The next two thoughts are motivated by the two complementary aspects of contemporary research in logic, proof theory and model theory. As I try to emphasise to my students, there are two broad ways you can define logical concepts like validity. Following the way of proofs, an argument is valid if there is some proof leading from the premises to the conclusion. Following the way of models, an argument is valid if there is no model in which the premises are true and the conclusion is not. In proof theory, validity is vouchsafed by the existence of something: a proof, which certifies the claim to validity. Invalidity is the absence of such a certificate. In model theory, invalidity is vouchsafed by the existence of something: a model — a counterexample to the claim of validity. Validity is the absence of any such counterexample. It was a great intellectual advance to understand that these are two very different ways to define logical concepts, such as validity, and it was a further advance to be able to rigorously prove that (on certain understandings of logic, such as classical first order predicate logic), these two different kinds of definitions can coincide to determine the same concept. A soundness theorem (relating an account of proofs and a account of models) shows that these two notions don’t clash: you never get both a proof showing that some argument is valid, and a model showing that it is invalid. A completeness theorem (also relating an account of proofs and a account of models) shows that these two notions cover the whole field — for each argument, we either have a proof (showing it is valid) or a counterexample (showing that it isn’t).
Having a sound and complete account of proofs and models for a particular understanding of validity gives you a very powerful toolkit: you can approach a question concerning validity in two distinct ways, by the way of proofs (attempting to build a bridge from the premises to the conclusions, or showing that there isn’t any) or by the way of models (attempting to show that there is a chasm between the premises and the conclusions by showing that there is some way to make the premises true and the conclusions untrue, or again, showing that there isn’t any). These two ways of accounting for validity have very different affordances, they are good for different things, both mathematically or technically, and philosophically or conceptually.
I am particularly interested in the kinds of conceptual gains that are possible when applying notions of proof and notions of model, and the modes of thinking that are involved when using these different tools.
One connection that I am beginning to learn is the intimate connection between proof and necessity, between logical consequence and the hardness and fixity of the logical must. It is one thing to think that an argument is valid, in the sense that it happens to fail to have a counterexample. It is another to have an account of why it is valid. What a proof gives you is some kind of account of how you can get from the premises to the conclusion. This kind of thing is quite powerful, especially given the generality of logical concepts. The power of concepts like conjunction, negation, the quantifiers, etc., (I think) is that our norms and rules for using them apply under the scope of suppositions (whether those suppositions are subjunctive alternatives — suppose that
This brings logic up close to issues in metaphysics, in epistemology and in philosophy of language. In metaphysics, we ask questions about the ultimate nature of reality, and the bounds of what is possible, or what is necessary. Of how reality is and how it must be. The kind of necessary connection between premises and conclusion of a valid argument must bring us up to the boundary of metaphysical necessity. If something is metaphysically possible, then it must count as at least logically possible. If there is a way the world is that makes
Similarly, proofs can also play an epistemic and dialogical role. Provided that you and I agree on the norms governing our logical vocabulary, then if we possess a proof from
I love the way in which the necessity of the logical must brings us right up to concerns of metaphysics and epistemology, of the nature of reality and what options we have as we attempt to understand it.
Necessity is the ninth of twelve things that I love about philosophical logic.
I’m Greg Restall, and this is my personal website. ¶ I am the Shelby Cullom Davis Professor of Philosophy at the University of St Andrews, and the Director of the Arché Philosophical Research Centre for Logic, Language, Metaphysics and Epistemology ¶ I like thinking about – and helping other people think about – logic and philosophy and the many different ways they can inform each other.
To receive updates from this site, subscribe to the RSS feed in your feed reader. Alternatively, follow me at @consequently@hcommons.social, where most updates are posted.