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An inferential chain is a series of inferences where each depends on the previous in sequence. "From A we conclude B, from B we conclude C, and from C we conclude D." That would be a chain of length three, as it has three inferences.

In contrast, "A supports D, B supports D, and C also supports D" is not a length 3 inferential chain, because the order of the three inferences can be rearranged without altering the argument. The longest inferential chain in that argument is only length 1.

To be more specific, we can consider an argument as a directed acyclic graph from premises to intermediate conclusions towards final conclusions, with an arrow from each justifying statement pointing towards the statement it justifies. An inferential chain is a directed path in this graph.

In mathematics we do have long inferential chains. Mathematical proofs may run into the hundreds of pages, and the longest inferential chain through such a proof would also be very long. This is possible because each step of mathematical reasoning is very certain. So we can put many links in the chain with a low risk of it breaking. (Still, very long and complicated mathematical proofs are not infrequently found to have flaws.)

In ordinary common-sense reasoning, inferential chains tend to be much shorter. "It's raining today, so I'll bring an umbrella," has only length 1. One good explanation for why common-sense inferential chains are short, is that common-sense inferences are uncertain. If each inference is only 90% sure, then if you chain 10 of them together, the chain is only 0.9^10 = 35% likely to be intact without broken links. So in ordinary common-sense reasoning we prefer to build broad, rather than deep, arguments. We prefer to consider the pros and cons that add or subtract directly from a claim, rather than extrapolating too deeply in too many steps from the available evidence.

In fact, often, common-sense arguments are neither deep nor broad; people find it difficult to subjectively weigh too many pros and cons, especially once there are more pros and cons than will fit in short-term memory. And so we often fall back on picking out one or a few pieces of evidence as most important, allowing ourselves to ignore or override lesser evidence.

How can we characterize the inferential chains used in philosophy? Are they long, like mathematics, or short, like common-sense reasoning?

It's my feeling that legitimate inferential chains in philosophy are short. Because in philosophy (except for logic) we don't have the luxury of reasoning with perfect certainty, and rather must appeal to intuition and analogy most of the time, long inferential chains have a high chance of being flawed. The situation is more like common-sense reasoning than mathematics.

Philosophical papers are often very long. But famous specific arguments in philosophy can usually be summarized within just a few paragraphs. Without the luxury of logical certainty, an excessively long argument becomes unwieldy and unsure, and we are unable to fit the whole thing in our heads or reason effectively about it. The length of philosophical papers often comes from comparisons to other works and examples or hypotheticals, rather than from having a long central argument. Is this perception of the state of affairs correct?

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  • For sure, very long arguments... IMO very seldom they can be "formalized" as strict inferential chains. Oct 24, 2023 at 7:38
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    The shortness of common sense is deceptive. Much of the length is folded into premises taken for granted, and the omitted arguments for them replaced with intuitions and unanalyzed surmises from shared background. Another difference with mathematics is that the reasoning is heuristic/abductive, not formal, hence not as sensitive to long chain errors. Philosophy gets to unfold all of that and weave in scientific and other sources on top of it. The "famous specific arguments" are just toy models for novices with the substance likewise folded into premises, which are what really is argued for.
    – Conifold
    Oct 24, 2023 at 9:50
  • @Conifold I would rather say that the premises in common sense are not arrived at by logical means at all. We've tried to mimic common sense via logical deduction, e.g. Cyc, and it does not work well. GPT-3 works much better for common sense than Cyc, and relies on pattern recognition rather than long chains of logic. Why do you say heuristic/abductive reasoning would be less sensitive to long chain errors? There don't seem to be long inferential chains in science - except where the foundational elements are subject to exact mathematics, e.g. chemistry, physics.
    – causative
    Oct 24, 2023 at 13:20
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    I don't see how "common-sense" inference chains are even inference chains. If they are analogistic or probabilistic- then how are they a "series of inferences where one follows the other" .... it's apples to oranges. Calling them inference chains is a mistake. IMO there are practically no inference chains in common parlance or non formal logic based philosophy. However, beyond that caveat which is easily fixed by calling them informal inference chains, I agree with your assessment. The length comes from trying to justify the jumps in the informal inference chain, if the inference chain Oct 24, 2023 at 14:38
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    I guess we have a different idea of what an argument is, discursions, clarifications, and examples are an essential part of informal arguments, as I understand it. Deductive chains of formal mathematics are just a degenerate special case and a poor model for logic in the full sense, which is why Cyc-style theorem provers are lousy. This is Toulmin's view in The Uses of Argument, for example. Informal arguments are less error-prone exactly because they do not rely on checking elementary steps, the same way mathematicians work in practice.
    – Conifold
    Oct 24, 2023 at 21:04

5 Answers 5

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My recollection of books, papers and talks on philosophy is that very few of them reason step by step through a long chain of reasoning to reach a remote conclusion. It would be interesting, IMHO, to see an analysis of philosophical texts broken down by the amount of space they dedicate to different facets of a case, eg background, rubbishing, sorry, assessing the works of other philosophers, defining terms, presenting examples, reaching interim conclusions to form the foundation for the next step, and so on. The point about common-sense reasoning is that it can be concise because it takes so much for granted. If you start to unpick the implicit premises underlying much common-sense reasoning, you quickly get into a mire of ambiguity and unsubstantiated assumptions. In any event, much of philosophy- like much of other academic work that is inherently not mathematical- is not about chains of reasoning, per se, but about promoting a particular perspective on a topic, or a particular schema for breaking a complicated subject down into more digestible chunks. Take, for example, literary analysis, and the different flavours of it that have risen and fallen in influence over the last hundred and fifty years. How they differ is mainly in perspective and focus- where one focuses on the purported meaning of the author, another focuses on meaning as a creation of the reader. You cannot say that one is more 'correct' than the other, that one has done a better job of taking logical steps- all you can really do is decide whether one has taken you along a more informative route to a more advantageous vantage point for overlooking the subject matter.

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Three major examples of philosophical treatises—Plato's Republic, Immanuel Kant's Critique of Pure Reason, and John Rawls' A Theory of Justice—all involve fairly long chains of reasoning, principally deductive but admixed with an implicit appeal to what Rawls identified as reflective equilibrium (a more coherentistic function, in counterpoint to the implicit foundationalism of well-founded deductive trees). Of those three examples, Plato's is the most free-form, being a dialogue after all, but both on its own terms and in relation to the extended Platonic corpus, it expresses Plato's interest in inferential understanding and apprehension by the by.

Now, Rawls in his book says that (moral) philosophy should aim for all the precision/rigor of geometry; he weaves simple, if perspicuous, geometrical illustrations into his theory (the curves in §§12-13), works out the premises from which the two principles of justice can be derived, and then reinforces those premises and principles in Part 3 by further combined appeals to both strait deduction and more Quinean web-of-belief considerations. Altogether, he does say (§87) that he has not actually used a purely, rigidly deductive method, neither only a more "naturalistic" one, but per his overarching inclusivism he has brought in whatever reasoning both intuition and intellection support as resonant with the goal of moral philosophy (which he says is, among other things, to find possibilities of agreement where none are believed to exist).

Next, consider other seminal (or at least naively "canonical") works such as Descartes' Meditations on First Philosophy. If one arranged such a text as numbered sentences, one would have a document with as many steps as many a solid mathematical proof (c.f. Rules for the Direction of the Mind). So too does Wittgenstein's Tractatus Logico-Philosophicus proceed in a fairly regimented fashion, albeit with a quixotic outcome that presages Wittgenstein's eventual amorphism of thought.

Finally (here), consider Kripke's writings on modal logic, or John Duns Scotus' argument for the existence of God. That latter is notoriously intricate, and though not as epic as, say, Wiles' proof of Fermat's Last Theorem, it still represents an example of where philosophically informed reasoning can be rigorously deductive (of course, the premises are not so stable, it turns out, but neither are those of set theory).

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  • Well, you've offered a challenge. I suppose the only way to settle it is to draw a complete argument map of one of those works. You say Scotus' work is "notoriously intricate" and it does have a clear single conclusion (that God exists); would you say that is the best candidate to be diagrammed?
    – causative
    Oct 25, 2023 at 16:01
  • @causative probably the easiest to extract from the original text (Scotus' nickname was "the Subtle Doctor"), but from Rawls' text would be the most rewarding and it has a strong index (IIRC he and his wife had a passion for indexes) that would help a lot in the attempt. Oct 25, 2023 at 16:23
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If one takes the formalist approach to be an abstraction of the use of conditionals in natural language and a chain is two or more links, then good philosophy often proceeds along chains of reasoning. To see the form of such natural language argumentation takes, see Toulmin's method of using a warrant, rebuttal, and so forth to arrive at a conclusion for inference. The nice thing about an axiomatic approach is that is foundationalist, and if formal language is used instead of natural language, much simpler to see. That being said, however, a formalist presentation of inference can be viewed through the Tarskian lens of meta and object languages such that for every meaningful formalism as an object language, there must a natural language metalinguistic complement that makes the symbols meaningful. Through that reasoning, any formal language chain of inferences has a natural language counterpart. Whether or not a philosopher is systematic in their reasoning and the construction of their language determines whether or not seeing the inference is easy. Some philosophers like Descartes are very methodical precisely because they attempt to imitate Euclid; among modern philosophers, some philosophers like Searle are appreciably observant of keeping the presentation simple and building small chains that interconnect, and others, like Dennett, for instance, take a much more verbose and meandering approach to presenting their text. The Oxfordian Nicholas Shea in his Representations in Cognitive Science (GB, the entire text is available through the site for free) actually has an appendix where he presents his whole book with a summary that would allow you to diagram the inferences rather easily. That sort of courtesy, however, is extremely rarely given to the reader.

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Philosophy isn't (generally) linear or teleological. It's more at hermeneutics: a system of interpretation and analysis that weaves back and forth within a conceptual domain to create a comprehensive and self-consistent understanding. Sometimes inferential chains are useful to get to specific points; inferential chains are linear and teleological, so appropriate to show specific relations. But the endpoint of an inferential chain in philosophy isn't an end in itself, but something that must be put in relation to other points to build a bigger picture.

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Does philosophy involve long inferential chains?

The Cogito?

I think; therefore, I am.

Seems very short to me.

What characterises at least Western philosophy is it analytical nature. You could say that Western philosophy is analytical hubris: All problems should be solvable through the rational analysis of common sense notions.

This is what Western philosophers have tried to achieve. They failed, most notably because there is still today no rational analysis of the Liar Paradox, 2,500 years after it was first minted.

Still, analysis seems to provide for long inferential chains, but this is nothing specific to philosophy. Mathematics and all sciences are bound to include long inferential chains as well.

Analysis is also present in mathematics. The difference is that mathematics does not purport to describe reality. Mathematicians can start from whatever system of axioms they please, irrespective of its potential denoting value.

Analysis is present in science as well and it is of course crucial to the scientific method that theories should be meant to refer to the real world by grounding its analyses on empirical data.

Philosophy is also meant to refer to the real world, although one might be sceptical of particular philosophies, this is nonetheless always the intention. Now the distinction between philosophy and science is that philosophy starts from common sense notions rather than empirical data, and is ordinarily conducted using some natural language, often extended with plenty neologisms, while science may start with natural language but will hopefully progress towards a mathematical expression.

So, philosophy may be characterised as analytical hubris conducted in some natural language with the intention of describing some real aspect of the real world by starting from common-sense notions.

Incidentally, rational analysis is also what distinguishes philosophy and science from religion. Most religions are not rational and not even analytical, but Catholic theology at least is analytical. The Catholic Church is the only institution to have ever tried to apply logic in a systematic and methodological way to its fundamental assumptions about the "real" world.

The difference, however, is that even the Catholic theology doesn't start from empirical evidence, like science does, or from common sense notions, like philosophy does.

Bear in mind that the Cogito is probably the most recognised of all philosophical wisdoms ever.

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