Online research in philosophy

I.1. Two reasons for studying inference. Inference is studied for two distinct reasons: for its bearing on justification and for its bearing on learning. By and large, philosophy has focused on the role of inference in justification, leaving its role in learning to psychology and artificial intelligence. This difference of role leads to a difference of conception. An inference based theory of learning does not require a conception of inference according to which a good inference is one that justifies its conclusion, whereas, obviously, an inference based theory of justification does require such a conception.1 Because of its focus on normative issues of justification, philosophy has taken a retrospective approach to inference, whereas a focus on learning naturally leads to a prospective approach. A focus on learning leads us to ask, "Given what is known, what should be inferred? How can what is known lead, via inference, to new knowledge?" A focus on justification has led philosophers to concentrate instead on a retrospective question: "Given a belief, can it be validly inferred from what is known? How can what is known justify, via inference, a new belief?" Thus, for philosophy, inference can be regarded as permissive: one needn't worry about what to infer, only about whether what has been arrived at somehow or other is or can be inferentially justified. A theory of learning, on the other hand, requires a conception of inference that is directive, for the problem of inference based learning is precisely the problem of what to infer.

Poincaré was a persistent critic of logicism. Unlike most critics of logicism, however, he did not focus his attention on the basic laws of the logicists or the question of their genuinely logical status. Instead, he directed his remarks against the place accorded to logical inference in the logicist's conception of mathematical proof. Following Leibniz, traditional logicist dogma (and this is explicit in Frege) has held that reasoning or inference is everywhere the same — that there are no principles of inference specific to a given local topic. Poincaré, a Kantian, disagreed with this. Indeed, he believed that the use of non-logical reasoning was essential to genuinely mathematical reasoning (proof). In this essay, I try to isolate and clarify this idea and to describe the mathematical epistemology which underlies it. Central to this epistemology (which is basically Kantian in orientation, and closely similar to that advocated by Brouwer) is a principle of epistemic conservation which says that knowledge of a given type cannot be extended by means of an inference unless that inference itself constitutes knowledge belonging to the given type.

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.

Our article identifies and describes the metaphoric fallacy to a deductive inference (MFDI) that is an example of incorrect reasoning along the lines of the false analogy fallacy. The MFDI proceeds from informal semantical (metaphorical) claims to a supposedly formally deductive and necessary inference. We charge that such an inference is invalid. We provide three examples of the MFDI to demonstrate the structure of this invalid form of reasoning. Our goal is to contribute to the set of known informal fallacies.

According to the main tradition, knowledge is either direct or indirect: direct when it intuits some perfectly obvious fact of introspection or a priori necessity; indirect when based on deductive proof stemming ultimately from intuited premises. Simple and compelling though it is, this Cartesian conception of knowledge must be surmounted to avoid skepticism. Seeing that the straight and narrow of deductive proof leads nowhere, C. I. Lewis wisely opts for a highroad of probabilistic inference. But how can one arrive at a realm inaccessible through direct knowledge having set out from one thus accessible? How could probabilistic inference offer any help? There are two different answers to these questions in Lewis's writings, and he moves from one to the other under pressure of well known objections from perceptual relativity. Our action divides into three acts, which we review in turn.

Non-monotonic inference is inference that is defeasible: in contrast with deductive inference, the conclusions drawn may be withdrawn in the light of further information, even though all the original premises are retained. Much of our everyday reasoning is like this, and a non-monotonic approach has applications to a number of technical problems in artificial intelligence. Work on formalizing non-monotonic inference has progressed rapidly since its beginnings in the 1970s, and a number of mature theories now exist – the most important being default logic, autoepistemic logic, and circumscription.

Towards the middle of the eighteenth century Hume asked: Why should we accept non-deductive inferences? Strangely enough he didn’t ask the corresponding question: Why should we accept deductive inferences? This was not due to an oversight but rather to the belief, widespread even today, that there is a basic difference between deductive and non-deductive inferences: while non-deductive inferences cannot be justified, deductive inferences can be justified. Though widespread even today, such belief has been challenged by a number of people, from Sextus Empiricus to Lewis Carroll. However, although their arguments raise doubts about the possibility of justifying deductive inferences, many people still believe that, while non-deductive inferences cannot be justified, deductive inferences can be justified. The question of the justification of deductive inferences is all the more important as it is strictly connected with the question: What is a deductive inference? and a non-deductive inference? This paper provides a new answer to these questions.

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?

For some authors, at least in some contexts,1 the distinction between inference and consequence is minimal. An inference can then be regarded as an ordered pair 〈Γ,φ〉, where Γ is a set of sentences or propositions and φ is a sentence or proposition.2 And then an inference 〈Γ,φ〉 can be said to valid just in case φ is a consequence of Γ (analogously for logically valid and logical consequence). For some other authors, the distinction between inference and consequence..

It has been common wisdom for centuries that scientific inference cannot be deductive; if it is inference at all, it must be a distinctive kind of inductive inference. According to demonstrative theories of induction, however, important scientific inferences are not inductive in the sense of requiring ampliative inference rules at all. Rather, they are deductive inferences with sufficiently strong premises. General considerations about inferences suffice to show that there is no difference in justification between an inference construed demonstratively or ampliatively. The inductive risk may be shouldered by premises or rules, but it cannot be shirked. Demonstrative theories of induction might, nevertheless, better describe scientific practice. And there may be good methodological reasons for constructing our inferences one way rather than the other. By exploring the limits of these possible advantages, I argue that scientific inference is neither of essence deductive nor of essence inductive.