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The Amateur Logician Tutorial:
An Invitation | Arguments | Evaluating Arguments | Laws of Thought | Ontology & Logic | Concepts, Signs, & Names | Categorical Propositions | Negations & Complements | Distribution | Euler’s Circles & Venn Diagrams | Predicables | Categories | Definitions | Square of Opposition | Equivalent “Immediate Inference” | Traditional “Immediate Inference” | Informal “Immediate Inference” | Categorical Syllogism | Syllogisms & Venn Diagrams | Moods & Figures of Syllogisms | Polysyllogisms & the Sorites | Enthymemes | Compound Propositions | Conditional Propositions & Conditional Syllogisms | Conditional Contrapositions, Reductions, & Biconditionals | Conjunctions, Disjunctions, & Disjunctive Syllogisms | Dilemmas | Modal Propositions | Reductio ad Absurdum Arguments | Deduction & Induction | Inductive Abstraction | Inductive Syllogisms | Mill’s Inductive Methods | Science & Hypothesis | Formal Fallacies | Testimony & Unsound Authority | Informal Fallacies & Language | Diversion & Relevancy Fallacies | Presumption Fallacies | Causal & Inductive Fallacies
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Formal Fallacies
Formal Fallacies versus Informal Fallacies
Categorical Syllogisms and Fallacies
– Fallacy of the Undistributed Middle
– Guilt by Association Fallacy
– Reductio ad Hitlerum
– Illicit Process of the Major
– Illicit Process of the Minor
– Fallacy of Exclusive Premises
– Fallacy of Four Terms
– Equivocation and Ambiguous Middle
– Fallacy of Affirmative Conclusion from a Negative Premise
– Fallacy of Negative Conclusion from an Affirmative Premise
Conditional Arguments and Fallacies
– Fallacy of Affirming the Consequent
– Fallacy of Denying the Antecedent
– Counterexample Method
– Fallacy of Illicit Commutativity
Introduction to Fallacies
How could AmateurLogician.com not have a page (or more) on logical fallacies?
Given man’s nature to want to categorize or classify, people have come up with various lists of common fallacies. There’s no “definitive” list or “definitive” way to categorize the common fallacies, though. What can be claimed for sure is that there’s a distinction between “formal fallacies” and “informal fallacies.”
It’s important to be able to spot fallacies!
To be reasonable and earnest, all of us need to make an effort to not make them ourselves. We have to form good habits of reasoning. We should, moreover, never intentionally mislead someone else with a bad argument. It’s also a matter of self-defense. We don’t want to be fooled by another person’s argument. It may look good; it may even have a conclusion we agree with, yet it may be fallacious.
Logic is normative. It’s about how we ought to think.
Flipping things around, it’s likewise about how we ought not to think!
Assessing arguments demands that we (1) check to see if the argument is valid and (2) check to see if the premises are true with clear and consistent use of terms. The first is about the “form” or “structure” of the argument. Formal logic can help us in determining this. The second often depends upon informal logic. That is, is there sufficient reason to believe the premises are true? And do all the terms in the premises make sense?
Formal Fallacies
Briefly put, a valid argument follows the rules of logic.
Insofar as we are dealing with a formal argument, — for example, a categorical syllogism, a mixed hypothetical syllogism, a disjunctive syllogism, or whatnot — any invalid argument has committed a formal fallacy of one kind or another.
The premises, whatever their truth or falsity, simply do not lead to the given conclusion.
Formal Fallacies of the Categorical Syllogism. . .
Recall, a categorical syllogism has six basic rules. Violating at least one of those rules automatically results in an invalid syllogism. It is then a syllogism which is fallacious.
Fallacy of the Undistributed Middle. . .
“All racists are those who support border controls.”
“This politician supports border controls.”
“Therefore, this politician is a racist.”
The middle term is not distributed at least once.
Thus, the categorical syllogism is invalid.
“Supporters of border controls” is not referred to in its entire extension. Premise one tells us that all racists support border controls, although it leaves open the possibility that other groups (i.e., non-racist groups) support border controls.
What’s more, the given argument gives us no means to relate racists and “this politician.” Both support border controls, but this doesn’t tell us that they do so in the same way or identically.
Notice that this is related to the “Guilt by Association Fallacy!”
Interestingly, guilt-by-association is usually thought of as an informal fallacy. The structure of such arguments, however, are usually formally invalid! All X support Z. All Y support Z. Thus, all Y are X.
Leo Strauss, a political theorist, coined the term reductio ad Hitlerum. “Hitler’s allies are against smoking. My bishop is against smoking. Ergo, my bishop is an ally of Hitler.”
Illicit Process of the Major. . .
and Illicit Process of the Minor . . .
Example 1)
“All Protestants are Christian.”
“No Roman Catholics are Protestant.”
“Therefore, no Roman Catholics are Christian.”
Example 2)
“All squares are rectangles.”
“All squares are quadrilateral.”
“Therefore, all quadrilaterals are rectangles.”
Any term that’s distributed in the conclusion must be then distributed as a term in a premise. We can group together the “Fallacy of the Undistributed Middle,” the “Illicit Process of the Major,” and the “Illicit Process of the Minor” as all fallacies dealing with distribution.
It’s vital to know the rules of categorical syllogisms!
With that will come a greater understanding of these formal fallacies.
With example one, the predicate of the conclusion (i.e., Christian) is distributed but it’s not distributed in premise one. We’re told that “all Protestants are Christians,” but nothing is implied about all Christians being Protestant with the exclusion of Catholics. Note that the predicate of the conclusion is called the “major term.”
With example two, the subject of the conclusion (i.e., quadrilaterals) is distributed but is not distributed in premise two. Like example one, a lack of distribution applies that the term’s full extension is not referenced. Premise two doesn’t reference every quadrilateral. “All squares are quadrilaterals” leaves open if every quadrilateral is a square or if there are things that are not squares (including rectangles) which are nevertheless quadrilateral. Note that the subject of the conclusion is called the “minor term.”
(We can reach a valid conclusion with example two. What is it?)
Fallacy of Exclusive Premises. . .
“No angels are material.”
“Some material beings are not those which move on their own power.”
“Thus, some of those which move on their own power are not angels.”
There’s a misleading saying: “you cannot prove a negative.” That statement is false. You can prove a conclusion that is negative.
For example, “No man is immortal. Socrates is a man. Therefore, Socrates is not immortal.” That’s a negative conclusion! It’s valid. And since it also has true premises, the conclusion must indeed be true.
What is the case, nonetheless, is that we cannot prove a conclusion in a categorical syllogism with only negative premises.
How can we get something genuinely new with the two negative premises? We cannot. Although “no angels are material,” that has nothing to do with some material beings that do not move themselves. These propositions are independent. It’s impossible to relate them to get the example’s conclusion.
We might be under a false impression that the middle term (i.e., material) in premise one is used in a way so that we can make a link with its use in the second premise. That is not happening.
Consider the six rules of inference. The example remarkably passes each rule but the one that demands one premise to be positive. Thinking through the reason why this rule exists will greatly enhance our understanding of logic.
Additional Fallacies. . .
—> Fallacy of Four Terms <—
Recall the “constitutive rules” of the categorical syllogism. There can only be three terms, no more or less than that. Hence, four terms is formally invalid.
In practice, this fallacy tends to occur more as an “informal fallacy.” It takes more than understanding the “form” of the syllogism, it takes an understanding of the so-called “matter.” That is, we have to understand the concrete meaning behind each term being used.
Equivocation is an informal fallacy. It’s also called an ambiguous middle. There appears to be three terms, but the middle term is being used in two different ways. Since it is being used in two different ways, there are technically four terms, not three terms, being used. Hence, it is invalid.
W. Jevons gives this example in his Elementary Lessons in Logic (p. 131):
“All metals are elements.”
“Brass is metal.”
“Therefore, brass is an element.”
It looks great formally, but bad materially. The middle term (i.e., metal) is used in two different senses. First, it’s used in a way a chemist defines metal. Second, it’s used in a more loose sense that’s found in the arts. Brass is not a metal in the first sense, as it is an alloy (of copper and zinc) and thus not an element.
—> Fallacy of Affirmative Conclusion from a Negative Premise <—
—> Fallacy of Negative Conclusion from an Affirmative Premise <—
Both violate the rules of a categorical syllogism.
Realize that these rules are not arbitrary constructions. Scroll down to review “Reflecting on the Rules (once again)” for a proof on the categorical syllogism page. Mastering the rules will allow us to spot all formal fallacies.
“All Mario games are Nintendo games.”
“Some Mario games are not on the Switch.”
“Therefore, some Nintendo games are on the Switch.”
This is invalid because it draws an affirmative conclusion from a negative premise. The conclusion is true, but it doesn’t follow from the given premises.
When not being careful, that argument can trick us! The premises are true and the conclusion is true; but, that doesn’t mean the argument must be valid.
Formal Fallacies of Conditional Arguments. . .
Recall, a mix hypothetical syllogism operates on the antecedent being sufficient (not necessary) on the consequent and the consequent being necessary (not sufficient) for the antecedent. With the form “if P, then Q,” there are accordingly two formal fallacies.
Fallacy of Affirming the Consequent. . .
“Whenever too many customers log into our Wi-Fi, our Internet goes down.”
“Our Internet went down.”
“Therefore, too many customers must have logged into our Wi-Fi.”
This argument takes on the invalid form. . .
If P, then Q.
Q.
Ergo, P.
A sufficient condition is not a necessary condition. If P were a necessary condition for Q, then Q could not occur without P. Confusing sufficient and necessary conditions produces this fallacy.
Q can be the case without P being the case.
Consider the above example…
It’s not necessary that there be too many customers logging in to make the Internet go down. Premise one only states it is a sufficient condition, not a necessary condition. When considering the material content of the example, it’s pretty clear that other things can result in the Internet going down.
To be sure, to spot that the example makes this formal fallacy depends upon understanding that this argument, as it is materially given, is of this form.
There’s a sense in which, therefore, understanding formal fallacies is always in the backdrop of informal logic and informal fallacies.
Fallacy of Denying the Antecedent. . .
“If you graduate a university, you will get a good paying job.”
“You didn’t graduate a university.”
“So, you won’t get a good paying job.”
This argument takes on the invalid form. . .
If P, then Q.
Not P.
Ergo, not Q.
This fallaciously treats the antecedent as necessary for the consequent.
Can we “corroborate” this? An interesting way to “test” to see if an argument is formally invalid is through the “counterexample method.” With the fallacy of denying the antecedent, think of two uncontroversially true premises and an uncontroversially false conclusion.
“If this animal is a man, then he is mortal.” [true]
“This animal — e.g., my pet bird — is not a man.” [true]
“So, this animal is not mortal.” [false]
Clearly, then, denying the antecedent is a fallacy. A valid argument with true premises will only produce a true conclusion. This argument has true premises but a false conclusion; therefore, it must be an invalid argument!
Fallacy of Illicit Commutativity. . .
“If it’s Saturday, then it’s the weekend.”
“Therefore, if it’s the weekend, then it’s Saturday.”
“If P, then Q” is not equivalent to “If Q, then P.” Since order matters, such propositions are not “commutative.” We cannot infer the former from the latter or the latter from the former.