On the Minimal Non-Fregean Grzegorczyk Logic

The paper concerns Grzegorczyk’s non-Fregean logics that are intended to be a formal representation of the equimeaning relation defined on descriptions. We argue that the main Grzegorczyk logics discussed in the literature are too strong and we propose a new logical system, MGL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{MGL}}$$\end{document}, which satisfies Grzegorczyk’s fundamental requirements. We present a sound and complete semantics for MGL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{MGL}}$$\end{document} and we prove that it is decidable. Finally, we show that many non-classical logics are extensions of MGL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{MGL}}$$\end{document}, which makes it a generic non-Fregean logic.


Introduction
There are plenty of logical systems discussed in the literature which are focused on different aspects of formal representation of real world problems. Over the years, the question as to which logic is adequate has been a matter of lively discussion. One of the fundamental controversies is the philosophical status of classical two-valued logic, and in particular the paradoxes of material implication and equivalence. In order to remedy the dissatisfactory aspects of classical logic, many logical systems have been introduced, such as modal, intuitionistic, relevant, non-Fregean, and paraconsistent logics, among others.
In [3] Andrzej Grzegorczyk introduced yet another logical system intended to cover the fundamental properties of descriptive equivalence of sentences. His main assumption is that the human language is used primarily to create descriptions of reality and logical connectives are the main tool to represent them formally. In order to be able to speak about interactions between descriptions-as Grzegorczyk argues-a logical language should contain at least the following four connectives: negation (¬), conjunction (∧), disjunction (∨), and descriptive equivalence (≡). According to Grzegorczyk, neither classical propositional logic nor any of its non-classical extensions can serve as an adequate formal language of descriptions. As stated in [3], classical logic "restricts itself to considering only one, admittedly the most important, parameter of the content of a claim, namely its truth value" which is the source of many paradoxes of material implication and equivalence. In particular the classical equivalence cannot properly express the reality of descriptions. Indeed, the expressions "2 + 2 = 4" and "Warsaw is the capital of Poland" are classically equivalent, but clearly they do not describe the same. As pointed out by Grzegorczyk in [3, p. 477]: Such combined propositions, established as equivalences according to classical formal logic, do not lead to anything interesting. The paradoxical nature of the classical concept of equivalence arises because we may expect equivalence to open the possibility of linking the content of one claim with that of the other. We would like equivalent sentences not only to be equally true, but also to speak about the same subject. It seems (from a philosophical point of view) that claims that are not connected by a common subject cannot be treated as fully equivalent.
Thus classical logic cannot be accepted as a basis of a proper logic of descriptions, and hence Grzegorczyk defines his new propositional logic from scratch (for more details on Grzegorczyk's motivations see [1]). He rejects all the classical tautologies except the law of contradiction and adds axioms reflecting intended properties of the descriptive equivalence, which are: (LD1) ≡ represents an equivalence relation and satisfies the extensionality property (equal descriptions can be substituted for each other).
(LD2) ≡ joins some of the boolean combinations of descriptions, like: associativity, commutativity, and idempotency of conjunction and disjunction, distributivity of conjunction (resp. disjunction) over disjunction (resp. conjunction), involution of negation that additionally satisfies de Morgan laws.
The new propositional logic obtained in this way has been called the Logic of Descriptions and denoted by LD. As shown in [1], the logic LD is indeed a new propositional logic as it cannot be thought of as a subsystem or extension of any of the known non-classical logics. In the last years the logic LD has been extensively studied. A sound and complete semantics for LD introduced in [1] shows that the logic LD is non-Fregean in a sense that it rejects the main assumption of classical Fregean logic, according to which sentences with the same truth value have the same denotations. This implies that LD has much in common with the non-Fregean logic SCI introduced by Roman Suszko in [4] (cf. [5,6]). Recall that SCI is the logic obtained by expanding the language of classical propositional logic with the connective ≡ (called in Suszko's logic the identity connective) and adding to the axiomatization of classical logic four axioms for ≡. Even though the philosophical motivations for introducing logics LD of Grzegorczyk and SCI of Suszko were similar, i.e., the necessity of separating logical values from denotations, these logics differ considerably in construction. Suszko's logic SCI is an extension of classical logic, while Grzegorczyk's logic LD is built from the ground up. Suszko accepts the classical implication, Grzegorczyk removes it from the language (so the classical modus ponens rule is not present in the logic LD). Contrary to SCI, in the logic LD the connective ≡ is not an additional operator, but rather it supersedes the classical implication and equivalence. 1 It means that the extensionality property cannot be expressed in Grzegorczyk's logic in terms of the classical implication as it is in Suszko's logic SCI. As a consequence, the logics LD and SCI are completely different realizations of the similar philosophical assumptions on the ontological status of sentences. The difference is evidently manifested on the formal level: there are theorems (resp. models) of one of the logics which are not theorems (resp. models) of the other. Hence, LD and SCI are uncomparable. More details on differences between LD and SCI can be found in [3].
Further results presented in [1] showed that the logic LD has many peculiar and strange formal properties, which led to some modifications of LD studied in detail in the paper [2]. Nevertheless, the philosophical interpretation of some axioms of Grzegorczyk's logics remains unclear and controversial. In fact, after many discussions at the seminar devoted to the logic LD, professor Grzegorczyk admitted that LD may be too strong and he was inclined to amend his axioms, but he insisted that (LD1) is the fundamental property of descriptive equivalence which a proper logic must reflect. The question whether any of the proposed logics is a good realization of this fundamental assumption of Grzegorczyk on descriptive equivalence is still open. In this paper we argue that the main three logics of Grzegorczyk are too strong. We propose a very weak logical system, which satisfies (LD1) and can serve as a generic non-Fregean logic that can be extended to non-Fregean logics constructed in Grzegorczyk or Suszko style.
The paper is organized as follows. In Sect. 1 we present Grzegorczyk's non-Fregean logics, their axiomatizations and some formal properties. Section 2 is concerned with a discussion on philosophical difficulties with an interpretation of some of Grzegorczyk's axioms. In Sect. 3 we present a very weak Grzegorczyk's non-Fregean logic, MGL. We construct denotational semantics for MGL and we show its soundness and completeness. Furthermore, we show that MGL is decidable. In Sect. 4 we argue that MGL can be seen as a generic logic by showing that many non-classical logics are just extensions of MGL. Conclusions and prospects of future work are presented in Sect. 5.

Grzegorczyk's Non-Fregean Logics
In this section, we present the basics of Grzegorczyk non-Fregean approach. We start with some general definitions, and then we present in detail the three representative Grzegorczyk's logics.
By a logic we would mean a formal system determined by a language, set of axioms, rules of inference, and the provability relation. All the logics considered in the paper are propositional. where n, k ≥ 1 and φ 1 , . . . , φ n , ψ 1 , . . . , ψ k are L-formulas. An L-formula ψ follows from the formulas φ 1 , . . . , φ n by φ 1 ,...,φ n ψ 1 ,...,ψ k if ψ ∈ {ψ 1 , . . . , ψ k }. Definition 1.3. (Provability) Let X be a set of L-formulas. A formula ϕ is said to be L-provable ( L ϕ for short) whenever there exists a finite sequence ϕ 1 , . . . , ϕ n of L-formulas, n ≥ 1, such that ϕ n = ϕ and each ϕ i , i ∈ {1, . . . , n}, is an L-axiom or follows from earlier formulas in the sequence by one of the L-rules of inference. A formula which is L-provable will be referred to as an L-theorem. A formula ϕ is said to be L-provable from X (X L ϕ for short) whenever there exists a finite sequence ϕ 1 , . . . , ϕ n of formulas, n ≥ 1, such that ϕ n = ϕ and each ϕ i , i ∈ {1, . . . , n}, is an L-axiom or a formula from X or follows from earlier formulas in the sequence by one of the L-rules of inference. Therefore, each logic is uniquely determined by its set of propositional operations, axioms, and rules. Definition 1.5 ((Strong) Admissibility). Let L be a logic and let φ i , ψ j be L-formulas, for all i, j ≥ 1. A rule of the form φ 1 ,...,φ n ψ 1 ,...,ψ k , for n, k ≥ 1, is said to be L-admissible whenever the following condition is satisfied: If φ 1 , . . . , φ n are L-provable, then ψ 1 , . . . , ψ k are L-provable.
By the L-validity of an L-formula we mean its truth in all possible semantic interpretations, provided that the semantics for L exists. Definition 1.6. (Correctness) Let L be a logic and let φ i , ψ j be formulas, for all i, j ≥ 1 A rule of the form φ 1 ,...,φ n ψ 1 ,...,ψ k , for n, k ≥ 1, is said to be L-correct whenever the following condition is satisfied: Grzegorczyk's non-Fregean logics discussed in this paper belong to a family of non-Fregean propositional logics with the connective ≡. All the logics share a common language, but differ in axioms. Definition 1.7 (Language of Grzegorczyk's non-Fregean logics). The language of Grzegorczyk's non-Fregean logics is determined by the set of propositional operations of negation ¬, disjunction ∨, conjunction ∧, and descriptive equivalence ≡. We also admit the following three shorthand notations: Below we present axioms and rules of the three basic Grzegorczyk's non-Fregean logics which are: the Logic of Descriptions LD, introduced in [3] and widely discussed in [1], the Logic of Equimeaning LE and the Logic of Descriptions with Suszko's Axioms LDS, both introduced in [2]. All these logics share most of their axioms and all of their rules. Their common axioms are the following: The rules have the following forms: As in the classical case, the rule (Sub) may be applied only to axioms. This restriction is not essential when no additional assumptions are used in the proof.
Logics LD, LE, and LDS differ on the last three axioms. They have the following form in the original logic LD: The last three axioms of LE are: In the logic LDS the three axioms are descriptive copies of specific axioms of the Sentential Calculus of Identity introduced by Suszko in [4].
A sound and complete semantics for LD has been presented in [1]. Models of LD are non-Fregean structures based on the Grzegorczyk algebras which are de Morgan bisemilattices satisfying some additional conditions (see [1]). In [2] it is shown that LD-models can be weakened to a paraconsistent version, thus the logic LD is contradiction-tolerant: there are formulas φ and ψ such that (φ ∧ ¬φ) LD ψ.
The following proposition proved in [1] shows that ≡ is not the classical equivalence: Proposition 1.8. The following formulas are not provable in LD: Further results from [1] show that negation and disjunction behave in LD in a non-classical way: the formula p ≡ ¬p is unsatisfiable in any LD-model, In the paper [2] it has been proved that the last three axioms of LD are essentially different from their corresponding forms in logics LE and LDS: The above proposition implies that the last three axioms of logics LD, LE, and LDS express the extensionality in a different way. The further question is whether any of the trio adequately realizes the intended extensionality principle for descriptions. We discuss this issue in the next section.

Extensionality
All the Grzegorczyk's logics have been defined to represent fundamental properties of a relation having the same meaning defined on sentences. On the formal side this relation is represented by the operation ≡ called the descriptive equivalence or the equimeaning operation. Thus, possible intended readings of an expression "ϕ ≡ ψ" are: ϕ and ψ have equal meaning, ϕ and ψ have the same content, ϕ and ψ describe the same reality, ϕ and ψ are equal descriptions of the world, ϕ and ψ denote the same situations. According to Grzegorczyk, formal properties of the operation ≡ should guarantee that it represents an equivalence relation. Furthermore, it is expected that a logic designed to adequately reflect properties of equimeaning should meet the following Extensionality Principle: Sentences that have equal meaning (have the same content or are the same descriptions) are interchangeable in all possible contexts.
Is any of the logics LD, LE, and LDS a good realization of these requirements? Observe that last three axioms of LD, LE, and LDS can be reformulated as: where # ∈ {≡, ∧, ∨}. As we mentioned in the previous section, LD, LE, and LDS are formally uncomparable. But what is the philosophical difference between these three logics? What do the last three axioms actually express? Is it the case that in all these logics the operation ≡ is a formal representation of the equimeaning relation?
It is quite clear what the axioms (Ax1)-(Ax12) mean if we read them in terms of having the same meaning-relation. They just express some of its elementary properties. For instance, the axiom (Ax2) can be read as: 'p' means the same as 'it is not the case that not p'. Although some or all of the axioms (Ax1)-(Ax12) can be thought of as too strong (e.g., it is not obvious that changing the order in a conjunction leads to sentences with the same content), their interpretation as 'the sentence on the left means the same as the sentence on the right' is intuitive, natural, and unquestionable. The problem arises when we try to do the same with the other axioms.
Note that in all the axioms (Ax1)-(Ax12) the connective ≡ occurs once and it binds formulas in which ≡ does not occur explicitly. It is not the case with the axioms (Ax13), (Ax14), (LD) ≡ , (LE) ≡ , and (LDS) ≡ , in which the operation ≡ is nested. The axioms (Ax13) and (Ax14) are not so problematic as the last three axioms. In particular, the axiom (Ax13) we can read as: The case that p has the same meaning as q means the same as The case that q has the same meaning as p.
Such a reading of (Ax13) is understandable and even easily acceptable. The same concerns the axiom (Ax14).
The equimeaning of (1) and (2) is not so evident as in the previous example, but can be explained as: The concept of equimeaning brings into play the information that sentences having the same meaning are interchangeable in the conjunction context. That is: if the sentences '2 + 2 = 4' and '1 + 1 + 1 + 1 = 4' actually mean the same, then the meaning of (1) already contains in itself the meaning of (2).
The problem, however, is that axioms should be universal and hold in every case, in particular when propositional variables are substituted with sentences which are not true or do not have the same meaning. It seems that the justification of (Ax16) LD given above works because we used sentences which can be easily seen as having the same meaning. Nevertheless, the last three axioms are extremely hard to accept if propositional variables are substituted with sentences that do not have the same meaning.
By way of example, consider now the axiom (Ax16) LE of the logic LE: This axiom forces us to accept that: (3) The case that 2 + 2 = 4 means the same as 2 + 2 = 5, and in addition, 2 + 2 = 4 and 1 = 1 means the same as (4) The case that 2 + 2 = 4 means the same as 2 + 2 = 5, and in addition, 2 + 2 = 5 and 1 = 1 Such a consequence is highly controversial. It is very difficult or even impossible to find any reason for accepting that (3) and (4) say the same thing. They are, of course, equivalent, but cannot have the same meaning. For better intuition, let us denote "2 + 2 = 4 means the same as 2 + 2 = 5", "2 + 2 = 4 and 1 = 1", "2 + 2 = 5 and 1 = 1" by False 1 , True, and False 2 , respectively. Then, the following is an instance of the axiom (Ax16) LE : The reason of all these problematic cases is that the axioms (LD) ≡ , (LE) ≡ , and (LDS) ≡ are intended to express the extensionality property which is a meta-property, but they make it in the object language instead of the metalanguage. In fact, to guarantee the extensionality it would suffice to assume that whenever the left side of an axiom holds then so does the right side. To express this we need a rule rather than an axiom. Although axioms (LD) ≡ , (LE) ≡ , and (LDS) ≡ enable us to prove all required formal properties of the equimeaning relation, they also express some strong additional properties, which are even difficult to capture in the natural language. Since axioms (LD) ≡ , (LE) ≡ , and (LDS) ≡ are too strong, none of the logics LD, LE, LDS is a good realization of the Extensionality Principle. Hence, the best solution is to replace the axioms with a rule that appropriately expresses the extensionality.
Following this line of thought, in the next section we will define a new logic MGL-the minimal Grzegorczyk's logic in the non-Fregean style. As we will show, the Extensionality Principle is represented in the logic MGL in an adequate way. Furthermore, since we are searching for a generic logic, we will reject all the axioms that express non-trivial properties of ≡, in particular, the axioms (Ax2)-(Ax14), which are not theorems of Suszko's logic SCI. In this way, we will obtain a logic which is as weak as possible and can be seen as a general logical framework for studying and comparing all the non-Fregean logics under consideration.

The Minimal Grzegorczyk's Non-Fregean Logic
The minimal Grzegorczyk's non-Fregean logic, MGL, is meant to be a minimal formal system satisfying the fundamental assumptions of Grzegorczyk's approach, according to which ≡ represents an equivalence relation on descriptions and satisfies the Extensionality Principle. At this point, it should be emphasized that the main motivation for logic MGL-apart from fulfilling Grzegorczyk's fundamental assumption on the extensionality of the descriptive equivalence-is to have a generic non-Fregean logic that can be used as a general framework for representing, investigating, implementing, and comparing theories with incompatible languages and/or semantics. We will argue that the logic MGL is able to play such a role.
Therefore, the logic MGL is not meant to be a logic representing a specific philosophical theory, but it should rather be seen as a starting point for investigating one's favourite non-Fregean theory of descriptions. As a consequence, the logic MGL does not include any specific axioms for the classical connectives, except the law of contradiction and the rules which are common for all the logics in question. This concerns in particular the disjunction connective. Actually, the connectives ∧ and ∨ are not treated in LD in the same way. Both are present in the language, but their status in deduction is different: there are rules for the conjunction, while nothing is said about reasoning with disjunction. Although in LD, due to its specific axioms, the formula p∨¬p is a theorem, the disjunction does not behave in a classical way, in particular it cannot be treated as a dual form of conjunction (for details see [1]). The logic MGL inherits this peculiar property of LD. It must contain ∨ in the language, since otherwise we could not interpret logics that contain the disjunction. However, as MGL aspires to be a generic non-Fregean logic, it cannot assume the symmetry between ∧ and ∨, between the law of contradiction (which is an axiom of LD and MGL) and the law of excluded middle, which is rejected in intuitionistic logics, among others. In this way the logic MGL may also serve as a base for studying non-Fregean intuitionistic logics.
The language of MGL is defined as in Definition 1.7. The axioms of MGL are (Ax0) and (Ax1): The rules are (MPE), (Sub), (∧ 1 ), (∧ 2 ) presented in Sect. 1, and the extensionality rule of the following form: where ϕ, ψ are arbitrary formulas, α is a formula in which ϕ is a subformula, and α(ϕ/ψ) is a formula obtained from α by replacing some or all occurrences of ϕ with ψ.
The notions of MGL-provability of a formula, MGL-theorem, and (strong) MGL-admissibility of a rule are defined in a standard way as in Definitions 1.3 and 1.5.
Proof. Let X be a set of MGL-formulas and assume that ϕ and ψ are MGLformulas such that ϕ ≡ ψ is MGL-provable from X. Then, the following proves ψ ≡ ϕ in MGL from X: 4. ψ ≡ ϕ (MPE) to 2. and 3.

ϕ ≡ θ (MPE) to 2. and 4.
Hence, the rule (tran) is strongly MGL-admissible. Now, assume that ϕ ≡ ψ and θ ≡ χ are MGL-provable from X. We will show that (ϕ#θ) ≡ (ψ#χ) is MGL-provable from X, for any # ∈ {∧, ∨, ≡}.  Note that MGL-structures are paraconsistent, in the sense that there may exist a ∈ U such that both a and ∼a belong to D. A trivial example of an MGL-structure is a structure based on the two-element Boolean algebra. Now, we will present more interesting example of an MGL-structure. Let  • v(¬ϕ) = ∼v(ϕ), As pointed out in [1], structures like MGLs can be thought of as variants of Suszko's non-Fregean structures. We may think of the universe U as consisting of denotations of sentences. Usually the elements of U are referred to as situations or states of affairs. The set D of factual situations consists of the correlates of true sentences. The correlates of false sentences, i.e., sentences whose negations are true, can be referred to as counterfactual situations. However, there may be sentences which are neither true or false. Their correlates can be referred to as undetermined situations.
The following can be easily proved:  Assume ϕ ≡ ψ and ϕ are satisfied in M by v.
Assume X MGL ϕ. Let M be an MGL-structure and v be a valuation in M such that M, v |= ψ, for all ψ ∈ X. Since ϕ is MGL-provable from X, there is a finite sequence ϕ 1 , . . . , ϕ n of MGL-formulas such that ϕ n = ϕ and for all k ≤ n, ϕ k is either an MGL-axiom or ϕ k ∈ X or ϕ k is obtained from earlier formulas in the sequence by one of the MGL-rules of inference. Clearly, all ϕ k , for k ∈ {1, . . . , n}, which are MGL-axioms or formulas from X must be satisfied in M by v. Note also that the rule (Sub) can be applied only to MGL-axioms, so each application of this rule must result with a formula that is satisfied in M by v. Furthermore, due to the strong MGL-correctness of the other rules given in Proposition 3.4, each application of an MGL-rule different than (Sub) to formulas ϕ k 's in the sequence results with a formula satisfied in M by v. Hence, for each formula ϕ k in the sequence, M, v |= ϕ k , and so M, v |= ϕ. Therefore, X |= MGL ϕ.
In order to prove completeness, we will define a relation R on the set of all MGL-formulas. Let X be a fixed set of MGL-formulas. Then: ϕRψ if and only if ϕ ≡ ψ is MGL-provable from X (i.e., X MGL ϕ ≡ ψ). Proof. Let ϕ, ψ, ϑ be arbitrary formulas. By the substitution rule applied to the axiom (Ax1), ϕ ≡ ϕ is provable in MGL, so in particular it is MGLprovable from X. Thus, R is reflexive. Assume that ϕRψ. Then, ϕ ≡ ψ is MGL-provable from X, so by the rule (sym) the formula ψ ≡ ϕ is provable in MGL from X. Hence, ψRϕ, which means that R is symmetric. Now, assume that ϕRψ and ψRϑ. Then, ϕ ≡ ψ and ψ ≡ ϑ are provable in MGL from X, so by the rule (tran) the formula ϕ ≡ ϑ is provable in MGL from X. Therefore, ϕRϑ, and hence R is transitive. Now, we show that R is compatible with all connectives. Indeed, if ϕRϕ , then ϕ ≡ ϕ is provable in MGL from X, so by the rule (¬), is so ¬ϕ ≡ ¬ϕ . Thus, ¬ϕR¬ϕ , so R is compatible with ¬. Let # ∈ {∧, ∨, ≡}. Assume ϕRψ and ϑRχ, that is ϕ ≡ ψ and ϑ ≡ χ are provable in MGL from X. Then, due to the rule (com), the formula (ϕ#ϑ) ≡ (ψ#χ) is provable in MGL from X, thus (ϕ#ϑ)R(ψ#χ). Hence, R is also compatible with any # ∈ {∧, ∨, ≡}.
The above proposition enables us to define the quotient MGL-structure. • For all |ϕ| R , |ψ| R ∈ U : Due to Definition 3.7 and the rule (MPE) we have the following: Proof. Clearly, U and D are non-empty sets satisfying D ⊆ U . The definition of the structure M X implies that for any formula ϕ, |ϕ| R ∈ D if and only if ϕ is provable in MGL from X. Let |ϕ| R , |ψ| R ∈ U . Then, In a similar way, we show the condition for ⊗. We have: |ϕ| R ⊗ |ψ| R = |ϕ ∧ ψ| R ∈ D iff ϕ ∧ ψ is provable in MGL from X iff (due to the rules (∧ 1 ) and (∧ 2 )) ϕ and ψ are provable in MGL from X iff |ϕ| R ∈ D and |ψ| R ∈ D. Hence, we have proved that |ϕ| R ⊗ |ψ| R ∈ D if and only if |ϕ| R ∈ D and |ψ| R ∈ D.
Due to the axiom (Ax0), for any formula ϕ, ¬(ϕ∧¬ϕ) is an MGL-theorem, so in particular ¬(ϕ ∧ ¬ϕ) is MGL-provable from the set X. Hence, we get ∼(|ϕ| R ⊗ ∼|ϕ| R ) ∈ D. Hence, M X is an MGL-structure. Proof. Suppose X MGL ϕ. Let M X = (U, ∼, •, ⊗, ⊕, D) be the canonical structure with respect to X and let v : V → U be the function defined by v(ψ) = |ψ| R . It is easy to prove that v is a valuation on M X . Thus, by the assumption and the definition of On the other hand, clearly, X MGL ψ, for every ψ ∈ X, which means that M X , v |= MGL ψ, for every ψ ∈ X. Hence, X |= MGL ϕ.
Hence, we have proved that MGL is strongly complete with respect to the class of MGL-structures. Propositions 3.5 and 3.10 imply: In what follows, MGL-structures will be referred to as MGL-models. By the size of an MGL-structure M = (U, ∼, •, ⊗, ⊕, D) we will mean the size (cardinality) of its universe U . The cardinality of a set X will be denoted by |X|. Let Sb(ϕ) be the set of all subformulas of a formula ϕ defined recursively as: Clearly, Sb(ϕ) is finite, for all formulas ϕ.
• For all a, b ∈ U , operations ∼ , • , ⊗ , ⊕ are defined as: The structure M defined above is an MGL-model. Indeed, by the definition U , D are non-empty and D ⊆ U . Observe also that U \ {1, 0} ⊆ U and D \ {1} ⊆ D.
Now, we will prove that M satisfies all the other required constraints. First, we prove that a Now, we show that ∼ (a ⊗ ∼ a) ∈ D , for all a ∈ U . Indeed, by the definition of ∼ , either ∼ (a ⊗ ∼ a) = 1 ∈ D or ∼ (a ⊗ ∼ a) ∈ U \ {1, 0}. The latter means that ∼ (a ⊗ ∼ a) = ∼(a ⊗ ∼a). Since M is an MGLmodel, ∼(a ⊗ ∼a) ∈ D, hence ∼ (a ⊗ ∼ a) ∈ D. So then ∼ (a ⊗ ∼ a) ∈ D . Therefore, we have proved that M is an MGL-model.
The above proposition implies the decidability of MGL: Theorem 3.14. The logic MGL is decidable.
Proof. We need to show that given an MGL-formula there is a procedure that decides in finite number of steps whether the formula is MGL-valid or not. Let φ be a given formula and let K be the class of all MGL-models (U, ∼, •, ⊗, ⊕, D) such that U ⊆ {1, 2, . . . , k} for some k ≤ |Sb(φ)| + 2. Clearly, every finite MGL-model of size at most |Sb(φ)| + 2 is isomorphic to an element of the class K. Given M ∈ K, let V M = {v Sb(φ) | v is a valuation on M}.

Conclusion
We have discussed philosophical interpretations of the three Grzegorczyk's logics LD, LE, and LDS. We concluded that these logics do not realize the Extensionality Principle in an adequate intended way, since the readings and interpretation of some of their axioms are obscure and unintuitive. As a remedy a new Grzegorczyk's non-Fregean logic MGL has been proposed. A sound and complete semantics for MGL was constructed and the logic has been proved to be decidable. Finally, we showed that some other crucial non-classical logics are extensions of MGL, so it can be treated as a common base for logics with different languages, semantics, and philosophical constraints. The results presented in the paper lead to some further questions. In particular, as the logic MGL is decidable, it is natural to search for a deduction system which would be a decision procedure for MGL. Furthermore, it would be interesting to complete the picture and determine whether intuitionistic, many-valued, relevant, and other paraconsistent logics are extensions of MGL.