Online research in philosophy

Inference rule deflationism is the thesis that the nature of truth can be explained in terms of the inference rules governing the word "true". This paper argues, first, that, in light of the semantic paradoxes, the inference rule deflationist must reject some of the classical rules of inference. It is argued, secondly, that inference rule deflationism is incompatible with model theoretic approaches to the definition of logical validity. Here the argument focuses on the question whether the number of primitive referring expressions in a natural language is denumerably infinite. Finally, it is argued that these conclusions pertain to T-schema deflationism and Horwich's minimal theory as well.

Knowledge can be transmitted by a valid deductive inference. If I know that p, and I know that if p then q, then I can infer that q, and I can thereby come to know that q. What feature of a valid deductive inference enables it to transmit knowledge? In some cases, it is a proof of validity that grounds the transmission of knowledge. If the subject can prove that her inference follows a valid rule, then her inference transmits knowledge. However, this only pushes the question back to the inference that was made in this proof. What feature of that inference enables it to transmit knowledge? A vicious regress looms here. Every proof requires a valid inference, and every valid inference must follow at least one rule of inference. So every proof must follow at least one rule of inference. Therefore not every valid inference that transmits knowledge can acquire this power through a proof, on pain of vicious infinite regress. So it must be possible to transmit knowledge by making an inference that follows an underived rule. A deductive inference that follows an underived rule is what I will call a basic deductive inference. It must be possible to transmit knowledge by making a basic deductive inference. But how is this possible? What feature of a basic deductive inference gives it this power to transmit knowledge?

In the nineteenth century, the separation of naturalist or psychological accounts of validity from normative validity came into question. In his 1877 Logical Studies (Logische Studien), Friedrich Albert Lange argues that the basis for necessary inference is demonstration, which takes place by spatially delimiting the extension of concepts using imagined or physical diagrams. These diagrams are signs or indications of concepts' extension, but do not represent their content. Only the inference as a whole captures the objective content of the proof. Thus, Lange argues, the necessity of an inference is independent of psychological accounts of how we grasp the content of a proposition.

1. Transmission Jim’s teacher has just given him his marked maths exam. Jim knows (because he is looking at it) that his mark is 7.25 out of 22. He also knows (because the teacher just said it) that the pass mark is 35%. Does Jim know he has failed? No, he doesn’t. Not yet. As you would expect from his mark, Jim is not very good with numbers. He’ll need a few minutes with pencil and paper to work out that 7.25 is less than 35% of 22. Only then will he know that he has failed. This case exemplifies a common and important phenomenon: someone recognises the validity of an inference from a set of premises that he knows, and in so doing he acquires knowledge of the conclusion. Jim knows that his mark is 7.25 out of 22 and that the passmark is 35%. Then, by virtue of his calculation, he comes to recognise the validity of the inference from these premises to the conclusion that he has failed, thereby coming to know the sad truth. It is undeniable that there are many cases, like Jim’s, in which recognising the validity of an inference from known premises brings about knowledge of the conclusion, as this is the most natural characterisation of what goes on when we acquire knowledge by deductive inference. But can knowledge always be acquired in this way? Does recognition of the validity of an inference from known premises always bring about knowledge of the conclusion?

What is an inference rule? This question does not have a unique answer. One usually finds two distinct standard answers in the literature; validity inference $(\sigma \vdash_\mathrm{v} \varphi$ if for every substitution τ, the validity of τ [σ] entails the validity of τ[φ]), and truth inference $(\sigma \vdash_\mathrm{t} \varphi$ if for every substitution τ, the truth of τ[σ] entails the truth of τ[φ]). In this paper we introduce a general semantic framework that allows us to investigate the notion of inference more carefully. Validity inference and truth inference are in some sense the extremal points in our framework. We investigate the relationship between various types of inference in our general framework, and consider the complexity of deciding if an inference rule is sound, in the context of a number of logics of interest: classical propositional logic, a nonstandard propositional logic, various propositional modal logics, and first-order logic.

This major contribution to the study of F.H. Bradley, the most influential member of the nineteenth century school of British Idealist philosophers, offers a sustained interpretation of his Principles of Logic. After explaining how it is possible for inferences to be valid and yet have conclusions containing new information, James Allard describes how this solution provides a basis for Bradley's metaphysical view that reality is one interconnected experience. In the process he uncovers a new problem as to the nature of truth.