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There are lots of operators that act on sentences. Here are a few examples:

  • P and Q
  • not P
  • forall x.P
  • necessarily P
  • eventually P
  • x believes that that P
  • it is obligatory that P

etc. The first two examples are truth functional, meaning that all you need to know is the truth value of the arguments and you know the truth value of the expression. The rest are not truth functional. There are lots of logics that deal with specific versions of these operators: predicate logics, modal logics, deontic logics, temporal logics, doxastic logics, etc. Sometimes all of these logics other than predicate logic are grouped under the single name "modal logics".

I've also seen some logics that combine several of these sorts of operators into a single logic. What I haven't seen is any work on discussing how all of these operators would fit together in general. Suppose you had a logic that let you define new sentence operators. What sorts of restrictions, if any, would be needed to ensure that a new operator "plays nice" with existing operators? That is, you could combine them freely and not cause any problems or inconsistencies?

UPDATE:

I think it would help to clarify what I mean by "playing nice". What I had in mind is formal logical properties, not just the intended meanings of the operators. First let me define several terms:

  • A logic is consistent if no sentence is both true and false.
  • A logic is nontrivial if there are both true and false sentences.
  • If S1 is a sentence operator a proposition is "S1-isolated* if it contains no other sentence operators other than S1 and truth-functional sentence operators.

Then what I mean by "playing nice" includes things like the following: Suppose L is a base logic enhanced by unrelated sentence operators S1 and S2 (what I mean by "unrelated" is that there are no axioms containing both of them). Then if L+S1 and L+S2 are both consistent and nontrivial, but L+S1+S2 is not, then S1 and S2 do not play nice together. Also if L+S1+S2 entails S1-isolated propositions that are not entailed by L+S1, then S1 and S2 do not play nice together because S2 has in effect added axioms about S1.

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    The term "fibring logics" is relevant. Commented Oct 22, 2023 at 20:30
  • Doesn't the play-nice clause (subsequently elaborated by you) answer your own question? In addition, you seem to be violating the novacula Occami (pluralitas non est ponenda sine necessitate).
    – Hudjefa
    Commented Oct 23, 2023 at 6:11
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    To nicely combine different modal logics is like to form fiber product (pullback) of different categories in the sense of category theory. Since most modal logics are Boolean lattices with your sentence operator as a (topological open) interior operator in the sense of their algebraic semantics, thus they can be embedded and expressed as some decidable fragment of classic FOL. Then you can further evaluate their consistency or other niceness entirely within the complete FOL as a rough idea… Commented Oct 23, 2023 at 8:02

4 Answers 4

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There isn’t a general procedure to determine the restrictions required for combining certain operators. There isn’t even agreement on how the standard connectives of propositional logic should interact. For example, Intuitionistic Logic interprets propositional operators so that, among other things, a disjunction is provably true just if at least one of the disjuncts is provably true. Other logics put different restrictions for different purposes.

First-Order Logic (FOL) does not play well with modal operators, especially since the Barcan Formula

∀x□A→□∀xA

and its converse are valid in all Kripke frames with the usual semantics. In this case, Free Logic may be used to require that a name for a specific object a must actually exist at a world w in order to perform “Existential-Introduction” and “Universal Elimination” on that name.

The point is that care must be taken in order to preserve the semantics at the level of syntax, and similarly, care must be taken to preserve the intended meaning of the operators in a semantics. So, I’d argue that choosing which restrictions to place on a semantics so that the operators reflect their intended meanings is roughly the same process as choosing the right axioms for a theory.

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To my knowledge, the phrase occurrent in the literature such as covers this topic is "combining logics":

The subject of combinations of logics is still a young topic in contemporary logic. Besides the pure philosophical interest offered by the possibility of defining mixed logic systems in which distinct operators obey logics of different nature, as for instance erotetic logics (the logical analysis of questions) which require combining epistemic and deontic logics, there also exist many pragmatical and methodological reasons for considering combined logics. In fact, the use of formal logic as a tool for knowledge representation in Computer Science frequently requires the integration of several logic systems into a homogeneous environment.

To some extent, and directly or indirectly (given the fluidity of the logic/category-theory distinction), concerns such as over operadic multicategories should overlap/intersect the issue of adding operations from multiple kinds of logic together into a single system. Universal algebra is pertinent, albeit more in the mathematical direction (again, distinctions become blurry as we move to and from e.g. Boolean algebras in our considerations).

Now, I can't say this on grounds of technical certainty, but I would venture to wonder whether a sufficiently complicated logical system can be confirmed noncontradictory or not. Second-order logic is incomplete, for instance, due to meta-issues with Gödel's relevant theorem.


Addendum:

Pursuant to Noah Schweber's comment, here's a link to an article about fibring in logic, the abstract for which reads:

Logics of a combined nature were abundant in the literature when, in the mid 1990s, the study of general mechanisms for combining logics developed into a well-posed research area [5, 6]. Several motivations, both theoretical and applied, concur to justify such a new line of research, but the increasing number of ever more complex logics and logical features appearing in application areas ranging from software engineering to linguistics was certainly amongst the most important. The idea of combining logics had been cooking in a low flame for more than a decade, namely in the particular context of modal logic [32, 63, 64, 74, 50, 29], and within the theory of institutions, with emphasis on equational logic [43, 56]. However, Dov Gabbay's 1996 article Fibred Semantics and the Weaving of Logics -Part I: Modal and Intuitionistic Logics [34], included in this anthology, brought the whole enterprise to a new era. There are two principal reasons to justify the impact of the paper. First, among various application examples, the paper put forward the notions underlying the general mechanism for combining logics known as fibring. Reinterpreting the original phrasing, the general problem of fibring two logics L 1 and L 2 is: (P0) Characterize the logics L built over the combined language that conser-vatively extend the two, and in particular the minimal such logic L 1 * L 2 . Perhaps even more importantly, the paper also clearly outlined for the first time the main objectives and subproblems that should guide a systematic study of the general problem, namely: (P1) Characterize the notion of a logical system. (P2) Present methodologies for combining any two logics. (P3) Investigate transfer properties. (P4) Compare the combined logics obtained with known logics. (P5) Study possible interactions between the logics being combined. * This work was partially supported by FCT and EU FEDER, namely via the KLog project PTDC/MAT/68723/2006 of SQIG-IT. The authors are indebted to many colleagues with whom they have collaborated, over the years, on the topic of combining logics.


Recent updates to the SEP: today (12/11/2023) the SEP article on philosophical aspects of multi-modal logic was updated. I'm in the process of reading it now, but I will quote from the opening epigraph to exemplify its resonance with your inquiry:

Here is what I consider one of the biggest mistakes of all in modal logic: concentration on a system with just one modal operator. The only way to have any philosophically significant results in deontic logic or epistemic logic is to combine these operators with: tense operators (otherwise how can you formulate principles of change?); the logical operators (otherwise how can you compare the relative with the absolute?); the operators like historical or physical necessity (otherwise how can you relate the agent to his environment?); and so on and so on. —Dana Scott (1970: 161)

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  • +1 "I would venture to wonder whether a sufficiently complicated logical system can be confirmed noncontradictory or not." My money is on not which is why Quine's underdetermination theses are so challenging, and why he gravitated towards "web of belief" and which gives the wiggle room to deal with contradiction in an ad hoc manner as it arises when reconciling theoretical constructs. So, in an intuistionist type theory, for instance, judgement of types, the categories of the foundational substrate of the system, are rife for contradiction because the adoption of ontological categories...
    – J D
    Commented Oct 22, 2023 at 20:36
  • That is, categories with semantic content, start across separate domains of discourse and then generate contradictions that stem from the basic nature of reconciling types during relations such as reduction, identity, and generalization which are free of the logic proper. Neurath's boat is a metaphor for spending one's life drilling down into the contradictions of categorization, more than the logical rule set which governs the categories proper.
    – J D
    Commented Oct 22, 2023 at 20:37
  • I guess you could say that any sufficiently complicated logic, one built on natural language semantics, such as Montague's, or Ranta's type-theoretic grammar, are an empirical exercise because the categorical structure of categories themselves expressed in natural language semantics is unavoidably idiolectic on the level of experience. This would seem consistent with why there are no private languages. Languages are conventions developed to negotiate experiences expressed as ontologies and function as a social mechanism of Quinean semantic ascent.
    – J D
    Commented Oct 22, 2023 at 20:43
  • Why negotiate the language of reality from the ground up, if you can simply adopt wholesale the system your society already uses? Anyone is free to add, substract, or play with that system as they see fit, and that is how we can account for the diachronic transformation of language over time.
    – J D
    Commented Oct 22, 2023 at 20:44
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What sorts of restrictions, if any, would be needed to ensure that a new operator "plays nice" with existing operators? That is, you could combine them freely and not cause any problems or inconsistencies?

Strictly speaking, when you are talking about designing any sort of formal logic, you typically are expressing an aspect of natural language that the designer wants to express. For instance, modal logic expresses epistemic modality present in language, sentential logic captures the logic that governs the interaction of propositions given classical logical presumptions, predicate logic the logic of relationships in sentences in phrasal syntax, etc. Therefore, the constraints placed on a logic are teleological in nature.

What is important to note is that logics conform to a grammar in the way that formal languages and automata define them, as a series of transformation rules that move from initial symbols to terminating symbols using an alphabet in a formal system, and the transformational power of any logic conforms to the general mechanics of a lambda calculus which involves three acts: transformation, definition, and application (the three axioms of the calculus). So, the constraints on a logic must include compliance with the general form of any logic as expressed by the axioms of the lambda calculus.

But the answer to your question isn't found in the grammar of the logics proper, rather the goals of the designer of the grammar. When George Boole created Boolean logic in 1854, he was trying to create a grammar that caught truth conditional semantics that captured the essence of the Laws of Thought. Brouwer's constructivist logic sought to weaken those constraints by modeling natural language use that ignores LEM and DN, a fact that allowed Martin-Löf to extend Russell's simple type theory. And in fact, "avoiding inconsistencies" in terms of truth-semantics itself isn't even necessarily something one has to do when one designs a logic, since paraconsistent logic shows that a logic can capture the essence of logical inconsistency itself to show how a language can tolerate contradiction.

Therefore, the design of a logic and the playing nice of operators is really is a function of the intent of the designer, and the transformation rules abstracted from natural language incorporated into the logic through grammatical codification which are subject to the whim and utility of the inventor of the logic. Historically, the constraints were to show how propositions and then predicates related according to the Laws of thought, but logics have moved to embracing more sophisticated semantics that we find in natural language, a fact noted by Montague post-UG when he built his grammar which attempted to capture the essence of "material logic" which has been borrowed by software engineers in their more sophisticated NLP systems.

EDIT for explication of the conclusion of the first half:

A "general sentence operator" is just a syntactic unit in a formal grammar to express an action semantics that inheres to natural language. If you are looking for understanding how to reconcile two operators, the constraints are in the coherence of the action semantics, not in the syntax, and the semantics exist in a possible world as extensive as the natural language semanatics which exist in the mind of the engineer of the logic.

In other words, as PW_426 notes "There isn’t a general procedure to determine the restrictions required for combining certain operators." The operator is an expression of the logic that inheres to the metalinguistic description of the logic system as expressed in natural language. Therefore to get two operators to "play nice" merely means that they express a coherent semantics in the metalinguistic framework of the logic, the... this-is-what-those-symbols-mean-ness.

Exemplification is clarification. Consider the box operator. What is it's relation to a negation operator? How do they play nice in regards to being a constituent or a modifier of a proposition. It has nothing to do with the formal logic, and everything to do with the natural language semantics of "necessity" and "negation". In fact, they are distinct concepts, therefore they play nicely because they are complementary. The inclusion of a diamond operator in a logic dovetails nicely with the box operator because "possibility" and "necessity" have a relationship in natural language.

What is implicit here is that all operators in all formal languages express an aspect of the logic that inheres in natural language using the formal grammar as an abstraction, and the lambda calculus as the rules for the formal system. Coherence of "general sentence operators" exists as a function of the coherenece of the natural language description of them, and is not subject to objective and external constraints related to the syntax or the logic proper.

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    I don't see how this addresses my question at all. Commented Oct 22, 2023 at 16:54
  • I've made the thesis more explicit in an edit.
    – J D
    Commented Oct 22, 2023 at 20:29
  • OK, I think I get the disconnect now. It was that i did not explain clearly what I meant by "playing nice". I've updated the question to clarify. Commented Oct 22, 2023 at 21:39
  • I'm with Agent Smith that your question as now posed answers the question (whether well or not is unknown to me). By offering your precising definition, you have changed the question to What formal properties of logical systems can be used to determine if the combination of logical operators? which is less ambiguous than How can I make logical operators "play nice"? But at the same time, you provide three axioms about formal properties that are the criteria you ask after which seem prima facie quite reasonable by suggesting that incompatible operators result in a fundamental change to the...
    – J D
    Commented Oct 23, 2023 at 9:40
  • necessary properties of some adequate notion of logic itself. I do have concerns that your criteria would necessarily exclude paraconsistency as a property of logical systems, and I wonder if truth without falsity is meaningful. (Are there monovalent logics, and if it is so, is it possible to transform a bivalent logic into a monovalent logic?)
    – J D
    Commented Oct 23, 2023 at 9:43
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I'm going to file a second response since you have offered a precising definition of "playing nice" which now changes the nature of the question substantially by presuming that formal logical properties are indeed a way of providing a rational and formal basis for appraising the compatibility of the products of combining logics (SEP). Obviously, a logician like Bumble would likely have an answer far more authoritative than what I offer, but I'm not sure the formal properties you list are either adequate or meaningful. For instance, requiring consistency of a product of combination seems to rule out the use of paraconsistent logics entirely at first glance.

That said, however, I think you have proposed the properties with an implicit question mark indicating your uncertainty, and that edit wasn't intended to answer your own question so much as exemplify the type of desired response. Therefore, I think the strategy involved is much broader an effort and seems to require a formal metalanguage to describe the two object languages of the objects logics to be combined and then use properties of the metalanguage to assess the results of the combination. That suggested there is a logical evaluation system, and low behold there is an article called logical framework which claims just such a thing. From WP:

A logical framework is based on a general treatment of syntax, rules and proofs by means of a dependently typed lambda calculus. Syntax is treated in a style similar to, but more general than Per Martin-Löf's system of arities.

and

In the case of the LF logical framework, the meta-language is the λΠ-calculus. This is a system of first-order dependent function types which are related by the propositions as types principle to first-order minimal logic. The key features of the λΠ-calculus are that it consists of entities of three levels: objects, types and kinds (or type classes, or families of types).

Since CMU, no small fish, has a project that implements this theory, taking a look at it might help to resolve additional questions you have about combining logics which seem to be treated as first-class citizens of the language. This is consistent with the notion that assigned types in intuitionist type theory can themselves grounded in logical processes and logical objects.

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    @KristianBerrry Oh, and given your preternatural hunger for logical grammars and an understanding of their interrelation, LF and it's lamda-pi calculus as a metalanguage for logical combinations probably would interest you.
    – J D
    Commented Oct 23, 2023 at 11:55
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    Oh boy, I still gotta learn the original lambda calculus itself :P I'm so used to the syntax they use in set theory (which gets mixed up with a lot of English syntax once they start floating off into large-cardinal land) that type theory has been an extra challenge (it seems more precise/compact). I've been getting the hang of some proof-theory notation, though. Commented Oct 23, 2023 at 18:31

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