Inductive logic

“Sean Gerety said:

Nice blog. Now I see that along with an incoherent and analogous view of truth and a contradictory view of Scripture, Van Tilians do not think asserting the consequent is a fallacy. Quite amazing. Thanks Tavis.”

Nice comment. Now I see that along with resorting to assertions in lieu of arguments, Clarkians don’t know anything about modern logic. Quite amazing. Thanks Sean.

BTW, the name of the commenter to whom Sean refers is not “Tavis,” but Travis.

Since Sean doesn’t believe in sensation, we must attribute his error to occasionalism. I guess that occasionalism isn't such a vast improvement over sensation after all.

This is some of what one professional logician (James Hawthorne) has to say about inductive logic:

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An inductive logic is a system of reasoning that extends deductive logic to less-than-certain inferences. In a valid deductive argument the premises logically entail the conclusion, where such entailment means that the truth of the premises provides a guarantee of the truth of the conclusion. Similarly, in a good inductive argument the premises should provide some degree of support for the conclusion, where such support means that the truth of the premises indicates with some degree of strength that the conclusion is true. Presumably, if the logic of good inductive arguments is to be of any real value, the measure of support it articulates should meet the following condition:

Criterion of Adequacy (CoA):
As evidence accumulates, the degree to which the collection of true evidence statements comes to support a hypothesis, as measured by the logic, should tend to indicate that false hypotheses are probably false and that true hypotheses are probably true.

This article will primarily focus on the kind of the approach to inductive logic most widely studied by philosophers and logicians in recent years. These logics apply classical probability theory to sentences to represent a measure of the degree to which evidence statements support hypotheses. This kind of approach usually draws on Bayes's theorem, which is a theorem of probability theory, to articulate how the implications of hypotheses about evidence claims redound to the credit or discredit of the hypotheses. We will examine the extent to which this kind of logic may pass muster as an adequate logic of evidential support, especially in regard to the testing of scientific hypotheses. In particular, we see how such a logic may be shown to satisfy the Criterion of Adequacy.

http://plato.stanford.edu/entries/logic-inductive/

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And this is some of what another professional logician (Branden Fitelson) has to say about inductive logic:

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The contemporary idea of inductive logic (as a general, logical theory of argument evaluation) did not begin to appear in a mature form until the late 19th and early 20th centuries. Some of the most eloquent articulations of the basic ideas behind inductive logic in this modern sense appear in John Maynard Keynes’ Treatise on Probability. Keynes (1921: 8) describes a “logical relation between two sets of propositions in cases where it is not possible to argue demonstratively from one to another.” Nearly thirty years later, Rudolf Carnap (1950) published his encyclopedic work Logical Foundations of Probability in which he very clearly explicates the idea of an inductive-logical relation called “confirmation” which is a quantitative generalization of deductive entailment. (See also CONFIRMATION THEORY.) The following quote from Carnap (1950) gives some insight into the modern project of inductive logic and its relation to classical deductive logic: Deductive logic may be regarded as the theory of the relation of logical consequence, and inductive logic as the theory of another concept which is likewise objective and logical, viz., ... degree of confirmation. (43) More precisely, the following three fundamental tenets have been accepted by the vast majority of proponents of modern inductive logic

1. Inductive logic should provide a quantitative generalization of (classical) deductive logic. That is, the relations of deductive entailment and deductive refutation should be captured as limiting (extreme) cases with cases of ‘partial entailment’ and ‘partial refutation’ lying somewhere on a continuum (or range) between these extremes.

2. Inductive logic should use probability (in its modern sense) as its central conceptual building block.

3. Inductive logic (i.e., the non-deductive relations between propositions that are characterized by inductive logic) should be objective and logical.

It is often said (e.g., in many contemporary introductory logic texts) that there are two kinds of arguments: deductive and inductive, where the premises of deductive arguments are intended to guarantee the truth of their conclusions, while inductive arguments involve some risk of their conclusions being false even if all of their premises are true (see, e.g., Hurley (2003)). It seems better to say that there is just one kind of argument: an argument is a set of propositions, one of which is the conclusion, the rest premises. There are many ways of evaluating arguments. Deductive logic offers strict, qualitative standards of evaluation—the conclusion either follows from the premises or it does not; whereas, inductive logic provides a finer-grained (and thereby more liberal) quantitative range of evaluation standards for arguments. (One can also define comparative and/or qualitative notions of inductive support or confirmation. Carnap (1950: §8) and Hempel (1945) both provide penetrating discussions of quantitative vs. comparative/qualitative notions of confirmation and/or inductive support. For simplicity, our focus will be on quantitative approaches to inductive logic, but most of the main issues and arguments discussed below can be recast in comparative or qualitative terms.)

Let {P1, ..., Pn} be a finite set of propositions constituting the premises of an (arbitrary) argument, and let C be its conclusion. Deductive logic aims to explicate the concept of validity (i.e., ‘deductive-logical goodness’) of arguments. Inductive logic aims to explicate a quantitative generalization of this deductive concept. This generalization is often called the “inductive strength” of an argument. (Carnap (1950) uses the word “confirmation” here.) Following Carnap, the notation (C, {P1, ..., Pn}) will denote the degree to which {P1, ..., Pn} jointly inductively support (or “confirm”) C. As desideratum (2) indicates, the concept of probability is central to the modern project of inductive logic.

• Inductive Logic, (in PDF), by Branden Fitelson, forthcoming in Philosophy of Science: An Encyclopedia, (J. Pfeifer and S. Sarkar, eds.), Routledge.

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For further online resources, cf.

http://socrates.berkeley.edu/~fitelson/148/syllabus.html

I would warmly invite Sean to strike up an email correspondence with some modern logicians like Dr. Fitelson and Dr. Hawthorne, then post it for the benefit of all Clarkians and Van Tilians alike so that we benighted Van Tilians can see Sean explain to the community of contemporary logicians how inductive logic is fallacious.

I’m sure that Sean won’t disappoint us by failing to rise to the challenge. After all, what does he have to lose?