Is truth-in-a-model understood as consistency-of-discharged-antecedents?

Question

I was trying to understand the thing about connexive logic, the fundamental ideas that ~(A then ~A) and ~(~A then A), or Boethius' playing of the theme, and so on. (One case I couldn't accept: "If B implies A, then ~B doesn't imply A": but so why can't one conclusion be supported by multiple/separate premises?) Mostly: why is this not given in classical logic? Is it because of stuff about truth-tables? I'm assuming so, per David Gudeman's answer to my previous question in this connection.

However, the reasoning from ~(A & A) to a contradiction, at least via the more complicated premise (A & ~(A & A)), seems reasonable to me... So likewise, I would think:

1. Assume that if A, then ~A.
2. Assume A.
3. Then ~A.^A
4. (Conjunction introduction, {2,3}) A & ~A.^B
5. But ~(A & ~A).
6. Therefore, ~A.

I get that, like with A = (A & A), if we set A = (A then A), then denying (A then A) reduces to denying A. And concluding ~A doesn't move us to concluding A also, afterwards, here.

But so then is that what they do in model theory? Because I could not for the life of me really grasp what this talk of "truth in a model" was meant to be. There was some recognizable, appreciable invocation of topics like "reference" and "quantification" and so on, of course, but not enough for me to know why a theory could have multiple models in the intended sense. Is what they do a matter of starting from sets of conditionals, maybe existential conditionals even (or is there even an option?), and then seeing if they can get a contradiction from assuming all the antecedents of those conditionals? So that truth-in-a-model is not identical to the structural truth afforded by the logic. And then this is how the matter of connexive logic is indirectly rejoined in this context? (As if to say that Aristotle's/Boethius' formulae are justifiable with respect to explaining model theory?)

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^AIn symbolic logic, we would infer just A, here, not "then A." And even in ordinary language, it is a little "superfluous" to distinguish between, "If A, then B," and, "If A, B" (e.g. "If the cow jumped over the moon, then the square root of this potato is orange," and, "If the cow jumped over the moon, the square root of this potato is orange"). However, it is much less clear, in ordinary language, that two sentences are being conditionally juxtaposed when the antecedent does not have at least an "if" prefacing it. Accordingly, there is some support, in ordinary language, for the idea that part of the conditional action in logic is mediated by both an inter-sentential connective (like the arrow) as well as a unary connective at least attaching to the antecedent, if not also the consequent (as in inferring "then A" vs. just "A").

^BOr should it be "A and then ~A," and how does this not transmute the proposition into a sort of "demi-conditional"? Or: although A and ~A be contradictory, are A and (then ~A) contradictory? But so see the preceding endnote.

Answer 1

As you say, in connexive logic, ¬(A → ¬A) and ¬(¬A → A) are both theorems. They are not theorems of classical logic, using the material conditional, since the first is logically equivalent to A and hence is false when ¬A is true, while the second is logically equivalent to ¬A and so is false when A is true.

These two logics give a different sense to the conditional and this is why they have different theorems. In connexive logic, the sense of "if A then not-A" that Aristotle objects to seems to be something like: I can never start from the premise that A is true and reach the conclusion that A is false without inconsistency. In classical logic, the conditional somewhat surprisingly has a kind of were/would sense. "If A then not-A" means that if A were true then A would be false, so A must be false. Or alternatively: On the supposition that A is true, A is also false, but since it cannot be both, the supposition must be rejected and so A is false.

As you mention, connexive logic does not allow the inference: "if A then B; if not-A then B; therefore B". Also, the logic of connexive conditionals is complicated by the fact that you cannot have transitivity of entailment and uniform substitutivity without inconsistency. Those are rather nice features of classical logic, so it is understandable that we don't want to abandon them without good reason.

Incidentally, I have given a rather longer answer to a question about "if A then not A" that covers more logics.

Your derivation of ¬A from "if A then ¬A" is correct. But it is not correct that denying "if A then A" reduces to not-A. "If A then A" is a tautology in nearly every logic I can think of, so denying it yields a contradiction.

Model theory provides another way of approaching logic using the concept of truth under an interpretation. Model theory runs in parallel with proof theory and we use soundness and completeness metatheorems to link them. The model theory of classical logic is nice and simple. The model theory for connexive logic is more complex and is often handled using Routley-Meyer semantics, similarly to relevance logic. There is some material on connexive model theory in sections 4.2 to 4.5 in the Stanford Encyclopedia article on connexive logic.

Answer 2

You ask:

But so then is that what they do in model theory? Because I could not for the life of me really grasp what this talk of "truth in a model" was meant to be.

Modern logic systems are conducted in the abstraction of formal systems which use the notion of formal languages. Being highly abstract collections of symbols that denote a syntax, it is necessary to understand how the syntax is used, otherwise known as the formal semantics of the system. There are different approaches to interpreting the various grammars of those syntax, of which the first and most dominant of those is Tarski's model theory. WP's article on formal semantics provides a summary:

The archetype of model-theoretic semantics is Alfred Tarski's semantic theory of truth, based on his T-schema, and is one of the founding concepts of model theory. This is the most widespread approach, and is based on the idea that the meaning of the various parts of the propositions are given by the possible ways we can give a recursively specified group of interpretation functions from them to some predefined mathematical domains: an interpretation of first-order predicate logic is given by a mapping from terms to a universe of individuals, and a mapping from propositions to the truth values "true" and "false". Model-theoretic semantics provides the foundations for an approach to the theory of meaning known as truth-conditional semantics...

In a formal system, there is an alphabet, WFFs, sets of axioms, rules of inference, and theorems, but it is possible, using for instance the lambda calculus as a notation, to perform a series of inferences where the symbols are devoid of any meaning and are simply transformations based on the "shape" or "form" of the inputs and outputs. Lambda calculus has application, alpha conversion, beta reduction and so on, so that the inputs are processed into outputs simply by pairing up the formal symbols. Computer scientists, for instance, make their bread and butter by building and running compilers and interpreters which execute these sorts of formalisms.

But a model is more than a set of symbols because those symbols and the grammars in which they function in have meaning, its formal semantics. Meaning is provided by an interpretation function mapping those symbols to other symbols. It's a formal system with certain properties that lend it to describing either an intensional logic say of a possible world semantics over individual instances in a domain of discourse, or a mathematical structure which requires an interpretation function that maps meanings of symbols to more explicit computational definitions.

For instance, to be a mathematical structure, there has to be an interpretation (which naturally is expressed as a function!). From WP:

Formally, a structure can be defined as a triple A'=(A,σ,I) consisting of a domain [of discourse] A', a signature σ, and an interpretation function I that indicates how the signature is to beinterpreted in the model.

The example given is understood by the differing of interpretations of a field (in the group-ring-field sense) over reals and a field over the complex numbers. Intuitively, you can sense that there must be a difference of meaning in symbols like '0', '1', '+', and '×' since the notions of identity and operations must differ whether one is dealing with reals or complex numbers. To interpret the mathematical structure, one simply has to apply different definitions of the additive and multiplicative identity or addition or multiplication. Thus, there is a difference in structural interpretation of (R,+,×) and (C,+,×) even though the two operations specified are the same + and ×.

In the second case, the one to which you refer, the model theoretic interpretations is about showing that the model has different interpretations under a differing set of premises of proofs so that we can show that a full range of proofs might hold in some domain of discourse or there are invariant properties of some sort in the logic system. Mauro provides an example from geometry:

In a non-technical but precise sense "true-in-a-model" means that the logical consequences (theorems) of the axioms of a theory hold (a satisfied) in every mathematical structure that satisfies the axioms (a model of the theory). Example: if we omit the parallel postulate from the set of axioms of usual Euclidean geometry we have that the sphere is a model of the (restricted) theory.

So, if we treat axioms as true or false in our model of geometry, and we let go of the parallel postulate (it is not true), then we have a different mathematical structure, but can show that many of the consequences of the remaining axioms are preserved. Spherical, planar, and hyperbolic geometries have many similarities, but they differ in their truths with respect to what they model. That a triangle has three angles which can be used to construct a straight angle only is a consequence or theorem in one of the cases, for instance. This theorem is only true of triangles on a Euclidean plane. And yet other Euclidean truths hold outside of the Euclidean plane.

More succinctly by WP:

In model theory, interpretation of a structure M in another structure N (typically of a different signature) is a technical notion that approximates the idea of representing M inside N. For example, every reduction or definitional expansion of a structure N has an interpretation in N... Many model-theoretic properties are preserved under interpretability. For example, if the theory of N is stable and M is interpretable in N, then the theory of M is also stable.

Thus, a model in this sense is a generalization of geometry over a set of axioms that we can vary and examine both individually and collectively and observe results. M_E, M_SPH, and M_HYP are all part of a greater structure N or, if you prefer, G. In one world, the parallel postulate takes form PP_E, in another form PP_SPH, and in the last, form PP_HYP. Thus, we have the truth of the parallel postulate in its respective model because we can vary the truth with respect to different models. That is, in one possible world, parallel lines are Euclidean, in two possible worlds, they are not.