The joy of subsumption

December 16, 2006 | By | 2 Replies More

All my life I’ve been suspicious of the alleged power of syllogisms.  Here is an often-cited syllogism:

• All men are mortal
• Socrates is a man
• Therefore Socrates is mortal

Syllogisms can be expressed in this logical form:

• All B’s are C
• A is a B
• Therefore, A is C

The above example is a perfect syllogism: the conclusion naturally follows from the premises.  Syllogisms constitute deductive reasoning (from a given set of premises the conclusion must follow).

Many excellent thinkers and writers have stressed the need to present one’s arguments in terms of syllogisms.   For example, in his excellent book on legal writing, The Winning Brief, Bryan Garner advises lawyers to frame every legal issue as a syllogism (see p. 88).

But what is really behind the power of syllogisms?   It turns out that they are actually based on a metaphor—the metaphor of objects in a box.  Consider this diagram in tandem with the classic syllogism (“All men are mortal; Socrates is a man; therefore, Socrates is mortal”)


As Judge Posner points out (in “The Jurisprudence of Skepticism,” 86 Mich L. Rev 827, 830 (1988)), the first premise presents a box labeled “all men,” in which each of the contents are each labeled “mortal” and one of those objects is labeled “Socrates.”  Posner notes,

“The second premise tells us that everything in the box is tagged with a name and that one of the tags says “Socrates.”  When we pluck Socrates out of the box we know that he is mortal because the only things in the box labeled “all men” are mortals.  Notice that we find the syllogism compelling by virtue of a metaphor, the metaphor of the box. (odd that one should “prove” the truth of logic by a metaphor!)”

Yes, how odd that we should find this device so very persuasive when it is based on such a simple metaphor.  But it works.  Syllogisms constitute popular forms of reasoning because they are so very effective.  They are effective because they are models.  Like all models, they focus our attention toward certain aspects of the thing being considered, (necessarily to the detriment of other aspects).  Syllogisms trigger a feel of logical movement from the general rule to a sometimes surprising conclusion. They take the audience from a starting point and cause an epiphany.  They often give rise to a dramatic “a-HA!” moment. 

That syllogisms work is yet another tribute to the limited attentional abilities of human animals.  They are also a tribute to the power of conceptual metaphors.

Thinking about the power of syllogisms reminded me of the great power of covering law explanations. Also known as the “deductive-nomological” (“D-N”) explanations and “subsumption theory,” covering law theory (developed by Carl Hempel in the 1940’s) holds that an explanation occurs when a description of the empirical phenomenon to be explained is logically deduced from statements of antecedent conditions in combination with general laws (“assumption of general regularities). See C.G. Hempel, P. Oppenheim, “Studies in the Logic of Explanation,” Philosophy of Science, vol. 15, pp. 567-579 (1948).  In other words, explanations are derivations.

Here’s an example of a covering law explanation (from a site called HBP):

The basic structure of a DN explanation consists of a deduction of the explanandum from the explanans, which must contain at least one “universal law of nature” and which all must be true. There are other constraints, but the basic structure is all that is necessary for my purposes.
Ultimately, a DN explanation looks like this:
C1, C2, …, Cn      (antecedent conditions)
L1, L2, …, Lp      (laws of nature)
E                             (explanandum)
Suppose one wants to explain the length of a shadow cast by a flagpole. That is, one wants to know why the shadow is the length it is. On the DN model, one cites the relevant antecedent conditions, such as the height of the flagpole, the position of the sun in the sky, then cites, e.g., the law of the rectilinear propagation of light. From these statements, the current length of the shadow may be deduced. And there you have it: The length of the shadow is explained. The length of the shadow is expected given the antecedent conditions and general law cited.

It turns out that the D-N model is flawed—limited in its application:  Many derivations didn’t really seem to explain; they fail to grant explanatory power to nascent theories; they don’t account for the many compelling explanations that don’t make reference to any covering laws (“I’m late because I couldn’t find my keys”).  Further, covering laws can’t cope with problems involving as few as three moving bodies.  Many complex situations invite the application of many different laws at many different levels of analysis. In those cases, which particular laws should one integrate into such an explanation?   Here is a site that discusses many of the problems with the D-N model. 

Explanations are often contrasted with descriptions, the claim being that explanations somehow go further than descriptions.  But the truth is that it is often difficult to tell explanations from descriptions.  This conflation often arises in the context of dynamic systems analyses, where behaviors are represented by graphed trajectories.   DST has been criticized for providing only “descriptions,” and not true explanations, but this is like the pot calling the kettle black.  It seems that the subject matter (whether in DST or in classic Newtonian physics) can be said to constitute a description; those descriptions might take either of two forms: A) symbolic formulae or B) those oftentimes-beautiful attractor plots.

There is usually more heat than light generated by the many attempts to distinguish real explanations and “mere” descriptions.  For instance, J.A Scott Kelso writes:

Explanation demands theory and a coupling of theory to experiment.   No matter how refined a formal description of behavior is (e.g., dance notation), there is no guarantee (indeed it seems highly unlikely) that a purely formal approach will provide any deep insights into the organization of behavior.

Dynamic Patterns, J.A. Scott Kelso, p. 33 (1995).  I agree with Kelso that merely citing formal laws doesn’t do much explaining.  But, then again, what can we possibly know about the “organization” of any system, other than descriptions or, at least, descriptions of descriptions?  For this reason, Bas Van Fraassen appears correct to make strange bedfellows of the two—explanations do seem to be descriptions that (somehow) result in that irreducible feeling.  For Van Frassen, an explanation is an answer—an informative description evaluated pursuant to the context established by a particular question—a request for a specific kind of information. See The Scientific Image, Bas C. Van Fraassen, p. 154-157 (1980).  Explanations might best be described as descriptions that make us feel satisfied.  

Why do D-N explanations (and syllogisms) so often work so well.  It’s difficult to say. Eleanor Rosch goes so far as to charge that D-N “explanations” are circular, in that the outcome is already present in the ground.  “[The D-N account] adds nothing to that which one already knows and which one wants to explain, predict or control.” “Is Causality Circular?  Event Structure in Folk Psychology, Cognitive Science and Buddhist Logic,” Journal of Consciousness Studies, Vol. I, No. 1 p. 57(1994).

The bottom line:  covering laws don’t necessarily make for compelling explanations, and many compelling explanations don’t involve any formalized laws.

Despite all of these problems with the D-N model, and despite the simple metaphorical basis for syllogisms, humans often do find the subsumption of events under general laws to be quite satisfying as explanations.

Why did the balloon rise?  The answer would go something like this:

hot air exerts greater air pressure per particle than cold air, so you don’t need as many air particles to build to the same pressure level. So a hot air balloon rises because it is filled with hot, less dense air and is surrounded by colder, more dense air.

That explanation satisfies us, even though it is a partial story.  All scientific explanations are partial stories.  Good partial stories are satisfying, a point made by Nietzsche:

Just beyond experience!– Even great spirits have only their five fingers breadth of experience – just beyond it their thinking ceases and their endless empty space and stupidity begins. 

Nietzsche (Daybreak, s. 564)

To conclude this admittedly meandering post:

  • All humans are satisfied by the subsumption of events under general laws. 
  • I am a human.  
  • Therefore, I am satisfied. 

The end.


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Category: Science

About the Author ()

Erich Vieth is an attorney focusing on consumer law litigation and appellate practice. He is also a working musician and a writer, having founded Dangerous Intersection in 2006. Erich lives in the Shaw Neighborhood of St. Louis, Missouri, where he lives half-time with his two extraordinary daughters.

Comments (2)

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  1. Scholar says:

    Very informative, this is the stuff they teach in Logic/Philosophy class, not exactly easy to just dive into. Don't you think you might be taking this whole "knowledge" thing a little to far? Kidding, I have enjoyed the last few stories and articles. Just because I haven't commented doesn't mean I haven't been learning a lot and using the links etc. Keep it coming!

  2. hogiemo says:

    "Satisfaction is death. As long as I have a want, I have reason for living" G.B. Shaw

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