply-typed -calculus

The language in which our logic and its meta-logic are formulated is the simply-typed -calculus with (binary) products, with [?] and the present series of papers taking the following deviations from standard terminology, notation and conventions.

  • The word ‘‘term’’ is used in the sense of first order logic, and consequently we speak of -expressions rather than -terms. The word ‘‘arity’’ is used is the traditional sense (of logic and mathematics), and consequently we speak of types (rather than arities) of -expressions. The word ‘‘closed’’ (as e.g. in ‘‘closed formula’’) is used in the sense of first-order logic, so we refer to -expressions that are closed in the sense of -calculus as -closed ones.

  • Abstraction is written in the style of mathematics, as , and not in the style of logic and computer science, . Function application is normally written as and only in some cases abbreviated as . The function type is denoted ; no associativity conventions for and are assumed.

  • We omit brackets in iterated products using standard isomorphisms, and use tuples , also written , which are defined recursively in terms of pairs. Projection on the th factor of a product is denoted . We also use multivariable abstraction , which is defined recursively in terms of abstraction and tuples (not just up to -equivalence; see [?]LABEL:int:simultaneous).

  • Substitution is denoted and is undefined whenever some variable is captured. The same goes for the simultaneous substitution . A -expression of the form may -reduce beyond ; if such a -reduction involves no -conversions, and its result is in -normal form, then this resulting -expression is denoted , and the tuple is called free for in S (see [?]LABEL:int:free_substitution).

  • The variables of a type are denoted . We generally use lowercase letters to write metavariables for variables and constants, and uppercase letters to write metavariables for arbitrary terms.

2.2. Language of QHC

QHC is a two-sorted first-order logic without equality. To describe its language, we need three basic types:

  • , the type of terms;

  • , the type of i-formulas (‘‘i’’ stands for ‘‘intuitionistic’’);

  • , the type of c-formulas (‘‘c’’ stands for ‘‘classical’’).

The language of QHC consists of the following sets of typed -expressions (variables and constants only), where ranges over :

  • the set of variables of type , called individual variables;

  • the set of variables of type , called -ary predicate variables;

  • the set of variables of type , called -ary problem variables.

Each of the sets (1), (2), (3) is a countably infinite set. Nullary predicate variables are also called propositional variables. For reasons of readability we will also use the alternative spelling for the first 26 individual variables , reserving an upright sans-serif font for this purpose. Similarly, we use the abbreviations for the first 26 predicate variables of each arity and for the first 24 problem variables of each arity, reserving a fancy (Euler) upright serif font for this purpose.

In using predicate and problem variables we follow the tradition of classic texts in first-order logic such as those by Hilbert–Ackermann, Hilbert–Bernays, Church and P. S. Novikov, who did include predicate variables in addition to predicate constants. Modern treatments of first-order logic usually do not include predicate variables in the language, and are content with predicate constants (even though they include propositional variables in the language of propositional logic). In fact, it is clear that the language of a logic in reality contains only predicate variables, whereas predicate constants are chosen differently for each theory over the logic, and so actually belong to the language of a theory and not to the language of the logic.

The sets (1)–(3) are common to any two-sorted first-order logic. Specific to QHC are the following constants. Connectives:

  • truth and falsity ;

  • classical negation ;

  • classical binary connectives ;

  • triviality and absurdity ;

  • intuitionistic negation ;

  • intuitionistic binary connectives ,

quantifiers:

  • classical quantifiers ;

  • intuitionistic quantifiers ,

and conversion operators:

  • ;

  • .

Some of the connectives and quantifiers are ‘‘syntactic sugar’’, i.e. they should not really be on the above list as they are definable in terms of others. Namely, the intuitionistic , and are definable in terms of the intuitionistic and [1]; and the classical , , , and are definable in terms of the classical and , and the classical is definable in terms of the classical , and . However it is convenient to regard all these symbols (4)–(11), including the redundant ones, as ‘‘connectives’’ and ‘‘quantifiers’’.

It should be noted that we do not differentiate graphically between classical connectives/quantifiers and intuitionistic ones, since they can be distinguished by the type of the -expressions that they act upon ( or ) — except for the nullary connectives, which we do take care to differentiate (classical: ; intuitionistic: ). This is based on the observation that lowercase Greek letters, which we use to denote problem variables, are visually distinct from lowercase Roman letters, which we use to denote predicate variables. Note, however, the difference between (classical or intuitionistic implication) and (function type).

If is a quantifier, is a -expression of type or , and is an individual variable, then abbreviates the -expression . More generally, abbreviates . Due to this abbreviation, -abstraction is only implicit in formulas.

Remark 2.1.

In the preceding paragraph, is a metavariable that stands for an arbitrary unknown -expression of type or . Accordingly, the symbol ‘’’’ can be read in two ways: as the first Roman uppercase letter or as the first Greek uppercase letter. We will use uppercase letters that are unambiguously Greek (from the viewpoint of TeX) to write metavariables that stand unambiguously for a -expression of type , and those unambiguously Roman for -expressions of type .

This completes the description of the pure language of QHC. However, the language of a theory over QHC (such as the plane geometry of [?]) may additionally contain the following sets:

  • a finite set of constants of type , called -ary function symbols;

  • a finite set of constants of type , called -ary predicate constants;

  • a finite set of constants of type , called -ary problem constants.

It should be noted that nullary predicate and problem constants are the same kind of -expressions as nullary connectives (i.e., constants of types and ). It is nevertheless convenient to distinguish them, since the latter belong to the pure language of QHC but the former do not.

Terms of the language are defined inductively, as built out of individual variables using the function symbols. Thus not every -expression of type is a term (for example, no term involves -abstraction). An atomic c-formula of is a -expression of type obtained by applying either an -ary predicate constant or an -ary predicate variable to an -tuple of terms; an atomic i-formula is a -expression of type obtained by applying either an -ary problem constant or an -ary problem variable to an -tuple of terms. A formula of is a -expression built out of atomic c-formulas and i-formulas using the connectives, quantifiers and conversion operators.

A formula of type is called a c-formula and a formula of type is called an i-formula. (Clearly, every formula is either a c-formula or an i-formula.)

A purely classical formula is a -expression of type built out of atomic c-formulas using classical connectives and classical quantifiers only; a purely intuitionistic formula is a -expression of type built out of atomic i-formulas using intuitionistic connectives and intuitionistic quantifiers only.

A -expression of the form , where is a formula and are pairwise distinct individual variables, is called an -formula. It can also be called an -c-formula or an -i-formula if is a c-formula or an i-formula.

2.3. Meta-logic

Introduction

Any kind of literature on first-order logic constantly deals with meta-logical concepts and assertions, but usually only implicitly. Why would one want to make them explicit, and discuss a first-order logic in terms of a formal meta-logic? One reason is that a pedantic verbalist, who ignores the implicit, must perceive the hidden meta-logic as an ever-present conflation and ambiguity. Here are two examples.

Example 2.2.

The literature on first-order classical and intuitionistic logics is accustomed to speaking of ‘‘the syntactic consequence’’; but the syntactic consequence in the sense of e.g. the textbooks by Schoefield and Mendelson is inequivalent to the syntactic consequence in the sense of e.g. the textbooks by Church, Enderton, Kolmogorov–Dragalin, Troelstra and van Dalen. Moreover, Kleene and Avron have considered the two notions simultaneously, as well as the corresponding notions of semantic consequence, pointing out that both are commonly used in elementary mathematics.

Kleene’s textbook contains the following example: the arithmetical formula begs to be understood as an identity (valid for all natural numbers ), whereas the arithmetical formula begs to be understood as an equation (i.e., as a condition on ). There is no special syntax to reflect this obvious distinction in meaning. Yet it is not illusory, as it is reflected in use. For, as noted by Avron, when ‘’dealing with identities […] the substitution rule is available, and one may infer from the identity . In contrast, […] substituting for everywhere in an equation is an error’’ (see references in [?]LABEL:int:formal).

In fact, the difference between the two variants of syntactic consequence is due to the implicit presence of a first-order meta-quantifier in one of them.

Example 2.3.

In intuitionistic logic, the principle of excluded middle is derivable from the double negation principle (due to the derivability of the schema ). Nevertheless, the schema expressing the principle of excluded middle is not derivable from the schema expressing the double negation principle (since for the latter is derivable, and the former is not). Thus the widespread practice of expressing principles by schemata is in a sense misleading.

In fact, the difference between principles and schemata is due to the implicit presence of a second-order meta-quantifier in principles.

But, actually, the explicit use of the second-order meta-quantifier makes the whole concept of schemata (i.e., the formal use of metavariables for this purpose) superfluous. Let us recall that early textbooks on first-order logic, such as those of Hilbert–Ackermann, Hilbert–Bernays and P. S. Novikov did not speak of any schemata, but only of formulas; instead, their derivation systems included a substitution rule. Some problems with this early approach are that inference rules were anyway stated in schematic form, and also that the substitution rule is, in contrast to other inference rules, not structural (i.e. it is not preserved itself by substitution without anonymous variables). Non-structurality is a serious complication in trying to treat rules as fully formal objects.

In fact, the use of both first-order and second-order meta-quantifiers enables one to state (structural) rules without using meta-variables; and one way to understand the substitution rule is that it is not an inference rule of the logic, but an inference meta-rule of the meta-logic. An advantage of this approach is that side conditions that normally occur in first-order logics, such as ‘’provided that is not free in ’’ or ‘’provided that is free for in ’’ effectively disappear (more precisely, they remain at the meta-level, but they disappear from what needs to be specified in order to state rules and principles). One consequence of not having to specify exactly which English phrases qualify as ‘‘side conditions’’ in rules and principles is that it becomes feasible to give actual formal definitions of these notions (a rule and a principle) as well as further notions such as a derivable rule, an admissible rule, a first-order logic, and (both variants of) syntactic consequence.

Meta-formulas

The language of the meta-logic8 of a two-sorted first-order logic involves, in addition to the basic types , and , a fourth basic type:

  • , the type of meta-formulas;

and consists of the following constants (common to all two-sorted first-order logics). Reflection operators:

  • , the i-reflection;

  • , the c-reflection,

meta-connectives:

  • , the meta-conjunction;

  • , the meta-implication,

and meta-quantifiers

  • , the first-order (universal) meta-quantifier;

  • , the -ary second-order (universal) i-meta-quantifier;

  • , the -ary second-order (universal) c-meta-quantifier.

Here ranges over . In practice, meta-quantifiers are written like the old-style (early 20th century) universal quantifiers, but with fancy parentheses so as to avoid visual confusion with the ordinary parentheses : if is a meta-quantifier (either of them), is a -expression of type , and is a variable of type , then abbreviates the -expression . More generally, abbreviates .

An atomic meta-formula is a -expression of type that is either of the form , where is a c-formula, or of the form , where is an i-formula. A meta-formula is a -expression of type built out of atomic meta-formulas using meta-connectives and meta-quantifiers. We usually omit and in writing -expressions of type ; thus atomic meta-formulas are effectively identified with formulas, keeping in mind that meta-connectives and meta-quantifiers cannot be used inside of formulas.

As usual, abbreviates ; ‘’’’ is called meta-equivalence. We stick to the following order of precedence of logical and meta-logical symbols (in groups of equal priority, starting with higher precedence/stronger binding):

  1. , , , and ;

  2. and ;

  3. and ;

  4. ;

  5. ;

  6. and .

Meta-rules

The inference meta-rules (i.e., the inference rules of the meta-logic) are the -conversion rule for meta-formulas:

, if is -equivalent to ,


and the usual introduction/elimination rules of natural deduction for , and the meta-quantifiers:

,


where and are meta-formulas;

, provided that does not occur freely in any of the assumptions;

, provided that is free for in ,


where is a meta-formula, and there are three ways to read and :

  1. is an individual variable and is a term;

  2. is an -ary problem variable and is an -i-formula;

  3. is an -ary predicate variable, is an -c-formula.

It should be noted that boils down to the ordinary substitution of -calculus in the case (1), but not in the cases (2), (3) (see §2.1 above).

Let us note that by using a meta-specialization (=meta-quantifier elimination meta-rule) immediately after the corresponding meta-generalization (=meta-quantifier introduction meta-rule), we get the meta-rules of substitution:

, as long as does not occur freely in the assumptions and is free for in .


A meta-formula is called deducible if using the meta-rules one can obtain (from the trivial deductions, in which a meta-formula is deduced from itself) a deduction of from no assumptions.

Syntactic meta-sugar

The first-order meta-closure of the meta-formula is , where is the tuple of all individual variables occurring freely in . The second-order meta-closure is , where is the tuple of all predicate and problem variables occurring freely in .

A rule, written , or, in more detail,

where and are formulas, is an abbreviation for the meta-formula

The formulas are called the premisses of the rule, and its conclusion.

If is a formula (and only in this case) we abbreviate by . A meta-formula of the form , where is a formula, is called a principle. In other words, a principle is a formula that is meta-quantified over all its free (individual, predicate and problem) variables. Rules with no premisses can be identified with principles, in the sense that each meta-formula of the form is deducible, as long as the empty meta-conjunction is defined as an abbreviation of some deducible meta-formula (for example, ).

The difference between formulas and principles is clear from Example 2.3: in (the meta-logical extension of) intuitionistic logic, the meta-formula

is deducible, whereas the meta-formula

is not deducible.

A derivation system9 is a meta-formula of the form

where each is a rule (possibly with no premisses) in the pure language of QHC. The with no premisses, or rather the corresponding principles, are called the laws, and the with at least one premise are called the inference rules.

A logic is a meta-equivalence class of derivation systems. In other words, derivation systems and are said to determine the same logic if the meta-formula is deducible.

A meta-formula is called derivable in the logic determined by a derivation system if the meta-formula is deducible (in the meta-logic). Clearly, adding a derivable principle or rule to a derivation system does not affect derivability of principles and rules in the logic determined by .

If is the logic determined by a derivation system , we denote by , or in more detail , the judgement that the meta-formula is derivable in the logic. The meta-meta-logical symbol is set to have lower priority than all logical and meta-logical symbols. The judgement is also abbreviated by

When this judgement is true, we also say that is a (syntactic) consequence of the . This yields two notion of syntactic consequence for formulas: is the traditional ‘‘fixed variables’’ one, as in the textbooks by Church, Troelstra and van Dalen; whereas is the traditional ‘‘varied variables’’ one, as in the textbooks by Schoenfield and Mendelson. There seems to be no standard notation for the judgement of interderivability for formulas:

so we will keep it in this form. Let us note that, due to the absence of the deduction theorem in QHC, it is weaker than the (object-level) equivalence (which makes sense when both if and are either i-formulas or c-formulas),

but stronger than the equivalence of principles,

which is in turn stronger than the equivalence of judgements:

2.4. Derivation system

When writing down a derivation system for a new logic, one has to engage in informal considerations, or else risk the new logic being entirely unmotivated.

To provide an informal mathematical meaning to the judgements of QHC, we interpret c-formulas by propositions/predicates and i-formulas by problems. More precisely, we instantiate predicate variables and problem variables by particular mathematical predicates and problems. Upon such instantiation, classical connectives and quantifiers are interpreted according to the usual truth tables; intuitionistic connectives and quantifiers according to the BHK interpretation, in Kolmogorov’s problem solving terminology (see below); and the conversion operators and are interpreted as in §1. The interpretation of the meta-logical constants and judgements will be discussed in part II.

Some laws and inference rules of the QHC calculus are immediate:

  • All laws and inference rules of classical predicate logic (see [?]LABEL:int:logics) applied to all c-formulas (possibly involving and ).

  • All laws and inference rules of intuitionistic first-order logic (see [?]LABEL:int:logics) applied to all i-formulas (possibly involving and ).

We will now discuss the remaining part of the derivation system.

From the problem interpretation

Let us recall Kolmogorov’s problem interpretation of intuitionistic logic [?] (with minor improvements largely due to Heyting; see [?]LABEL:int:BHK, §LABEL:int:about-bhk for further details).10

A prescribed class of contentful (e.g. mathematical) primitive problems is fixed, and it is assumed to be known what is a solution of a primitive problem. For instance, Euclid’s first three postulates are the following primitive problems:

(1) draw a straight line segment from a given point to a given point;

(2) extend any given straight line segment continuously to a longer one;

(3) draw a circle with a given center and a given radius.

We may thus stipulate that each of (1) and (3) has a unique solution, and describe all possible solutions of (2). (Euclid’s Elements will be discussed in some detail in part III of the present paper.)

Composite problems are obtained from the primitive ones by using contentual connectives , , , , [1] and quantifiers , . (They are ‘‘contentual’’ in that they provide a natural interpretation of, but should not be conflated with, the connectives and quantifiers in the formal language of intuitionistic logic.) What it is a solution of a composite problem is explained as follows:

  • a solution of consists of a solution of and a solution of ;

  • a solution of consists of an explicit choice between and along with a solution of the chosen problem;

  • a solution of is a reduction of to ; that is, a general method of solving on the basis of any given solution of ;

  • the absurdity [1] has no solutions; is an abbreviation for ;

  • a solution of is a solution of for some explicitly chosen ;

  • a solution of is a general method of solving for all .

A key element here is the idea of a ‘‘general method’’ (roughly corresponding to the idea of a ‘‘construction’’ advocated by Brouwer and Heyting), which Kolmogorov further explains as follows. If is a problem depending on the parameter ‘‘of any sort’’, then ‘’to present a general method of solving for every particular value of ’’ should be understood as ‘’to be able to solve for every given specific value of of the variable by a finite sequence of steps, known in advance (i.e. before the choice of )’’.

Let us observe that if denotes the set of solutions of the problem , then the above clauses guarantee that:

  • is the product ;

  • is the disjoint union ;

  • there is a map into the set of all maps;

  • ;

  • is the disjoint union ;

  • there is a map into the product.

Now the proposition ‘’ has a solution’’ can be rephrased as ‘’’’. It follows that the following propositions must be true for any contentful problems , and any contentful parametric problem :

  • ;

  • ;

  • ;

  • ;

  • ;

  • .

See [?]LABEL:int:about-bhk for a more thorough discussion of these propositions.

This motivates some laws of QHC (beware that some of these will turn out to be redundant):

  1. ;

  2. ;

  3. ;

  4. ;

  5. ;

  6. .

It should be noted that formulas with almost same appearance and motivation, but somewhat different meaning appear in [?]LABEL:int:weak_BHK.

Informally, () is saying that [1] is not just the hardest problem (as guaranteed by the explosion principle, ), but a problem that has no solutions whatsoever. This is just the first example of how some content found in the BHK interpretation and not entirely captured in the usual formalization of intuitionistic logic is more fully captured in QHC.

Some versions of the BHK interpretation include the well-known principle (see [?]LABEL:int:about-bhk), that every solution of a problem must be supplied with a proof that is it indeed a solution of . This principle was emphasized by G. Kreisel in connection with interpreting intuitionistic logic (in a somewhat different form) and also by the ancient Greeks, particularly Proclus, in the context of geometric construction problems, which as we now know can be seen as a model of intuitionistic logic (see [?]). This Proclus–Kreisel principle is usually considered to be relevant when one tries to make sense out the BHK interpretation in the context of first-order logic, rather than a constructive type theory (see references in [?]LABEL:int:about-bhk).

A consequence of this Proclus–Kreisel principle is that a solution of a problem yields a proof of the existence of a solution of . This is expressible in the language of QHC:

  1. .

From the proof interpretation

The remaining part of the derivation system is motivated by the proof interpretation of intuitionistic logic, given independently by Orlov and Heyting (see details in §6.2.1) and partially formalized in Gödel’s translation of intuitionistic logic into classical modal logic S4 (see [?]LABEL:int:provability). A remarkable attempt to clarify the informal notion of ‘‘proof’’ used by Orlov and Heyting occurs in Gödel’s sketch of a proof-relevant analogue of S4, which is found in his outline of a 1938 lecture, published posthumously in his collected works [?].

Gödel’s proposal is based on a ternary relation ‘’, that is, is a derivation of from ’’. But as a matter of fact he also uses a binary relation ‘’’’ which is presumably meant to abbreviate . Here stands for German Beweis (proof), and apparently refers to proofs ‘‘understood not in a particular system, but in the absolute sense (that is, one can make it evident)’’ (these words of Gödel appears earlier on the same page). Gödel’s axioms for are as follows (literally):

  1. ‘’’’;

  2. ‘’’’;

  3. ‘’’’;

  4. ‘’if has been proved and is the proof, [then] is to be written down’’.

Instead of attempting to clarify the meaning of this in Gödel’s original terms, let us consider something similar but more clearly described: the extension of classical predicate logic by

  • an operator    associating to every formula and every term a formula ;

  • an unary function that associates to every term a term ;

  • a binary function that associates to every two terms a term ;

  • an operator that associates to every formula a term ,

that satisfies all laws and inference rules of classical predicate logic along with the following additional ones:

    [label=()]
  1. ;

  2. ;

  3. ;

  4. .

S. Artëmov discovered that a further extension of this logic by an additional function (‘‘sum of proofs’’) and an additional law (not hinted at in any way by Gödel) is indeed a proof-relevant analogue of S4 in a sense one could expect [?].11 But we do not need these for our motivational purposes.

The logic just described has the following derived principles and rules:

  1. ;

  2. ;

  3. ;

  4. .

Here (i) is just the special case of (i) with substituted by the classical falsity . Next, (i) is derived from (i) by using two inference rules of classical logic: and . Of course, (i) is derived from (i) using the modus ponens rule. To establish (ii), let us first note that from the classical law we get , and if denotes the latter formula, then by (iv) we get . Now from (iii) and the modus ponens rule we get . Finally, (ii) follows from this and (ii), if we set .

Just like Gödel’s proofs ‘‘in the absolute sense’’, the ‘‘proofs’’ of propositions referred to in the intended reading of the problem , Find a proof of , are not supposed to be formal proofs. In the language of QHC, we have the following direct analogues of (i), (i), (ii), (iii), (iv) and (i):

  1. ;

  2. ;

  3. ;

  4. ;

  5. ;

  6. .

Here () is a kind of internal soundness: a proof of falsity leads to absurdity. Semantically (informally), this is pretty much like in Gödel’s system; but let us note that (i) is about the c-formula , whereas () is about the i-formula . In contrast, () is about the c-formula . Note that by the explosion principle, the reverse implications to () and () are trivial. Thus () identifies the classical falsity, , with the proposition ‘’[1] has a solution’’; and () identifies the intuitionistic absurdity, [1], with the problem ‘’Prove ’’.

This completes the list of additional inference rules and laws of QHC. Let us note that () can be dropped from this list since it follows immediately from (), () and (). Some other laws will be shown to be redundant in 3.6.

3. Symmetries and redundancy

3.1. Galois connection

Proposition 3.1.

The inference rule () is equivalent to the following inference rule:

  1. .

We will see in [?] that the converse rule, , is not derivable in QHC.

Proof.

Given (), we can derive () using (): and . Conversely, given (), we can derive () using (): and . ∎

The equivalence relations on i-formulas and on c-formulas yield the ‘‘Lindenbaum’’ poset of equivalence classes of i-problems, ordered by if , and the ‘‘Lindenbaum’’ poset of equivalence classes of c-problems, ordered by if . By () and (), and respectively () and () we have:

  • implies ;

  • implies .

Thus and descend to monotone maps between the two posets. Using the monotonicity of and and substitution, from () and () we also obtain:

  • ;

  • .

These identities resemble well-known properties of a Galois connection. Indeed, it turns out that our two monotone maps do form a Galois connection between the two Lindenbaum posets:

Theorem 3.2.

For an i-formula and a c-formula , if and only if .

The same argument works to prove a slightly stronger assertion, .

Proof.

If , then . So from () we get .

Conversely, if , then . So from () we get . ∎

Another standard fact on Galois connections takes the following form in our situation.

Corollary 3.3.

Let denote a c-formula and let denote an i-formula.

(a) is the least among all such that is an upper bound of ; and is the greatest among all such that is a lower bound of .

(b) is the least of all upper bounds of of the form ; and is the greatest of all lower bounds of of the form .

Proof.

The first assertion of (a) says that , and if , then . This is indeed so by () and by 3.2. The first assertion of (b) says that , and if , then . This follows similarly, using additionally the monotonicity of . The second assertions of (a) and (b) are proved similarly. ∎

3.2. Modalities

Let us write for the c-formula , and for the i-formula . Upon substituting problems and propositions for the atoms of and , these are interpreted by the proposition , ‘’There exists a proof of ’’, and the problem , ‘’Prove that has a solution’’. By another standard fact on Galois connections, the ‘‘provability’’ operator descends to an interior operator (in the sense of order theory) on the poset of equivalence classes of c-formulas, whereas the ‘‘solubility’’ operator descends to a closure operator (in the same sense) on the poset of equivalence classes of i-formulas. In the case of , this amounts to (i) the derivability in QHC of the principles

  • ;

  • ;

and (ii) the judgement

  • implies .

These are easy to verify directly: () is the same as (); () follows from () and the monotonicity of ; and () follows from the monotonicity of and .

In fact, () is a consequence of the derivability in QHC of the following principle and rule:

  • ;

  • ;

Here () follows from () and (), and () from () and (). We have proved

Proposition 3.4.

Sending to yields a syntactic interpretation of QS4 in QHC, which is the identity on QC.

We will see in §5.1 that this interpretation is exact. Before we get there, we need to distinguish two roles of the symbol ‘’’’: the modality of QS4 and an abbreviation for in QHC.

Similarly, that induces a closure operator on the poset of equivalence classes of c-formulas translates to (i) the derivability in QHC of the principles

  • ;

  • ,

and (ii) the judgement

  • implies .

Here () is a consequence of the derivability in QHC of the principle

  • ,

which follows from () and (). We also note that the following consequence of () and (),

  • ,

is equivalent (modulo () and ()) to (cf. §3.4 below), which can be considered to be dual to (). We define QH4 to be the logic obtained from QH by adding a new unary connective and additional laws ()–(). We have thus proved:

Proposition 3.5.

Sending to yields a syntactic interpretation of QH4 in QHC, which is the identity on QH.

It should be noted that the laws of QH4 mimic some properties of . In fact, by substituting for we get an interpretation of QH4 in QH. Indeed, under this substitution, () holds by ([?]LABEL:int:tautologies, (LABEL:int:double_negation)), () and () follow from ([?]LABEL:int:tautologies, (LABEL:int:triple_negation)), and () holds by ([?]LABEL:int:tautologies, (LABEL:int:neg-neg-imp)). Let us note that since the purely intuitionistic fragment of QH4 is fixed under this interpretation, this fragment is precisely QH (in other words, QH4 is a conservative extension of QH). We will see in §5.3 that the constructed interpretation of QH4 in QH factors through the interpretation of 3.5.

It is not clear to the author whether the interpretation of 3.5 is faithful (in other words, whether QHC is a conservative extension of QH4). Thus one should not conflate two potentially distinct roles of the symbol ‘’’’: the modality of QH4 and an abbreviation for in QHC.

The modal logic QH4 was studied by Aczel, who called the modality satisfying ()–() a strict lax modality [?], and more recently also by Artëmov and Protopopescu, who showed its completeness with respect to some Kripke models [?]. The intuitionistic modal logic given by the postulate schemes ()–() was studied by Curry (1952, 57), Goldblatt (1979, 81) and many others; in particular, categorical models of QH4 related to the sheaf-valued models of QH in [?] are known; see [?], [?], [?].

The properties of are also similar to those of the squash/bracket operator in dependent type theory (see [?] and references there).

3.3. Simplification

Proposition 3.6.

(a) The laws (), (), (), () and () are redundant.

(b) The following holds in QHC:

  1. ;

  2. ;

  3. ;

  4. .

Remark 3.7.

From the informal semantic viewpoint, the implication cannot be reversed. Indeed, let be the proposition is a rational number and the proposition is an irrational real number. The problem amounts to showing that is a real number. This problem is trivial: . On the other hand, the problem amounts to , where is the problem Determine whether is rational or irrational. This is not an easy problem.12 This shows incidentally that one cannot get an exact interpretation of classical logic in intuitionistic logic by just looking at problems of the form , where is a proposition.

Proof. Redundancy of ().

By an intuitionistic law, . Then by () and (), we get . By the generalization rule, we obtain . By another intuitionistic rule, we infer that . Now the variable can be renamed. ∎

Redundancy of ().

The implication in () is redundant similarly to the redundancy of (). Conversely, the intuitionistic validity can be rewritten, by the exponential law, as . Then by () and () it follows that . Again applying the exponential law, this time regarded as an inference rule of classical logic, we obtain . ∎

Proof of () and ().

This is parallel to the redundancy of (). In more detail, by a classical axiom scheme, . Then by () and (), we get . By the generalization rule, we obtain . By another classical rule, we get . Now the variable can be renamed. The case of () is similar. ∎

Redundancy of () and ().

The implication in () is redundant similarly to the proof of () or to the redundancy of (). Conversely, by the proof of () we have shown that using only (), () and classical logic. Substituting, we get . On the other hand, from () it follows that . By combining the two implications we get . By the proof of 3.2, we obtain from this the implication in (), using only (), () and (). The case of () is similar. ∎

Proof of () and ().

The implication in () is proved similarly to the redundancy of (). The converse implication is parallel to the redundancy of (). In more detail, () implies , and it follows from () that . Thus , hence by 3.2 . The case of () is similar, or alternatively can be treated similarly to the redundancy of (). ∎

Redundancy of ().

By the explosion principle, we have . Then by () and () we get . On the other hand, by () we have . Composing the two implications, we obtain . ∎

Corollary 3.8.

The meta-conjunction of the following meta-formulas is a deductive system for QHC.

  • A deductive system for intuitionistic logic;

  • A deductive system for classical logic;

  • ;

  • ;

  • ;

  • ;

  • ;

  • .

3.4. Negation

Proposition 3.9.

Some laws of QHC can be rewritten as follows.

(a) () is equivalent, modulo () and (), to and to ;

(b) () is equivalent, modulo () and (), to and to