How do we separate rules of logic from non-logical constraints?

Question

I think that very often the idea of 'constraint' appears in mathematics. For example, when a triangle is considered, 3 points are constrained not to be co-linear, and then we try to discover the implications of this initial constraint, when the rules of logic are applied to it.

Doing this, we discover implications which, although present from the beginning, are not immediately apparent, so essentially we discover what we have already assumed without knowing it.

So in this very general setting, my 2 questions are:

1) What is the most general way to define and separate "the rules of logic" from "the things to which the rules are applied" ?

2) If the above separation can be made, is there some "minimal constraint" on the things upon which logic is applied, enabling us to see the workings of logic in the most pure form, i.e. with minimal inteference with the properties of things upon which logic acts?

For example, suppose we start with the constraint: let S be an object. Nothing else is assumed. Is there something meaningful and "non-trivial" implied by this constraint?

Maybe some of the words used above are not as well-defined as needed, but i leave them unchanged hoping that the direction towards which i'm asking for guidance is not very unclear.

EDIT: (too long for a comment) based on the answer of Mauro ALLEGRANZA. As humans we have the innate ability to separate words and construct sentences with persistent meanings. This seems to me a very great capability, which pre-exists even relative to other very basic "mathematical" ones, e.g. the capability to duplicate things, to count things, to conceive concepts like "for every", to conceive the set of natural numbers...
So is there a universally accepted foundation of logic which clearly separates the ultimate basic concepts from all the rest?
Am i wrong in thinking that all known formalizations address concepts already well-understood relative to the still obscure foundations of logic, (e.g. the border between physical language and natural numbers) resulting in an ever-present foundational mist?

EDIT 2 (motivated by the answer of Bumble): I wonder whether logical constants include (besides concepts as 'and','or','if' etc.) the natural numbers too. They seem absolutely necessary for "primitive" logical processes like the separation of words and the formulation of sentences. If this is true, then the clear separation of "logic" from "objects upon which logic acts" becomes problematic, as natural numbers appear to be some sort of "common ground". Is it the only common ground? Could geometric shapes also exist in logic? (for example, we have the ability to spot "cyclical reasoning". Is this just a play of words, or is it produced by the idea of a cycle, or a closed curve in general?)

Answer 1

If I understand your question correctly, you are asking in effect how do we distinguish logic from non-logic? Logical expressions give rise to valid arguments and logical truths, that is, arguments where if the premises are true it is impossible for the conclusion to be false, and truths such that there is no way for them to come out as false. But this prompts us to enquire what it is about such arguments or sentences that warrants us to say that they determine what is possible, rather than merely what is true.

Consider an example. A simple valid argument is: grass is green and snow is white; therefore grass is green. This argument remains valid whatever we substitute for 'grass is green' or 'snow is white'. We can even emphasise the point by using propositional symbols: P and Q; therefore P. But we cannot substitute the 'and' with something else. It would not do to write: P or Q; therefore P. Our understanding of the validity of the argument depends on holding some features of the sentences constant, while allowing others to be variable. Those features of sentences that need to be held constant to support our understanding of what is valid and what is not are commonly called the logical constants. They include 'and', 'or', 'not', 'if', 'every', 'some', and can be extended to include 'equals', 'is a member of', 'is part of', 'contains', etc.

If we ask what justification there is for considering some terms to be logical constants and not others, then several different answers have been given. One is to say that logic is by its nature formal: it is concerned with separating the form of a proposition from its content. The logical constants are then those terms that serve this formal role. Another approach is rooted in grammar. Some words allow us to build long complex sentences from simple ones in such a way that we can systematically understand the complex sentences in terms of the simple ones. The logical constants are then the connectives that support this function. Another approach is to say that what distinguishes logic from non-logic is that logic is 'topic-neutral', i.e. it is not about any particular subject matter but applies universally. The logical constants are then those terms that allow sentences to remain true under all uniform substitutions of the words that have a subject matter. Another common approach is to introduce the technical concept of 'interpretation', which is a function that assigns referents to names, classes to predicates and truth values to propositions. On this approach, a valid argument is one such that there is no interpretatation under which the premises are true and the conclusion false. The logical constants are those terms that are not a feature of the interpretation function. Another approach emphasises that logical constants are not sensitive to the identity of particular objects. Logical truths are then those that remain true under all permutations of the objects within our domain, and the logical constants are the terms that allow us to express such truths.

There remains an issue that any of the above approaches could be considered question-begging. We still rely upon a fundamental understanding of what things or properties should be treated as variable and what as constant. This makes our collection of logical constants look like a laundry list rather than a principled distinction. Some have pursued the idea of identifying the logical constants by the inferential relationships they warrant. Others have questioned whether logical validity is essentially formal.

Answer 2

The "rules of logic" are the object of study of formal logic and mathematical logic.

They define languages and proof systems, like e.g. predicate calculus, that are "applicable" to any topics whatever.

The so-called "laws of logic" are formulas that are true irrespective of any possible interpretation, i.e. they hold in every interpretation.

In this sense, the laws of "pure" logic are applicable to every domain of discourse consisting of objects whatever.

In addition to the study of "pure" predicate logic, we are interested to add to the "logical engine" suitable non-logical constants, like ∈ ("in"), the binary relation of set theory, or + and × ("plus" and "times"), the basic arithmetical operations, with suitable axioms that govern their behaviour.

Thus, according to the specific mathematical symbols and axioms introduced, we have different mathematical theories; when the collection of axioms is the first-order version of Peano's axioms, we have PA, i.e. first-order theory of arithmetic.

The same for ZF, i.e. Zermelo-Fraenkel Set Theory.

In this sense, first-order theory of arithmetic and set theory are "applications" of logic to specific domains of inquiry (or universes).

We may say that formal mathematical theories are the study of topics where the universe is "constrained" to some specific properties : those defined by the non-logical symbol used (∈, +) and by the specific mathematical axioms postulated for the theory.

Answer 3

Here are the questions:

1. What is the most general way to define and separate "the rules of logic" from "the things to which the rules are applied" ?

The following definition of "theory" from Wikipedia may help clarify the separation:

A theory about a topic is usually a first-order logic together with a specified domain of discourse over which the quantified variables range.... [my emphasis]

The first-order logic would correspond to "the rules of logic"; the domain of discourse to "the things to which the rules are applied".

This Wikipedia article also provides a set of definitions of the rules or concepts used in first-order logic. This could be viewed as a "general way" to do this. Textbooks on deductive logic will also define these (perhaps slightly differently).

Two textbooks from the Open Logic Project provide more depth. The forallx text provides an introduction and The Open Logic Text provides a more advanced presentation.

2. If the above separation can be made, is there some "minimal constraint" on the things upon which logic is applied, enabling us to see the workings of logic in the most pure form, i.e. with minimal interference with the properties of things upon which logic acts?

Assuming that the domain corresponds to the things upon which this deductive logic is applied, the minimal constraint is that these things have to be individuated enough so that they can be members of a domain or set.

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The above applies to deductive logic. There is also non-deductive logic. Wikipedia describes defeasible reasoning as

In logic, defeasible reasoning is a kind of reasoning that is rationally compelling, though not deductively valid.

This is also called "presumptive" reasoning and this is sometimes studied through the concept of "argumentation schemes".

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Wikipedia contributors. (2019, January 28). Defeasible reasoning. In Wikipedia, The Free Encyclopedia. Retrieved 12:41, June 26, 2019, from https://en.wikipedia.org/w/index.php?title=Defeasible_reasoning&oldid=880695250

Wikipedia contributors. (2019, June 16). First-order logic. In Wikipedia, The Free Encyclopedia. Retrieved 12:41, June 26, 2019, from https://en.wikipedia.org/w/index.php?title=First-order_logic&oldid=902047532

Answer 4

1) What is the most general way to define and separate "the rules of logic" from "the things to which the rules are applied" ?

The basic distinction is obtained by the definition of propositions and of the logical operations done on them. "Separation" is achieved in the sense that only the truth of the propositions is relevant to the way the operations work.

2) If the above separation can be made, is there some "minimal constraint" on the things upon which logic is applied, enabling us to see the workings of logic in the most pure form, i.e. with minimal inteference with the properties of things upon which logic acts?

The "things upon which logic is applied" being the truth of the propositions, it is necessary to identify all constraints that may exist between the truth of the different propositions to which logical operations are going to be applied.

This normally requires to spell out the propositions themselves, and to define all concepts that are relevant to their truth. So, to apply logic to the proposition "S is an object", in your example, it would be necessary to define first the concept of "object".

This can only work essentially on abstractions because most real-life situations and macroscopic objects cannot be so exhaustively described. Thus, deductive logic isn't generally conclusive outside of its application to abstractions, some areas of physics and ... personal beliefs.

When this is done, there is no "interference" outside that mediated by the truth of the different propositions involved.

So is there a universally accepted foundation of logic which clearly separates the ultimate basic concepts from all the rest?

The abilities that it is necessary to have to perform the required logical analysis will depend on the particular propositions involved. We can choose the propositions involved so as for example have no need for counting (e.g. A and B implies A). Beside the logically operational semantics of the conjunction, "A and B implies A" only requires to be able to distinguish between "A" and "B". No notion of counting or of number is involved. And, presumably, while counting requires the ability to distinguish, distinguishing does not require counting.

Thus, it is not the logical operations that presuppose counting.

The analysis of propositions is distinct from the logical calculus performed on their truth values. However, the analysis comes before the application of the operations. Any ability necessary to perform the analysis is therefore presupposed. However, it is presupposed by the analysis of the propositions involved, not by the application of logical operations.

Thus, the analysis of propositions involving sets, for example, presupposes an ability to analyse the relations between the kinds of set involved.

And the analysis of propositions involving numbers requires an ability to analyse the relations between the kinds of number involved.

No notion of number is required for "A and B implies A". A conjunction requires the ability to distinguish between a number of conjuncts. It does not involve and therefore require any counting or any notion of number.

Thus, the implication "A1 and A2 and A3 ... and A1000 implies A567" does not require any counting or notion of number. All is required is the semantics of "and" and the ability to distinguish A597 from the other premises.

Thus, where counting is required, it is required by the semantics of the propositions involved.