Why do we call logical concepts abstract when logic is specific to the situation?

Question

Things like propositions and predicates, which are specific to certain logics.

Example:

If only classical logic applied, not everything would be possible. (LEM, etc).

Many logicians say classical logic is not universally applicable, and/or are pluralists who say in some situations other logics apply.

But if classical logic prohibits anything being neither true nor false, yet another logic comes in and can handle it, we can say something can be which one or more logics prohibit.

Thus the logic is specific to the situation and there are no universally applicable logics. Or if there are, we certainly haven’t found them.

Am I wrong to think the monicker “abstract” interferes with this? Implicit in abstract is some notion of universalness, generality, timelessness, or acausalness. But no situation is free of specificity, time, and causality. No logic is universally applicable. And nothing is completely generalizable. That is, by omitting certain features we still don’t arrive at something universal. So even that use of abstract fails.

There really seems to be just specific situations we are free to choose which we want to study. I am confused by the uses I see of abstraction and I don’t see why I need it or how it helps. I am not confused by predicates and propositions abstracting natural language. I’ve done it. But that process stops there. We don’t generalize FOL into universal applicance. These are all limited abstractions. I believe (and I could be mistaken) implicit in many uses of abstract, some ultimate notion of applicability or existence outside the finite situation is being assumed. Why else are platonic objects called abstract objects?

Supplemental: It’s almost as if to recommend a specific logic, that is an additional piece of knowledge than the logic itself. But if you know which logic to recommend, you don’t need the logic directly. If I want to know what the tallest mountain on my island is, maybe it’s preposterous to believe someone can know what logic to use without knowing what the tallest mountain is. Just tell them the dang mountain.

It seems like ultimately everything collapses and stagnates into situation-> x. And then there I go again wondering why use abstraction. It’s like every specific question has a specific answer.

Answer 1

Suppose you are a logical pluralist.

Even so, you likely believe that the objects of some domain are best modeled by some logic L1. For example, you may believe that a quantum logic holds for physical reality, that a constructive probabilistic logic holds for epistemology, and that classical logic holds for mathematics.

*note that holds here is not in the logical sense, just a pre-formal notion.

But then, you are still considering each logic to be abstracting away from the particulars in their particular domain of application. hence the name.

Answer 2

So, abstraction is simply a form of generalization, which might be understood as a general and diffuse process of binding collections of things to an identifier. Linguistically, it is when graphemes map to morphemes in a succinct fashion. For instance, 'x' is a letter and can stand for a semantic expression, proposition, or even passage. From WP:

Abstraction in its main sense is a conceptual process wherein general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods. "An abstraction" is the outcome of this process—a concept that acts as a common noun for all subordinate concepts and connects any related concepts as a group, field, or category.

That means, logic is essentially the study of the patterns of inference, and when it comes to formal systems of logic, that generally means focusing on a highly symbolized version of logic where variables, relations, and values are represented with a very bare syntax which is distinct but related to semantics in that it represents the idea economically. This has implications because the human mind is capable of keeping so much in mind at a time, a psychological concept known as chunking. Fewer symbols allows for better retention, and forms the basis for mnemonic strategies. Let's take a simple example:

Socrates is in the kitchen. If Socrates is in the kitchen, we can conclude he is in his house, and likewise, if he is not in the house, we can conclude he is not in the kitchen. If he is in the house, and we know no more, we cannot conclude with certainty he is in the kitchen. He might be in his bedroom. And if he's not in the kitchen, we cannot conclude whether or not he is or is not in the house. This is a very true and meaningful text, and guides our every day thinking, when looking for Socrates, or anyone really.

But now, it just so happens that this can be expressed as the following:

1. p then q
2. ~q then ~p
3. q then p or ~p
4. ~p the q or ~q

The first example of our text is expressed in a very concrete way and in natural language. The second example is expressed in a succinct, abstract way and is the beginning of an artificial language (though languages are usually conceived of as having more complex grammars).

The mystery is why does this work? How is it we can distill natural language to a terse, symbolic form, perform transformations on the expressions, and then arrive at a form that translates back into natural language with an often guaranteed conclusion? And even more importantly, how can we continue to build formal systems that more closely approximate human reasoning?

One way is by observing more semantic features in argumentation and adding operators. For instance, moving from sentential logic to first-order predicate logic, it becomes possible not just to represent propositions with abstractions, but predication within in the proposition. One can now consider elements of linguistic modality of natural language by creating modal logics in the broad sense. And if one observes that mathematical theorization is largely an exercise in construction of language, one again can emulate natural language more fully by dropping some of the classic laws that were once seen as Laws of Thought. Intuitionistic logic is one such program (and my personal bias). As one continues to study examples of inferences, one can see deduction, induction, and abduction, too, as abstractions. And eventually, one gets away from the toy constructions of propositional logic, and has to admit there's a fair amount of defeasible reasoning that occurs in informal logic which tends to be much more domain-specific in terms of inferences. The Toulmin model functions much more in the semantic domain than in the syntactic one.

So, it's best to think of logic as ranging over the study of very natural, informal discourse that humans use and that is related to common-sense (which has serious implications regarding semantic relationships of linguistic categories) to very artificial, formal methods that accommodate description by mathematical functions (which does it's best to ignore semantics). Perhaps one of the most challenging sorts of formal systems are those of model theory which encompass abstract algebraic techniques combined with formal logical methods resulting in ways of coming to conclusions about a collection of structures whose inputs and outputs are the truth conditions. A supreme example of intellectual challenge that results in trying to process abstraction are Kripke semantics.

Answer 3

Abstraction is a matter of degree. Whenever we perform an abstraction, some features are held constant and others are generalised.

For example, in physics, we can speak of the electrostatic force between two particular charged particles. We can then abstract away from these particular particles and write down a general expression for the electrostatic force between any pair of particles, with the magnitude of the charges and the distance between them as variables. We can then abstract further and write some general field equations that apply for any number of particles. What remains constant here are the interpretations of the predicates 'charge', 'distance' and 'force'. What is being generalised is the domain of particles and circumstances in which the predicates apply.

When it comes to logic, we abstract away from the predicates, names, functions, and the domain of quantification, and we keep the logical constants fixed. For a particular logic, such as classical logic, the constants ¬ ∧ ∨ ⊃ ∀ ∃, etc., have a fixed meaning, while the predicates, names and functions require an interpretation in order to have a definite meaning. In this way, logic is sometimes described as being 'topic neutral'. It is not about forces and charges but about any predicates that we choose. This is one way of delineating between what is logic and what is not.

But now comes the kicker. As you point out in your question and comments, there are many logics. If logic is supposed to abstract away from all circumstances, properties, relations, etc., then why are there many of them, and why do some seem to apply in some cases and not others? Isn't logic supposed to be the ultimate level of abstraction?

From a pluralistic perspective about logic, the answer is no, or at least not uniquely. Different logics can be said to have different semantics. Classical logic is naturally understood as the logic of crisp, bivalent truth values, where every sentence is unequivocally either true or false, and where an argument is valid if it preserves truth from the premises to the conclusion. This idea of what is preserved from the premises to the conclusion of a valid argument is a useful way of characterising different logics. Intuitionistic logic can be understood as having the semantics of warranted assertability, where validity preserves assertability from premises to conclusion. Probability logic preserves a high degree of certainty from premises to conclusion. Three valued logics preserve designated values. Relevance logic can be given a natural semantics of information passing, under which a valid argument preserves the integrity of information. Provability logic preserves what is provable within a formal system. Linear logic has several kinds of natural semantics, one of which is the preservation of integrity of resource-bound interaction.

We could proceed to ask, is there a level of abstraction above all of these, a kind of metalogic that subsumes all others? Possibly a version of classical logic could fill this role, if it were fitted out with numerous modal extensions. Though such a claim would be highly contentious.