Anatomy of a Fallacy
CICESE, Depto. de Geología, Carretera Ensenada-Tijuana No. 3918, Fraccionamiento Zona Playitas, Ensenada, Baja California, C.P. 22860, firstname.lastname@example.org
This webpage draws from the paper: Cañón-Tapia, E., 2010, Origin of Large
Igneous Provinces: The importance of a definition,
in Cañón-Tapia, E., and Szakács,
A., eds., What Is a Volcano?: Geological Society
of America Special Paper 470, p. 77–101,
There are many ways in which formal errors in logical reasoning can be made, some of which might be difficult to identify since the argument may seem to be correct at first sight. These types of errors are known as fallacies (Copi & Cohen, 1994). Here I consider four types of fallacies that are commonly found in a context relevant to Large Igneous Provinces (LIPs).
Quaternio Terminorum (the fallacy of four terms)
A categorical syllogism should contain only three terms (two premises and one conclusion). If one of the premises contains a second premise that is presented in cryptic form, being embedded in the premise that is easily identified, a faulty syllogism has been constructed. This type of error is formally known as the “fallacy of four terms”. (The name remains regardless of the real number of hidden premises). In the context of LIPs, it is common to find this fallacy when examining evidence provided by seismic imaging. Although some workers might consider seismic imaging of the Earth’s interior an unbiased and very objective source of information, it turns out that there are several assumptions made in the interpretation of the actual data, and discrepancies concerning some of those assumptions can actually lead to discrepancies concerning the interpretation of the actual data in significant forms. For a recent and extensive discussion of such assumptions, see Thybo (2006).
For example, a commonly overlooked assumption is that the “anomalies” in seismic velocities are related to a “standard Earth model”, that was found either as an average of measurements made around the world, or as the result of theoretical predictions concerning seismic velocities measured in the lab for specific mineral phases. In the first case, the reds and blues in the seismic tomographic results indicate departures from the average of all the measurements used to calculate the standard. In the second case (departures from theoretical-lab predictions), the non-explicit assumptions are embedded in the model that is used to describe mineral composition variations as a function of depth. Again, this model usually will be an average, although in this case of both the presumed mineral composition and of the seismic velocity measured for that particular mineral. In either case, every tomographic image of the Earth includes a series of inexplicit assumptions, and neglecting their existence increases the risk of committing the fallacy of four terms.
Even if the mineral composition and seismic velocity were accurately and positively known, there is another commonly hidden premise embedded in the use of seismic imagery for the study of LIPs. Measured seismic data only contain information concerning the physical state of the rocks through which seismic energy actually traveled. If that physical state changes with time, then the conclusion reached by the seismic survey will be invalid for other times. Realistically, we do not expect that the physical state of large portions of the Earth interior to change over minutes or days. However, if time periods of thousands or millions of years are involved, such changes become a real possibility. Thus, in this case the effect of time becomes a hidden premise that may lead to the fallacy of four terms if not acknowledged explicitly.
The important point being made here is that completely different assertions can be made using the same database because the final interpretation depends on a series of additional assumptions. Failure to state such additional premises clearly opens the door to the construction of a fallacious argument.
The fallacy of drawing an affirmative conclusion from a negative premise
Use of hidden premises in a syllogism opens the door for other types of fallacies. For example, if the truth value of one of the cryptic premises turns out to be false, then the conclusion of the second syllogism necessarily must be false. Failure to acknowledge this possibility leads us to commit the fallacy of “drawing an affirmative conclusion from a negative premise”.
This error might be extremely difficult to identify because commonly we overlook the truth value of a premise that is not explicitly stated in the syllogism, and even worse, the truth value of the hidden premise might become accepted “de facto” more through habit than through a real exercise of logical inference. This type of error is very common in mythical thinking, as it promotes the selective acceptance of some facts and rejects any questioning of them (Dickinson, 2003).
There are many examples of this type of fallacy in the literature concerning the existence of mantle plumes in the Earth, some of which have already been examined by Anderson (2005) and Foulger (2010) and will not be re-iterated here. Many of those fallacies are related to extrapolations made on a relatively limited database that at some point in history seem to indicate a particular correlation of variables. The correlation is assumed to be true (or have become entirely demonstrated) based on such limited data base, and then all new observations are interpreted based on such premise, when in fact those observations should be used to re-estimate the veracity of the correlation. Time and again this approach can be found especially related to the geochemical aspects of supposed mantle plumes. It is extremely difficult to convince the advocates of one interpretation that a fallacious argument is being constructed because usually these workers tend to focus only on the premise that is entirely true. In this case, that could be the model of isotopic decay/evolution, or the compatibility vs. incompatibility of a given element in solids/melts. They often will not accept the fact that the first association between a mantle plume and the particular geochemical signature may have been based on a very limited database, and is not supported by a more complete one.
This type of fallacy nevertheless cannot be completely eliminated from any analysis of a natural system. As pointed out by Oreskes (1999) “the history of science demonstrates that the scientific truths of yesterday are often viewed as misconceptions, and, conversely, that ideas rejected in the past may now be considered true”. This situation is inherent in all natural systems, because a definitive proof is only possible in a closed system and therefore extrapolation unfortunately always has the possibility of leading to a fallacious syllogism. Consequently, the lack of correlation at some point in history also might be the beginning of a fallacious argument. Because of this, an adamant position that refuses to acknowledge that more data might point to a real correlation is as fallacious as the one described in the previous paragraph.
A well known example of this fallacy is provided by the age of the Earth estimated by Kelvin in 1863. Kelvin’s estimate was based on the false premise that all heat sources were accounted for, and therefore, although his calculations of heat dissipation were all correct (this was the true premise), the conclusion of the syllogism (the final age estimation) was false. Interestingly enough, this example gives rise to a second fallacy related to the idea that radioactivity is the sole factor missed by Kelvin. This assertion has become accepted de facto because of countless repetitions, although the historic evidence suggests that it is false. As it turns out, the role played by the discovery of radioactivity alone would have been insufficient to obtain an accurate age of the Earth, since one part of the argument sustained by Kelvin also included the age of the sun (and this age was not strongly modified by the inclusion of radioactivity). The full argument concerning the age of the Earth debate can be found in Stacey (2000), but in short it involves the discovery of fusion processes, and therefore of a fundamentally different physics from that used in Kelvin’s calculations. In any case, this example shows that the best way to avoid this fallacy is to keep an open mind and accept observations at face value, even when such observations might seem to be at odds with a seemingly well-established correlation.
Finally, it is important to comment that not all extrapolations are fallacies, although we need to be aware at all times of the original range for which the factual observations were used, as well as the assumptions behind the extrapolation. We should avoid the dogmatic support of extrapolations well beyond the limits of the factual information. Click here for discussion of extrapolations made between magma extrusion rate (observed) and magma creation rate (inferred).
The fallacy of equivocation
Another error related to the existence of a hidden premise in a syllogism is made when one of the terms is used in different senses in each of the two premises explicitly stated in the syllogism. This error, known as the fallacy of equivocation, can be found by examining the meaning of the term “melt” in any LIP definition.
Some workers have argued that the amount of melt produced in a given setting can be inferred from the composition of the erupted products, and that those estimates can be confirmed using seismology (e.g., Korenaga et al., 2002; White et al., 1992; 2001). Despite their apparent appeal, these works have the problem of combining two different sources of information (seismic and geochemical). The source of the problem (at a logical level) is that the term “melt” in each of the two disciplines has a slightly different meaning.
In seismic studies “melt” is assumed to denote “crustal thickness” which in turn has been assumed to result from the collection of a liquid phase that a) was extruded in its entirety from the region of origin, and b) remained trapped at depth to form the observed crustal thickness. In contrast, in geochemical studies, “melt” denotes the “cumulative volume of liquid” that was formed within the source region before eruption and which was expelled efficiently from that region to erupt from the surface. Therefore, when comparing seismic and geochemical evidence we are really comparing a) the inferred thickness of a solid layer we think was produced by a complex process of melt extraction which could not transport the liquid all the way to the surface to be latter eroded, with b) an inferred volume of liquid that was expelled all the way to the surface. It seems that many more factors were involved in the creation of the seismic “melt” than the geochemical “melt”, and any numerical agreement of volumes of “melt” determined by the two methods might be merely coincidence. Failure to recognize this possibility inherent in the subtle difference in meaning of the term “melt” in the two methods clearly leads to the “fallacy of equivocation”.
Fallacies in focus
Identification of fallacies should not be confused with undue criticism of any of the methods used to make inferences concerning the internal state of the Earth. For instance, although the equivalence between a seismically determined “thickness of melt” and one determined geochemically is a fallacy, identification of such fallacy does invalidate either method if considered independently from each other. This is so because the fallacy is formed when both types of information are combined in the same syllogism, and not because either is necessarily false.
In other words, it might be the case that the seismic method yields the true crustal thickness and the geochemical method yields a true fractional distribution of melt as a function of depth. The key point is that the crustal thickness determined seismically is not necessarily related to the volume of melt produced in a single region of partial melt at a given short-time interval, and that the geochemically inferred melt distribution does not necessarily correspond to the estimated crustal thickness measured by seismic methods. Consequently, it should be clear that the use of both methods to obtain information about the Earth’s interior is still valid (at a logical level), as long as the conclusions reached by each are not invoked as “corroborations” of the inherent truth value of the conclusions reached by the other method.
Taking all this into consideration, a critical step that needs to be taken to avoid mythical thinking in relation to the origin of LIPs is to be certain that we always compare the same type of evidence. This evidence should be gathered through equivalent means and with the same set of underlying assumptions for both LIPs and non-LIP provinces.
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Oreskes, N., 1999. The rejection of continental drift. Theory and method in American Earth Science. Oxford University Press, Oxford, 420 pp.
Stacey, F., D., 2000. Kelvin’s age of the earth paradox revisited. J. Geophys. Res., 105: 13 155 – 12 158.
Thybo, H., 2006. The heterogeneous upper mantle low velocity zone. Tectonophysics, 416: 53-79.
White, R., S., McKenzie, D. and O'Nions, R., K., 1992. Oceanic crustal thickness from seismic measurements and rare element inversions. Journal of Geophysical Research, 97: 19683-19715.
White, R., S., Minshull, T., A., Bickle, M., J. and Robinson, C., J., 2001. Melt generation at very slow-spreading oceanic ridges: Constraints from geochemical and geophysical data. Journal of Petrology, 42: 1171-1196.
last updated 13th November, 2012