When A Fallacy is Not a Fallacy:
Non-Logical Modes of Argumentation

[Introduction]
1. Ad Hominem Argument
2. Argument from Authority
3. Argument from Common Belief
4. Appeal to Antiquity
5. Fallacy of Origins
6. Fallacy of Composition
7. Post hoc ergo propter hoc
8. Affirming the Consequent
9. False Dilemma
10. Begging the Question, Circular Reasoning
11. Special Pleading or Changing the Premise
12. Appeal to Consequences
Conclusion

In modern mathematics, computer science and physical science, rules of argumentation are usually defined by a formal logic such as Boolean calculus or lower predicate calculus, or by a common language equivalent of such logical systems. Such logical calculus is but a formalization of what was called logic or dialectic in traditional Western philosophy. As I discuss in another work, modern symbolic logic does not adequately capture all the richness and comprehensiveness of classical logic, so there are many valid arguments in classical logic that would be invalid or incomprehensible in modern symbolic logic. Even on its own terms, as a basis for mathematical and physical science, symbolic logic is incomplete, as, for any set of axioms, there will always be true statements that cannot be deduced in a finite inference. Due to the exposure of this limitation by Kurt Gödel, symbolic logicism failed as a philosophical endeavor in the 1930s, and hardly any philosopher would now claim that all valid modes of argumentation are contained in formal logic.

Nonetheless, philosophical literacy cannot be assumed among students and teachers of the quantitative sciences, since formal philosophy has ceased to be a mandatory part of university curricula for more than a century, and what philosophy is taught tends to be a disjointed, unsystematic historical review of various thinkers, rather than a coherent body of knowledge. In the absence of higher philosophical criteria, academics are left with the faulty premise of scientism, namely that everything in existence can be explained by the methods of mathematical and physical science.[1] This leads to the untenable position that the only valid deductions are those of mathematical logic, and that the only valid inferences are those of formal mathematical deduction and of induction as used in the physical sciences. In actual practice, as sociologists since Thomas Kuhn have shown, scientists make use of other epistemologies besides deduction and induction, without being aware of it. The idealized scientific method of (1) observation, (2) hypothesis, (3) experiment often fails to reflect to actual procedure of researchers, who also rely on intuition, abstraction, bias, and sometimes just plain serendipity.

It is one thing to regard mathematical deduction and physical induction as the only forms of argument acceptable in a certain science, but quite another to insist that they are the only valid forms of argumentation at all. Modern scientists have a tendency to overgeneralize from the methodology of their specialization to universal rules of knowledge. A physicist, for example, might think the existence of immaterial entities to be unprovable, because his science only admits of what can be observed directly or indirectly by the senses. This confuses what is unprovable in physics and unprovable in general. It would be greatly fortuitous if absolutely everything in existence can in principle be observed through the five senses of human beings. We have no reason to expect such good fortune, and it is certainly not a matter of rational necessity. Similarly, the logical proof that is used in mathematical and computer science need not encompass every kind of proof that is conceivable in philosophy.

Those who mistake formal logic for all valid rational thought may develop the unfortunate habit of parsing every argument, mathematical or otherwise, in terms of some limited formalism. Any departure from this formalism will be derided as a logical fallacy, at which point the critic, acting more like a computer parser than a rational thinker, feels justified in dismissing the argument as invalid or worthless, so he need not trouble himself with it. This procedure has led to a great impoverishment of the intellectual life of scientific academics, who have needlessly closed off entire avenues of valid thought. To them it seems obvious that all that exists is what quantitative science can prove, since they have a blinkered conception of logic that only admits arguments of a certain mathematicized form.

The notion of a logical fallacy is no invention of modern symbolic logic. Many classes of fallacy were identified by classical and medieval philosophers, which is why most of these fallacies have Latin names. Traditional Western philosophy, however, recognized that logical deduction or dialectic was not the only valid mode of argumentation. There was also a science of rhetoric, which recognized modes of argumentation that did not depend on strict logical necessity, but appealed to physical and moral circumstances, as well as the opinions and positions of those to be persuaded. Rhetoric can have the negative qualification of smooth-talking or insubstantial wordplay, and it was admitted by classical rhetoricians that rhetorical art could be abused to persuade others of falsehoods. However, used correctly, non-logical argumentation can establish truths to a high degree of certainty, so much so that we entrust to it matters of guilt or innocence, life or death. The legal systems of the world all make use of the rhetorical arts and its epistemic assumptions, and to good effect. The world of valid reasoning is much bigger than the strict logician would care to admit.

For this reason it is a mistake to think an argument containing a formal logical fallacy is necessarily fallacious or without merit. Identifying a so-called logical fallacy in an argument only means the following:

An argument with a logical fallacy is one whose conclusion does not follow from its premises as a matter of logical necessity.

I emphasize the word logical, since there are other kinds of necessity. An argument with a logical fallacy does not establish logical necessity, yet it could still suffice to establish metaphysical, moral or physical necessity. Further, a good argument need not attempt to establish necessity at all, but only greater probability or likelihood.[2]

Empirical induction, which corroborates a hypothesis with repeated observations, is really a probabilistic argument. As a hypothesis receives many diverse experimental confirmations, we regard it as more likely to be true, though we can never prove it with absolute certainty. It is erroneous to speak of logical induction, for induction is not strictly logical, as it does not depend solely on the conceptual form of the argument, but on certain assumptions about probability. In some respects, these assumptions are dubious, leading to the famous problem of induction, namely that induction cannot establish, even probabilistically, a universal law. This is because a universal law would have to hold for infinitely many cases, while we can only establish the truth of finitely many cases, so we never enhance the probability of universality by repeated confirmation.

There are two kinds of invalid universal inferences by induction, shown below. The first is an inference about an infinite set of individuals A_i, and the second is an inference about the universal species A, abstracted from any determinate set of individuations.

We have observed that A_i is B for millions of individuals A_i.
Therefore (1) A_i is B for all i, and (2) A is B for the universal A.

We cannot logically prove anything about the universal, or even an infinite set of individuals, from facts about a finite set of individuals, no matter how large. This would not even enhance the a priori probability, unless we introduce some assumptions about the way the world works: e.g., nature is orderly; nature uses a minimum number of principles (Occam’s Razor); every sample is equally representative of the whole (the ergodic principle). These are all metaphysical assumptions, and they do not become logical rules simply because they are universally accepted. Since most scientists deny adherence to any school of metaphysics, they tend to regard their metaphysical assumptions as rules of logic or brute empirical facts, if they think about them at all.

Logic, dealing as it does with form and a priori knowledge, does not include empirical induction at all. Those who count empirical induction as a kind of logic do so only to spare scientific methodology from the accusation of logical fallacy, erroneously thought to be equivalent to irrationality. This special pleading for empirical induction as a kind of logic effectively renders ‘logical fallacy’ simply a derogatory label for any epistemology the writer does not espouse.

In mathematics, there is a type of proof called mathematical induction, but this is really a deduction of the following form.

(1) If n = 1, A(n) holds.
(2) If A(k) holds, then A(k + 1) holds.
Conclusion: A(n) holds for all n ≥ 1.

A few mathematicians question the validity of mathematical induction, since it seems to implicitly require an infinite number of steps. Even if it cannot strictly prove A(n) holds for all n ≥ 1, it can at least prove that it holds for any arbitrarily large but finite range of n, and this proof is certainly of deductive form, going from A(1) to A(2) to A(3), and so on.

Classical logic recognizes that there can also be modal or probabilistic deductions. For example:

(1) Most squirrels are brown.
(2) You saw a squirrel.
Conclusion: It was probably brown.

Here, probability is explicitly built into the premise (1), so it is conveyed in the conclusion. The axioms of probabilistic theory are also assumed (e.g., the ergodic principle) in order for the deduction to hold. The conclusion, since it explicitly includes probabilistic qualification, does follow from the premises, when we add the axioms of probability.

Modern symbolic calculus can be used to express modal logic, but most formalisms are too dependent on a strictly numeric representation of probability (ranging from 0 to 1). This notion of modality does not capture classical concepts such as necessary and contingent, which mean something more than probability = 1 and probability < 1. Again, excessive dependence on symbolic calculus can blind a person to entire fields of valid inferences.

For the purposes of our discussion, we will regard only invalid deductions as logical fallacies. This is not to disregard or deny the validity of empirical induction; on the contrary, we intend to expand our epistemic horizons even further, and describe other valid ways of knowing beyond deduction and induction. These ways of knowing, and their associated forms of argumentation, have been known to rhetoricians for millennia, and indeed by any person of sound common sense, including hard-boiled empiricists who rely on these alternate epistemologies constantly in their day-to-day affairs, including the conduct of academic research. We will not present here a complete system of rhetoric, but only illustrate some of the valid rhetorical uses of arguments that involve so-called deductive fallacies. Just because an argument's conclusion does not follow as a matter of logical necessity does not mean it cannot be established on other grounds. In the examples that follow, we will show how the use of so-called fallacies is consistent with, and sometimes essential to, the construction of a rational argument. By rational I mean an argument that give sufficient reason or cause to believe in the truth of the conclusion. As shall become clear, the sphere of the rational is much broader than that of the logical.

1. Ad Hominem Argument

Let us begin with what seems to be a very obvious fallacy, the ad hominem argument. This is not to be confused with the medieval argumentum ad hominem, which is a logically valid argument where you make the same assumption as your adversary, even though you do not hold that assumption to be true. In the modern sense of the expression, an ad hominem argument attacks the person making the argument rather than the argument itself. This is considered a fallacy, because of the following faulty deduction:

(1) Person A makes argument X.
(2) Person A is a bad person.
Conclusion: Argument X is wrong.

It is true that the conclusion does not follow from the premise as a matter of logical necessity, but this alone does not render the ad hominem argument rationally worthless. If it can be shown that the person making the argument has a reputation for mendacity or unscrupulousness, we should be well advised to view his arguments with skepticism, as we would avoid smooth talking salesmen. When we have no better means to verify or refute his contention, we are wise to look at a man’s character. His argument may be a valid deduction from false premises, or a cunning yet invalid argument, making use of subtle linguistic tricks. A chronic liar or charlatan surely deserves less trust, even if we cannot immediately perceive a flaw in his argument. Such instincts or common sense regarding whom to trust have served humans well for thousands of years, and continue to yield good results today. Our perceptions on such matters are fallible of course, but fallibility does not imply irrationality.

The ad hominem may have merit even when it does not pertain to the trustworthiness of the one criticized. In matters of ethics or politics, where we are trying to determine norms of behavior, we may place less value on the opinion of one whose values we find contemptible. Here there is not a question of true or false, so much as what we ought or ought not to do.[3] If a person is making an ethical argument about topic A when he has contemptible morals on a related topic B, we may reasonably distrust what he has to say about A, since it is likely to be congruent with his deplorable morals on B, which we wish to avoid. It is not a matter of strict logical necessity that someone who is immoral in one area will also be immoral in another related area. It is conceivable that a horse thief would object to robbing banks, or that a bank robber might find stealing horses to be immoral.

Nonetheless, human experience teaches us that a person’s moral stances on various subjects are often interrelated. From a person’s religious or political ideology, we can deduce many of their ethical positions on specific topics—fallibly, to be sure, but not irrationally (i.e., without reason or cause). It is reasonable to believe that someone promoting a specific ethical or political position may be trying to advance his broader worldview. If we know a person to be selfish, we may interpret his opinions as reflecting his own interests rather than a detached concern for the common good.

Generally speaking, we do not expect good things from wicked people, and the one who does otherwise is often taken for a fool. A man who has no scruples about cheating on his wife might also have little scruples about lying to voters; this is not a matter of logical necessity, but of practical wisdom grounded in experience. A person is not a collection of disjointed beliefs, but an integral whole, so we expect a certain degree of consistency in behavior informed by character. A man we deem to be of bad character will be rightfully viewed with greater skepticism in anything he says or does. As long as we do not make the error of supposing that this association is a matter of logical necessity, we are guilty of no fallacy. It might indeed be a matter of psychological necessity, if it could be shown there were no exceptions to such a rule. In practice, most people operate on the tacit assumption that the immoral man will probably be less trustworthy, or at any rate, as a matter of justice, he loses his right to be taken seriously and with public confidence, since he has falsified public mores.

These arguments make appeals to other ways of knowing: experience, knowledge of human nature, and ethical principles such as social justice. Identification of fallacies in these arguments simply indicate that their conclusions are not logically necessary. This does not mean the conclusion cannot be necessary on some other ground (metaphysical, moral, physical) or at least probable.

The ad hominem argument, used prudently, relies on an authentic insight into the human condition. The man who moralizes and the man who reasons are one and the same being; the two may be separated logically, but not psychologically. Thus the following sayings, paraphrased from the Gospels, constitute sage advice when dealing with human beings.

(1) By their fruits you shall know them.
(2) Good fruit does not come from a bad tree.
(3) If you cannot be trusted in small things, neither can you be trusted in great things.

In the first saying, we are advised that we should judge people by what they do or accomplish rather than by what they say or promise. If deeds do not match words, we should distrust the words, even if we cannot find a logical flaw in them. In the second saying, we are warned that nothing good can come from an evil person, insofar as he is evil. Since, as a general rule, we do not expect good fruit to come from a rotten source, we should not expect that good will result from words or deeds motivated by evil. This is not a strict rule of logic, but a general rule grounded in human psychology, whose proof is found in experience, which is often bitter. The last saying appeals to the unity of the person, for it is the same person who encounters small challenges and great ones. If he falters before the small moral challenges, he can hardly be expected to overcome the great difficulties. Again, we are appealing to psychological realities, and not abstract logical necessity.

These Gospel sayings, and others like them, are regarded as wise precisely because they are congruent with actual human experience. As strictly logical arguments, they could be said to contain the ad hominem fallacy, but it would be a mistake to disregard their merit on that ground, since this would suppose that there can be no wisdom in anything other than logical necessity or tautology. Those who would restrict valid argument to deductive logic alone will find that no moral system short of relativism, libertarianism or nihilism is sustainable, since you cannot arrive at normative principles solely from deductive logic, absent some thesis about moral reality.

2. Argument from Authority

The argument from authority is a positive version of the argument ad hominem. As a logical deduction, it is a fallacy of this form:

(1) Person A is an authority on X.
(2) Person A says Y is true.
Conclusion: Y is true.

There are circumstances when this is a more or less valid inference, though there are also innumerable cases where persons of authority are mistaken. For establishing statements of fact, we may regard this as at best a probable inference. The inference is more probable when (a) the statement Y is within the speaker's domain of authority X, and (b) the speaker’s authority on X is grounded in actual expertise or competence.

Authority can be derived either from reputation or from office. If the former, the argument from authority holds best if that reputation is grounded in real expertise, as verified by others with the capacity to judge such expertise. Since we cannot all be experts on everything, in practice we entrust most of our knowledge to the authority of others. This is true even for academic experts, who must defer to others on all matters outside their specialized field. Increased specialization has actually led to a greater need for reliance on expert authority, to which we entrust public health and safety on a regular basis.

The authority of experts may be reinforced by empirical verification of their theories, which is often simple enough for a layman to interpret, or by the practical efficacy of implementing their theories (e.g., the technological products of scientific developments), which can be seen by all. Still, there are many theoretical matters not easily verified by laymen, for which we tacitly rely on the opinions of experts, on the basis of their authority alone. Although a theory’s effectiveness in practical applications may give us reason to believe it is sound, history is replete with scientific and cosmological theories that gave good practical results but are now regarded as false.

When authority is derived from one’s office, we presume the existence of a competitive or meritocratic process for achieving that office, so occupation of the office implies a certain degree of expertise.

In all of the foregoing we have assumed that assertion Y is in the speaker’s domain of authority X, yet there are cases where a person’s domain of authority has nothing to do with his assertion. The most common example is a person of moral authority, who is sufficiently esteemed for his rectitude that he may be trusted with special insight on any testimony within his capacity. For normative or practical principles, one may obey authority out of deference to the office or the person, for we are not dealing with matters of simple truth or falsehood. A son may feel compelled to obey his father, to whom he owes life, not because he is convinced that his father’s practical advice is always correct, but because of a sense that duty compels him to repay his debt with obedience. One similarly obeys public officials not because we believe they are always correct, but because we are obligated to do so for the sake of public peace. These are not purely intellectual outcomes, but acts of the will where one knowingly chooses loyalties for the sake of some good. Thus such decisions are by no means necessarily irrational (without reason).

As with the ad hominem argument, we see that there are many circumstances where the argument from authority is a rational and practical mode of inference.

3. Argument from Common Belief

Another type of inference, argument from common belief, may be written deductively as follows:

(1) Most people believe X.
Conclusion: X is true.

As a strict logical deduction, this obviously fails, since the majority of people have been mistaken on countless matters throughout history. Nevertheless, there are circumstances where it may have a strong probabilistic validity. First, if X is a matter of verifiable fact within the competence of ordinary people, the inference certainly gains some weight. Public historical events are extremely likely to have occurred if the mass of humanity agrees it was so, the entire public having witnessed it directly or indirectly. Second, if X is a normative judgment rather than a statement of fact, it may deserve assent as a social principle, to the extent that social principles ought to be derived from the will of the people. Alternatively, the assertion may demand acceptance out of respect for others, as with rules of etiquette. For judgments regarding human nature or common sense, who better to judge what is human or common than the common mass of humanity?

Another basis for the argument from common belief is the thesis that it is improbable for many people to make the same erroneous judgment. This thesis needs to be weighed against (1) the competence of common people to make judgments about the particular subject and (2) the likelihood that most people adopt the belief only by imitation rather than independent judgment. The first possible objection disappears if the subject matter is simple or we are dealing with witnesses to a physical fact or event. To answer the second possible objection, we need not prove that all witnesses are absolutely independent and not influenced by each other, but only that they concur in their judgment by free and considered consent, rather than passive imitation. It also helps if most people would be in a position to contradict the belief if it were untrue, having potential access to facts to the contrary. This lessens the possibility of mass deception.

The argument from common belief might involve a direct appeal to natural desires that are found in most human beings. From this one could argue that it is improbable, or even impossible that there should be a natural desire without a real object. If human beings desire the Good, for example, then the Good is likely, even certain to exist. We may consider it likely if we take an evolutionary standpoint, namely that such a trait is unlikely to spontaneously arise and persist if it did not serve some useful function, and nothing could be more useless than a desire that cannot ever be fulfilled. If we add the premise, Nature creates nothing in vain, we may even say it is impossible for there to be a natural desire whose object does not exist. This ancient dictum, made famous by Aristotle, is an unspoken assumption in modern science, which often presumes a certain natural economy or parsimony.

One may object that an argument should state all its premises explicitly, and while this is certainly true as a formal ideal, in actual human communication we often leave our shared assumptions unstated, as it would be tedious to do otherwise. When a person makes an argument from common belief, he is usually assuming one of the premises or contexts we have described, so it is unfair to characterize the argument as a fallacious deduction from the single premise of common belief.

4. Appeal to Antiquity

Related to appeals to authority and to the majority of humanity is the appeal to the antiquity of a custom or belief as grounds for its continuation.

(1) Custom/belief A has been held since antiquity.
Conclusion: We should continue said custom or belief.

Obviously, the conclusion does not follow as a matter of logical necessity, but there are circumstances where such an inference may be rational. The long-term survival of a belief or custom shows that it has withstood the test of time; this consideration has special weight when dealing with practical social principles. Seemingly rational changes in law or custom may have unforeseen consequences, while existing customs have already been tested in society, and developed organically with the people. In a society permitting intellectual criticism, the survival of a theoretical proposition is an indication it has withstood many critiques, but such an open society may have little need to appeal to antiquity.

There are cases where the antiquity of testimony is germane to an argument, if the ancients were closer to the source of knowledge about the subject in question. This is obviously the case regarding ancient historical facts. Also, in belief systems based on a body of doctrine recorded or expounded in a definite historical period, more ancient witnesses and customs deserve greater credence, as they are closer to the source materials.[4] Another area in which the testimony of antiquity might be favored is regarding what is natural for man, on the anthropological assumption that the ancients lived closer to some primordial state of nature, being less encumbered with the artificial trappings of modern society.

5. Fallacy of Origins

The question of how primitive man behaved, though interesting, does not necessarily tell us anything about human ontology or teleology. If man evolved from brute beasts, this does not mean that he is now nothing more than a brute, nor that he ought to behave like one. This is to succumb to the fallacy of origins, which takes the following general form:

(1) X originated from Y.
(2) Y is Z.
Conclusion: X is or ought to be Z.

The fallacious conclusion can have one of two modes: ontological (X is Z) and teleological (X ought to be Z). The ontological conclusion may be validly inferred only if we add another assumption, such as the thesis that X retains the nature or essence of Y in the act of its generation. For example, we might assume that, in the act of procreation, the earliest humans imparted a human nature that we retain to the present day. Even when the ontological conclusion (X is Z) is validly inferred, we cannot thereby infer the teleological conclusion (X ought to be Z). Returning to our example, even if we know that modern humans have the same physical or psychological nature as their ancestors, this does not necessarily tell us what we ought to do, unless we can also find teleology or purpose in this nature.

Several philosophical systems, most notably those of Aristotle and the Scholastics, have purported to demonstrate a posteriori that there is in fact teleology in nature, and in human nature specifically. If we accept such a metaphysical thesis, then it may be valid to ground moral arguments in human nature, saying, Man is constituted this way, therefore he ought to act in a certain manner. The argument from origins, then, cannot always be dismissed out of hand, but we must take into account the metaphysical context in which it is asserted.

Related to the fallacy of origins is the so-called genetic fallacy, a specific case where the moral qualities of the originators of an idea or practice are imputed to the idea or practice itself.

(1) People X originated idea/practice Y.
(2) People X are good (evil).
Conclusion: Idea/practice Y is good (evil).

This manifestly does not follow logically, as morally bad people are not utterly bad, and therefore can generate many good things, or at least technical developments that are morally neutral.

However, there are circumstances where the inference does hold, as in the adage that sick trees necessarily bear bad fruit. Here the inference holds because the nature of the thing generated (the fruit) retains or shares the nature of the thing generating it (the tree). If, for example, an idea or practice were an essential or defining characteristic of Nazism, which we have judged to be bad, we could justly infer that the idea or practice is also bad. As a matter of prudence, we may regard anything coming from a bad source with distrust, testing it to see if it can be adopted by good people. Such distrust is not irrational, but is sound practical wisdom, as discussed in our treatment of ad hominem arguments, and it would be a mistake to dismiss such wisdom as fallacious.

Aside: The adage in the Gospel about the tree and its fruit has an additional meaning beyond mere psychology, referring to the spiritual origins of sin. Since all goodness comes from God, whoever does good does it of God, but whoever does evil is not of God insofar as he does evil. The tree/fruit example is but a simile to explain this spiritual truth. Here the argument from origins is supplemented by the revealed truth, accepted by the Jews in the time of Christ, that all goodness comes from God.

6. Fallacy of Composition

A more subtle fallacy is the fallacy of composition, namely that if a composite entity’s constituents do not possess a certain property, then the composite entity cannot possess that property. Stated more formally:

(1) Entity X is composed of constituents, A₁, A₂, A₃, etc.
(2) None of the constituents A_i have property Y.
Conclusion: X cannot have property Y.

As a matter of strict logical necessity, the conclusion clearly does not follow. For example, atoms are colorless and have no temperature, yet things that are composed of atoms may have color and temperature. How can this be? Are we getting something for nothing?

Let us step aside from physics, and take a more pure, abstract example from mathematics. A square is composed of four line segments, yet it has area, a property held by none of its constituent line segments. It is at once clear that the square is something more than a collection of four line segments. They are arranged in a certain way in a two-dimensional space. The space spanned is the something more than the constituents that makes possible the property of area. Suppose we simply considered the four line segments as a set of objects, without placing them in any geometric space. Even then, the composite entity (the set) could have a property not held by any of the constituents, namely the property of having four elements. Beyond this trivial property, however, we cannot generate any new properties unless the constituents are combined in some relation, or situated in some spatiotemporal medium, such that the metaphysical structure of the composite enables the manifestation of a new property. If these conditions are not met, then the fallacy of composition is not a fallacy, and we are right to complain that you cannot gain something for nothing.

It is easy—and wrong—to invoke the fallacy of composition in order to declare it always possible to reduce a property to microscopic constituents lacking that property. We cannot get emergent properties willy-nilly, but there are real metaphysical and physical constraints that limit what sort of properties may emerge from a system of constituents. Such limits include the physical powers of the constituents, as well as the ontological category and dimensionality of each constituent, and the way in which they are combined (spatially, temporally, etc.). It may be, for example, that as a rule of nature, the combination of substances A and B yields a substance with some new property, not because this property is derivable from the properties of the constituent, but it is simply a higher order law of nature. Thus the fallacy of composition cannot be invoked to prejudge the question of reductionism in physics.

There are even some logical (or at least, a priori) constraints on how we can derive properties from constituents. First, it is impossible to derive a quality from its contrary, without invoking some other principle. Black qua black cannot give us white, so if a black substance is able to produce a white substance, it must do so in virtue of some property other than being black. Generations of quantity or logically confined by arithmetic constraints, as well as the dimensionality of the geometric space considered. No spatial permutations of objects in a plane can yield a three-dimensional object, nor can any combination of motion in a plane yield motion in a third dimension.

The fallacy of composition resonates with us as if it had an air of truth, precisely because it does contain an intrinsic logical truth, namely, that a thing cannot come out of nothing, or of its opposite. Being cannot arise out of non-being as such. It is possible for a thing that once was not to come to be, so that non-being is replaced by being, in a sense, but it does not come into being by virtue of its prior non-being. Some agency or power or entity is needed to account for its coming into being. Properties and substances may be generated by the relations and interactions of prior existing substances. We are not then guilty of getting something for nothing, because the generated property is contained in potentia in some pre-existing order.

It is, however, a mistake to say that a property or aspect of reality can be generated by constituents and media that possess no dimensional or categorical semblance to that property, without appealing to some higher metaphysical order. This is the error of reductionism, which too often hides its absurdity behind the accusation that its opponents commit the fallacy of composition. As we have seen, this supposed fallacy actually contains some important logical and metaphysical insight, and it is unworthy of a philosopher of any subtlety to dismiss an argument that makes use of such insight. The question of when a property or substance may be validly generated by a set of constituents is not merely a physical question, but also has metaphysical and logical aspects. The discernment of the validity of a composition depends on an ontological analysis of the entities involved. This requires some understanding of ontological concepts, not merely parsing the form of an argument.

7. Post hoc ergo propter hoc

A common fallacy seen in the social sciences, medicine, and economics is that of post hoc ergo propter hoc, after this, therefore because of this. This has the form:

(1) First, A happened; then B happened.
Conclusion: Therefore A caused B.

This is not so much a logical fallacy as a metaphysically false inference, for it equates temporal sequence with causality. There have been, however, some important philosophers who denied that causality means anything more than temporal sequence. David Hume famously suggested that ‘A causes B’ simply means that B always observed to happen after A happens. Bertrand Russell found the notion of causation to be so hopelessly vague that he at one point sought to expunge the notion of causality from science, which should deal only with statistical and temporal correlations. This endeavor won few adherents, however, as most scientists accept causality as an operating assumption. It is a hallmark of scientific thinking to view a phenomenon and not merely accept it as an inexplicable fact, but to find out why it occurs the way it does, and to seek some underlying physical cause.

Most scientific researchers are sophisticated enough to recognize a distinction between correlation and causality. For example, if dark-haired people score better on a test than light-haired people, it does not follow that their dark hair caused them to perform better. It could be that this genetic trait is correlated to some other gene that causes the improved performance, or it could just be a coincidence, or perhaps dark-haired people have access to better resources due to cultural bias. For there to be causality, there must be a credible mechanism by which one physical occurrence generates another. It is not enough for them to be correlated statistically, nor does mere temporal sequence suffice. The fact that inheriting dark hair is followed by strong test performance does not establish that the first event caused the latter.

Even when a phenomenon B invariably occurs after some other phenomenon A, it does not follow that A causes B. For example, sunset always comes after sunrise, but this does not mean that sunrise as such causes sunset. Rather, the same orbital mechanics that causes sunrise will invariably cause sunset to follow. In other words, both A and B share a common cause C that generates effects A and B sequentially. Similarly, just because a human fetus always develops fingers and toes prior to developing ribs, it does not follow that the appearance of fingers and toes causes the development of the ribs. Rather, the two processes operate in parallel according to their own timescale, starting from a common origin (conception), and it happens as a rule of nature that the first process is always completed before the second.

Although statistical and temporal correlations do not prove causality, there may be cases where causality is at least suggested by correlation. If a government implements a radical new economic policy, and this is swiftly followed by an economic recession, we are justified for at least strongly suspecting that the policy was responsible for the recession, due to the swiftness with which one followed the other, especially if the implementation of the policy was preceded by a period of stability. When two unusual occurrences happen near simultaneously, it is not unreasonable to suspect that one may have caused the other. If the two highly unusual events are not independent of each other, we do not multiply their probabilities, so the combined occurrence becomes more plausible. This is more of a probabilistic argument than a proper proof of causality, but in empirical science we often have to deal with what is probable rather than what is certain.

It is unjustified, then, to automatically dismiss arguments for causality invoking temporal or statistical correlations on the grounds that these use the post hoc, ergo propter hoc fallacy. If a certain course of action is followed by consistently bad results for an extended period of time, it would be madness not to at least consider changing one’s course of action, even if it cannot be strictly proved that one’s current actions are causing the bad results.

8. Affirming the Consequent

A rather obvious logical fallacy, at least when stated formally, is affirming the consequent. It has the following form:

(1) If A, then B.
(2) B.
Conclusion: A.

We can easily see that the conclusion does not follow, yet when we consider concrete examples, we may find arguments of this form that have some validity. Consider the following:

(1) If unicorns were real, then they would be visible.
(2) Unicorns are visible.
Conclusion: Unicorns are real.

This is a valid argument only because unicorns are visible is understood to entail unicorns being real, not because of the logical structure or form of the argument as presented. We may say that there is a hidden premise, namely: (3) If unicorns were visible, then they would be real. The reason for this hidden premise is that, in ordinary speech, when we say If A, then B, we sometimes really intend logical equivalence, that is: If and only if A, then B. Thus, rational people may make valid arguments that have the linguistic form of affirming the consequent. When dealing with ordinary speech and writing, it does not suffice to simply parse language as if it were formal logical syntax.

There are cases where affirming the consequent has the strength of a probability argument.

(1) If my team scores more goals, we will win the match.
(2) We won the match.
Conclusion: We scored more goals.

It could be that, in the game in question, there are scenarios where the match is decided by a tie-breaker, but these are very rare. Therefore affirming the consequent would have the strength of a probabilistic argument. This is bad form for an argument, because we are not explicitly declaring all our assumptions (i.e., the rarity of tie-breakers), yet in ordinary speech, we often assume certain common knowledge on the part of the listener. There is nothing irrational in this practice; indeed, even the most logically rigorous will find that he can convey little of import without making unstated assumptions about his reader’s knowledge.

9. False Dilemma

Another supposed logical fallacy is the false dilemma. Formally stated:

(1) Either A happened, or B happened.
(2) B did not happen.
Conclusion: A happened.

This is not a logical fallacy at all, as this has the form of a valid argument. It is called a false dilemma when a critic disputes the first premise, arguing that there exist other possibilities besides A and B. Whenever B is not simply the negation of A, it is at least logically possible that the first premise is false. It is then necessary to invoke other metaphysical and physical premises to show that there are only two possibilities. Accusations of false dilemma are, strictly speaking, not accusations of faulty logic, but a challenge of one’s metaphysical or physical premises.

Often atheists and philosophical materialists will mistake their challenge of theistic or non-materialistic metaphysical assumptions as identifying an error in logic. This is because they are usually blind to their own metaphysical assumptions (since they nominally deny metaphysics), which they build into their conception of what is logical.[5] By erroneously dismissing certain theological and philosophical arguments as logically invalid false dilemmas, such thinkers fail to treat the underlying metaphysical assumptions (and their own unspoken denial of these assumptions) on their merits. This short-circuiting of the conversation prevents the possibility of ever moving beyond a metaphysically barren philosophy.

10. Begging the Question, Circular Reasoning

It is obviously of little probative value to assume the conclusion among the premises. This does not make the argument wrong or false, but simply logically vacuous. Such circular, reasoning is of the form.

(1) B (where B contains A)
Conclusion: A.

This is circular reasoning—assuming what you pretend to prove—only if B contains A intrinsically. By this I mean that the act of affirming B logically entails affirming A, not because A is consequent to B, but because A is a necessary supposition antecedent to B. In that case, the argument is reducible to the form:

(1) A.
Conclusion: A.

A circular argument is not an incorrect inference, but it is logically vacuous, adding no content beyond what we assumed. The conclusion is convincing only to those who assume the conclusion to be true, and so this trivial inference is useless as a persuasive argument. Begging the question, then, is as useless to the rhetorician as to the logician.

We should emphasize an important distinction. The argument is logically circular because the conclusion is intrinsically contained in the premise, not because it is unconvincing to those who reject the premise. All arguments are unconvincing to those who reject the premises, but this does not suffice to establish circularity or invalidity. It is a serious error, conflating human psychology with logic, to call an argument circular simply because no one who denies the conclusion would agree to the premise. Take the following example:

(1) Every effect must have a cause.
Conclusion: From this we can prove (after some ratiocination) the necessary existence of a First Cause (called God), that is not an effect.[6]

Some would say this sort of argument is circular because only a theist would agree to the first premise. This does not suffice to establish circularity. A person will naturally reject any premise that he sees will lead to an undesirable conclusion, yet the intellectual cowardice or evasiveness of an atheist does not establish the circularity of a theistic argument. It must be shown that the logical content of the statement, A First Cause exists, (or God exists,) is a necessary supposition underlying the assertion, Every effect must has a cause. This is a question of intrinsic, not extrinsic, necessity. It does not suffice to say that in all possible realities where (1) is true, the conclusion is also true, therefore (1) assumes the conclusion. That only establishes extrinsic necessity, and on that ground, we could say that all logically valid inferences are circular, since the conclusion is true in all realities where the premise is true. Further, this type of criticism assumes a purely conditional notion of logical inference (similar to Russell’s inaptly named material implication). Unless it can be shown that Every effect must have a cause, is logically unintelligible a priori without contemplating the statement, A First Cause exists, then there is no circularity.

To take a clearer example (without glossing over intermediate steps, as in the conclusion of the First Cause argument), consider the following:

(1) Child pornographers are evil.
(2) Senator Smith is a child pornographer.
Conclusion: Therefore, Senator Smith is evil.

One could credibly argue that only someone who already thinks Senator Smith is evil would be willing to believe accusations that he is a child pornographer. Even if this is so, it does not suffice to establish that the argument is circular. It is simply a question of people being disinclined to accept assumptions that will clearly lead to conclusions they find unpalatable.

The notion of begging the question finds censure in the Anglo-American legal system, which prohibits statements explicitly or implicitly pronouncing a formal verdict during testimony. One cannot declare in testimony that the accused committed the formal crime of, say, murder in the first degree, nor can counsel ask a witness plainly, Did the defendant commit murder?, since that is for the jury to decide. This would be like asserting Senator Smith is evil without first establishing that he is a child pornographer. One must establish the facts of the defendant’s actions (In a fit of anger, he shot the victim at point-blank range,) without declaring a definitive evaluation of them (e.g., first- or second-degree murder, voluntary manslaughter). The jury is to draw its own conclusion from the facts, without having the conclusion itself presented among the facts.

11. Special Pleading or Changing the Premise

We have noted that people are inclined to reject premises that lead to unpalatable conclusions. Sometimes the premise is a plain fact that cannot be simply denied, so more subtle maneuvering is needed. A common tactic is special pleading, whereby one hopes to show that the inconvenient fact is not really applicable to the subject, so the preferred conclusion is unscathed. Special pleading usually involves citing some contingency that explains away the contradictory fact either as inapplicable or as an acceptable exception to the rule. To take an example of each.

(1) No black man could get elected president of the United States.
(2) Barack Obama was elected president of the United States.
Conclusion: Statement (1) still holds, because Barack Obama is actually mulatto.

(1) Adultery should be a punishable crime.
(2) A good friend of mine committed adultery.
Conclusion: My friend should not be punished, because he was not thinking clearly at the time.

In the first example, the disputant posits that fact (2) does not really contradict assumption (1), as it is not really applicable. This argument holds weight only if we accept that Mr. Obama being mulatto (or whatever more fashionable term one chooses) rather than black significantly affected his ability to get elected. In the second example, the special pleader admits that the fact is applicable to the subject, for he does not deny that his friend committed adultery. Nonetheless, he is able to produce an extenuating circumstance that he claims should permit an exception to his rule (1).

Special pleading is not necessarily a fallacy, and the examples above may be persuasive rational arguments to some people. This depends on one’s assessment of the plausibility of each plea. If a person comes up with repeated and increasingly implausible excuses and exceptions, we have good reason to question his premise and his intellectual integrity. There is no hard and fast rule as to when special pleading is a fallacy. This depends on a consideration of the facts and premises, and an evaluation of the plausibility of the exceptions averred. No simple analysis of the argument’s form will suffice to evaluate its validity.

A variant of the first example, where one finds the contradictory fact to be inapplicable to the first premise, would be to effectively modify the meaning of the first premise so that the fact does not apply. This sleight-of hand was described by Anthony Flew in Thinking about Thinking (1975) as the No True Scotsman fallacy. It may be stated formally as follows:

(1) No Scotsman would commit crime X.
(2) A certain Aberdeen man (Y) committed crime X.
Conclusion: The Aberdeen man (Y) was not a true Scotsman.

In this type of special pleading, we effectively modify our definition of a Scotsman in order to accommodate assumption (1) in the face of fact (2). If the crime X is repeatedly committed by new offenders born and living in Scotland, we could just repeatedly deny that these criminals are true Scotsmen. By taking this position, we are effectively building the predicate of premise (1) into our definition of a Scotsman, so it is only trivially true that no Scotsman would commit crime X, by definition.

This type of argument uses terms in a manner contrary to their ordinary meaning, while appearing to declare the ordinary meaning. Thus it may also be considered a fallacy of equivocation. We will pass over other fallacies of equivocation, as these have no more merit in rhetoric than they do in logic, except for those who wish to learn to deceive.[7]

12. Appeal to Consequences

A rhetorical tactic often described as a logical fallacy is the appeal to consequences. It can have one of two contrary forms.

(1) If you affirm X, good consequences will result.
Conclusion: Therefore, X is true.

(1) If you deny Y, bad consequences will result.
Conclusion: Therefore, Y is true.

This is actually a rational form of argument in certain contexts. Most obviously, if X and Y are practical or normative principles about what one ought to do to achieve a good result, this is an eminently valid inference. X and Y, for example, could be social or economic policies, whose truth or validity are indeed defined by their results.

The appeal to consequences might also be valid if one accepts that there is an intrinsic goodness at the base of all reality. On this metaphysical assumption, the one who is alienated from goodness has fallen away from the truth, for if his premises were true, he would surely partake of the goodness that is at the base of all reality. An alternative metaphysical justification for the appeal to consequences is the tenet that good cannot come from falsehood.

The appeal to consequences may also have merit if it renders a position self-stultifying. For example, we may tell an academic who claims everything is a meaningless accident that this would imply all intellectual activity is pointless and his career is a waste of time (or at least indistinguishable from a waste of time), and we should not even care whether anything is meaningless or not. Here we are not simply saying, Your belief leads to bad consequences, therefore it is false, but are pointing out a certain incoherence in affirming such a belief and trying to convince others of it as if any of it mattered.

In a similar vein, strong psychological determinism can be validly attacked by an appeal to consequences. If one holds that all human behavior is strictly deterministic, it follows that no one, including the speaker, has any intellectual autonomy, not even while making the assertion of strong determinism. On the assumption that the assertion is true, paradoxically, the assertion would hold no more strength than an arbitrary utterance, since it emanates from an unwilling machine incapable of thinking otherwise. This line of criticism is the argumentum ad hominem in the medieval sense, where I take my opponent’s assumptions and show that they lead to confused results. Here I did not strictly prove that strong determinism is false, but only the incoherence of a strong determinist expecting to be taken seriously. If he is correct, then neither he nor anyone else should be taken seriously. Further, since logic presumes intellectual activity, an effective denial of its possibility undermines any basis for appealing to logic.

Appeals to consequences are also used in moral arguments such as Pascal’s Wager, which does not pretend to be a logical demonstration, but is a practical argument where it is assumed that the listener wishes to maximize his chances of felicity. Such arguments are not worthless on account of their appeal to consequences (though they may be challenged on other grounds); quite the contrary, moral decisions—i.e., decisions about what one ought to do—necessarily entail judgments about what is preferable, so an appeal to consequences is highly relevant.

In a Christian context, the appeal to consequences has special resonance, because in Christianity the act of faith or belief is itself a moral act. As such, it makes perfect sense to say, You should believe X because it yields good consequences, where X is some doctrine proposed as orthodox. If we trust, as a tenet of natural religion or as revealed to the Jews, that God is omnibenevolent, then it follows that any true religious doctrine ought to yield good results, and that a person is morally obligated to prefer such a doctrine. Non-Christians generally perceive such thinking as fallacious, because they do not share the perception that belief or faith is a moral act, and regard it merely as an intellectual act. This is why most criticisms of Christianity are obsessed with the epistemological aspect of blind faith, unfavorably contrasted with logic and empiricism. Such criticism misunderstands the basis of the Christian appeal to consequences.

Conclusion

In general, I have advanced the view that an argument containing a so-called logical fallacy can be quite rational and valid given the appropriate context or qualification. In some cases, logical fallacies are not fallacies at all, but are called such because the critic refuses to accept the argument’s premise. What I have hoped to bring out in this exposition is that it does not suffice to simply identify a formal fallacy in order to refute or dismiss an argument. Real philosophical criticism requires actual thinking and understanding, not just parsing syntax. It is intellectually lazy to simply scan the form of an argument and say, I found a logical fallacy, therefore I can ignore what this person says. Neither logic nor rhetoric is reducible to grammar and syntax. Any communication containing non-trivial insights will contain some unstated assumptions that need to be understood by the hearer. This is true even of mathematical logic, which, in order to be a logic and not just a meaningless string of symbols, requires real understanding of fundamental concepts by the reader. We cannot reject the necessity of unstated assumptions without making human discourse practically unworkable.

Students of mathematics, physical science, and computer science are accustomed to striving for syntactic precision in their expressions. Admiring the clarity and rationality of mathematical syntax, they would like to reduce the richness of human discourse to the quantified formalisms with which they are comfortable, most notably mathematical logic. As I show in this and other works, the domain of rational argument is much broader than what formalistic logicism will admit. Those who dismiss any argument not grounded in formal logical necessity will miss out on most of the conversation.

If we are not to retreat into a bunker where the only truths are the quantifiable and formalizable, we must not treat arguments as computer code to be debugged, but instead should follow the advice St. Ignatius of Loyola recommends in his Presupposition to the Spiritual Exercises:

…let it be presupposed that every good Christian is to be more ready to save his neighbor’s proposition than to condemn it. If he cannot save it, let him inquire how he means it; and if he means it badly, let him correct him with charity.

When dealing with anything difficult or subtle in philosophy, this principle needs to be observed, or it will be impossible to understand anything someone else has to say.[8] If we do not give our neighbor the benefit of the doubt, it is child’s play to interpret even the most sober argument in a way that renders it incoherent or fallacious. This is why those who reduce philosophical criticism to an exercise in identifying logical fallacies will end up with a highly restricted philosophical sphere, shorn of any metaphysical subtlety, and much of the Western philosophical tradition will remain a closed book to them, since they pretend to have dismissed entire schools of thought with a few facile rationalizations.

Notes

[1] Extreme positivism, extended even to ethical matters, is especially prevalent among academics in neuroscience and behavioral science, as well as among non-scientists who express a near-magical belief that science has the potential to resolve all of mankind’s problems.

[2] Grasping these distinctions is essential to comprehending all that follows in this essay. Once we accept that a rational argument may intend to establish only probability, or perhaps necessity in a given metaphysical, moral or physical context, it is senseless to complain that such arguments do not establish logical necessity and to deride them as fallacious.

[3] I do not assume that normative judgments are purely subjective or arbitrary evaluations, depending on nothing but common consent, nor do I assume the contrary. In either case, ethical theses may be treated as philosophical assumptions. It is their teleological aspect, rather than their supposed subjectivity, which makes them modally different, therefore meriting separate discussion.

[4] We are, of course, abstracting from the veracity of the original source materials themselves. When applied to historical inquiries, the argument from antiquity is but one of several factors for determining credibility, and by itself it is a highly fallible principle. Its inability to establish necessity, however, is not grounds for denying its legitimate use by historians, for it is not possible even in principle to establish necessary truths in historical or forensic inquiries.

[5] I neither say nor intend all atheists, but this sort of blinkered thinking is sufficiently prevalent in that ideological camp to deserve mention.

[6] This choice of example should not be construed as implying that the rhetorical arts are especially preferred by religious apologists. On the contrary, most of the notable anti-religious literature of the Enlightenment relied on rhetorical argument, since the philosophes were more literary men than formally trained philosophers (Hume being a notable exception). Even today, most arguments against the existence of God are really arguments against divine wisdom or omnibenevolence, e.g., How could a wise, just, and loving God allow X, Y, and Z? Such arguments have only rhetorical force, and do not establish logical necessity, but I would not suffer the theist to breezily dismiss such arguments as fallacious without engaging them seriously.

[7] I would include the man of straw among the fallacies of equivocation insofar as this entails misrepresenting someone else’s determinate argument. It is increasingly difficult to construct a man of straw in the stronger sense of an argument that nobody holds, since practically every humanly conceivable opinion, however bizarre or improbable, has its defender in this culturally diverse world of nearly 7 billion people.

[8] The acceptance of rhetorical arguments enables us to recognize rationality even in arguments that we find unconvincing. We can be unpersuaded because we disagree with the implicit assumptions or because we would have applied rhetorical principles differently (e.g., we place different weight on antiquity or authority). This is an improvement in understanding others, as opposed to supposing that everyone who thinks differently has committed some elementary logical fallacy. Instead of talking past each other, we can respect the intrinsic rationality of the other’s position, and by learning to identify the implied assumptions of the other’s argument, we can become more aware of our own unconsciously held contrary assumptions.

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