Logical Consequence, Proof Theory, and Model Theory

This article's main concern is the notion of model-theoretic consequence. What does it have to do with correct reasoning? The article takes on deductive consequence only by way of contrast. Do these two notions answer to different intuitive notions of consequence? Is one of them primary, and the other secondary? Or perhaps they are autonomous and independent. Maybe there are two distinct notions of correct reasoning, valid thought, and/or inference. For what it is worth, treatments of mathematical logic usually presuppose that the model-theoretic notion is the primary one. For example, one...

This article's main concern is the notion of model-theoretic consequence. What does it have to do with correct reasoning? The article takes on deductive consequence only by way of contrast. Do these two notions answer to different intuitive notions of consequence? Is one of them primary, and the other secondary? Or perhaps they are autonomous and independent. Maybe there are two distinct notions of correct reasoning, valid thought, and/or inference. For what it is worth, treatments of mathematical logic usually presuppose that the model-theoretic notion is the primary one. For example, one says that a deductive system is sound or complete (or not) for the semantics—not the other way around. If a deductive system is not sound for a given semantics, then that alone disqualifies the deductive system. It is because the deductive system allows us to deduce a falsehood from truths in some interpretation of the language.