Right-Wing Pseudo-Intellectualism and Basic Logic
An introduction to formal logic to combat the "facts and logic" right-wing
Introduction
This paper is a bit of a mix, discussing tendencies I see in modern right-wing politics, especially understanding its mixture of anti-intellectualism and pseudo-intellectualism, and trying to counteract this to some extent by providing a basic introduction to logic. There is some irony here in that, while the right-wing has made a catchphrase out of presenting itself on the side of “rationality” in contrast to the “emotional” left-wing. They do this even as the mainstream American right set themselves squarely against modern scientific knowledge on the climate, vaccines, gender, evolution, and many other issues. They have made “fact-checking” a dirty word, a conspiracy being leveled against them, even as they claim to be on the side of “facts and logic.”
I begin this paper by giving some general thoughts on how this discourse has shaped itself in the right-wing and how I view the connection between their science denial and how that helps drive them into developing pseudo-science and pseudo-philosophical arguments. Properly addressing these negative social tendencies requires a few things. While many people have focused on the very real need for people to recognize reliable sources of information and interact with them in a healthy way that isn’t blind acceptance or reflexive rejection, I am focused more on the need for clarity of thought, and therefore onto formal logic.
This is a somewhat jarring diversion to be part of a single paper, as if I switched to talking about Euclidean geometry. I hope this is made a little less jarring as I give a bit of attention to a piece of right-wing pseudo-formal logic from Stefan Molyneux, which you don’t see as frequently in these anti-intellectual circles and which I find morbidly fascinating on its own. Properly understanding just how badly Molyneux gets this wrong requires understanding some basic principles of formal logic that I introduce.
The final part though which this leads into is almost entirely devoid of political content, presenting a lengthy introduction to the fundamentals of formal logic. This is where I’d say most of the “meat” of this paper actually lies. Perhaps the best way to think of this paper is primarily an introduction to formal logic, but one where I start by trying to convince a more political audience that it is worth studying. While it starts as a general discussion of the conspiratorial madness that dominates US politics, especially in the age of the internet, the paper ends by leaving this discussion almost entirely behind to instead focus on a topic one might find in an “introduction to logic” class.
The Anti-Intellectualism of the Modern American Right
The culture of today’s American far-right is primarily driven by hatred and greed. There are many ways this can be exemplified by things like their war on compassion and declaring empathy to be a sin. For people in this camp, argument and facts can have little sway, as everything is made subordinate to increasing their own feeling of power, which they do primarily by acting in some way that is deeply repugnant on its face targeting the most vulnerable parts of the population. They want to “trigger the libs.” This culture, as it has spread so widely and become so firmly rooted online, that recently a woman raised over $600,000 for herself by calling a black 5-year-old child the n-word.
This is a despicable and terrifying tendency, enjoyed by these weirdos precisely because they enjoy the terror of their victims. The phrase “the cruelty is the point” has been something of a tired phrase at this point, but does still encapsulate how the right-wing can not only tolerate but praise these things. I am reminded of a Jean Paul-Sartre quote on the anti-Semitism he witnessed in France during and after the Nazi occupation:
The anti‐Semite has chosen hate because hate is a faith; at the outset he has chosen to devaluate words and reasons. How entirely at ease he feels as a result. How futile and frivolous discussions about the rights of the Jew appear to him. He has placed himself on other ground from the beginning. If out of courtesy he consents for a moment to defend his point of view, he lends himself but does not give himself. He tries simply to project his intuitive certainty onto the plane of discourse. I mentioned awhile back some remarks by anti‐Semites, all of them absurd: “I hate Jews because they make servants insubordinate, because a Jewish furrier robbed me, etc.” Never believe that anti-Semites are completely unaware of the absurdity of their replies. They know that their remarks are frivolous, open to challenge. But they are amusing themselves, for it is their adversary who is obliged to use words responsibly, since he believes in words. The anti-Semites have the right to play. They even like to play with discourse for, by giving ridiculous reasons, they discredit the seriousness of their interlocutors. They delight in acting in bad faith, since they seek not to persuade by sound argument but to intimidate and disconcert. If you press them too closely, they will abruptly fall silent, loftily indicating by some phrase that the time for argument is past. It is not that they are afraid of being convinced. They fear only to appear ridiculous or to prejudice by their embarrassment their hope of winning over some third person to their side. (Sartre, Anti-Semite and Jew)
This kind of mindset is, of course, common to other forms of hate, rather than being unique to anti-Semitism. The same kind of pattern can be found in white supremacy, Christian nationalism, Islamophobia, Zionism, homophobia, sexism, and so on. Someone doesn’t become, say, an Incel complaining about “femoids” without realizing on some level that they are being absurd, yet finding this to be a virtue, not a flaw, of their worldview.
As Sartre explains, it is a mistake to try and engage in honest and good faith debate with this kind of hatred. It is not a position they reasoned themselves into, and is not one they can be reasoned out of. Their fear is only towards being embarrassed, especially in front of an audience, since that is directly antithetical to the feeling of imperviousness and power. They feel no shame in being deplorable, as they want to be feared, which embarrassment undermines.
This hatred has been thoroughly mixed in with a deep level of anti-intellectualism. The American right-wing has been at war with education for quite a while. Donald Trump’s recent fights with Harvard University, the gutting of the Department of Education, and promotion of a “patriotic education” that erases the US government’s history of crimes and problems all exemplify just how extreme this attack is. In addition to this, the Republican Party has spent decades undermining their own faith in science with anti-evolution “creationism,” climate change denial, and reflexive dismissal of “fake news” for whoever practices responsible journalism. Their modern embracement of the anti-vaccine movement and attacks on transgender identities and healthcare show how this tendency is only getting worse.
However, these two aspects of the right-wing have also led to the creation of, or relied on the existence of, a kind of right-wing pseudo-intellectualism. It is not anti-intellectualism alone. Natural curiosity and demand for an answer drives even people immersed in nonsense to demand logic and coherence, or to “prove” that their side is correct. This requires at least a pretense of gesturing towards the principles of rationality and logic. They need “alternative facts,” and therefore need to at least claim to be standing up for Truth and rational discourse. In short, it leads to the creation of apologetics.
It is inherent to the nature of apologetics that the arguments are almost always bad. If they weren’t bad, the positions would be submitted to the scrutiny of scientific and philosophical scrutiny, or would be pulled from other sources who have done so. The target audience for apologetics though is not the general public or experts in a field, but for people who are already bought into an idea and looking to find any argument, no matter the quality, that sooths their fears and reassures them that their position is true and defensible. They are typically, as the Biblical scholar Dan McClellan so frequently puts it in his own critique apologetics, looking for a “sliver of not-impossible.”
At the most basic and least interesting level, this feeds into the right-wing’s online “debate” culture. Debate here is, as Sartre presented, not so much about actually doing research or being persuasive, but a way to dominate a conversation. “Debate” with bad-faith opponents becomes an exercise in just talking over others or controlling a microphone so you can “own” your opponents. It requires some level of argument, but that is always secondary to the mere spectacle, and you can “win” by just talking over someone with some well-placed insults if you need to, making them appear weak and you strong, or just suddenly changing the topic, even if all pretense of investigation is dropped.
You see this with right-wing pundits like Ben Shapiro’s “Facts don’t care about your feelings” or Steven Crowder’s “change my mind” as they try to “debate” college students in a platform where they have complete control over the microphone. The right-wing attack on empathy is especially useful here for casting their opponents as being driven by irrational “feeling,” while their performative confidence and ruthless attack can evoke a feeling of certainty and truth, even as they dismiss actual facts and evidence provided by scientists, newspapers, journalists, reports, and so on, which they can dismiss as “woke” sources with little effort.
I don’t care to engage with much of this content as it is deeply unpleasant and usually amounts to a waste of time when someone is not engaging in good faith. They are simply not putting in enough effort into actually trying to prove their case that would make disproving them satisfying or effective. That is why the few gems that I find require someone to buy their own BS at least a little bit, and attempt to create something approaching a rigorous system. Ayn Rand’s essay “Objectivist Ethics,” which I recently wrote a paper on, is an excellent example here. Born from her extreme kind of anti-intellectualism, handwaving away the entire history of ethical philosophy, she then has what she believes is a novel solution of trying to apply reason to ethics. From this she gives a few arguments which pantomime something like a rational argument, but which falls apart on close examination.
I say all this to emphasize that, against our enemy, logic and reason can only go so far. In the best-case scenario, some people in this area of the right-wing can really be persuaded by argument if they take these “apologetics”-type arguments seriously if they are shown the flaws in that reasoning and exposed to better alternatives. But all too often, the function of apologetics is merely social, and no argument could ever be sufficient because arguments and reason were never the determining factors in what they believe. I also say this to emphasize that, in our own practice, we need to be careful not to fall into the same traps.
This works as a long introduction for me to lead into some of the basics of formal logic. This is important, not because it will let us win debates or because we need to fit all of our arguments into syllogistic forms, but because it helps our own practice and promotes clarity of thought, showing how we can put something under stricter scrutiny.
Truth, Validity, and Stefan Molyneux’s Art of Confusion
The white supremacist Stefan Molyneux published his book “The Art of The Argument: Western Civilization's Last Stand” in 2017. In it he describes “The Argument” in extremely religious language, giving it a kind of mystical importance. He does this in its very opening lines:
A professional philosophical logician, Cian Chartier, took interest in Molyneux’s claim to be providing the fundamentals of a good argument and wrote a wonderful review that explores how flimsy and slippery Molyneux’s idea of “The Argument” really is. Molyneux uses terms that are common to formal logic, like an argument being sound compared to it being valid, but he defines these terms in ways that run entirely counter to their standard definitions, if they don’t contradict it entirely. As Chartier puts it:
Many newcomers to logic reading the book will not know the terminology being confused, and might not care that it’s being confused. Molyneux demonstrates little respect for virtually any academic philosophy written in the past 50 years, and neither do most of his avid readers. Why should they care about the right terms to use? Does it matter what you call a tool, like a valid argument, as long as it does the job?
In the teaching of philosophy, it matters. Ideally, if you want to teach people to reason with others in good faith, you should teach them in the same standard terminology that the rest of the English-speaking field uses, or at least mention where you significantly differ. Otherwise, you’re training people to confuse others.
Training people to confuse others is, of course, the point. Molyneux, like Ayn Rand or Murray Rothbard, wants his audience confused. He uses his anti-intellectualism to isolate his audience, and then further reinforces this isolation by inventing his own jargon, his own kind of New Speak, to make it impossible for anyone within his audience to engage with the outside world, and for the outside world give up on the effort of actually untangling his Gordian knot. It is to Chartier’s great credit that he took the time and effort to do so.
I highly recommend Chartier’s article and will be borrowing some of his observations about to help elaborate on formal logic, with Molyneux’s mistakes acting as an excellent contrast.
To begin, logic divides different kinds of arguments into different categories. To give two major examples, there is deductive reasoning and inductive reasoning. Deductive arguments attempt to show that certain conclusions necessarily follow from certain premises, while inductive reasoning attempts to show that something likely follows from their premises. You can think of deductive reasoning as being like arithmetic, showing that a certain answer absolutely follows from an equation, while inductive reasoning is more like statistics, showing you the probability of getting a certain answer based on the data.
To help illustrate this, we can look at a few examples of each kind of argument:
Inductive Reasoning
Premise 1: Almost all students did well on their history test.
Premise 2: Ann is one of the students that took this history test.
Conclusion: Therefore, Ann probably did well on the test.
Deductive Reasoning
Premise 1: All men are mortal.
Premise 2: Socrates is a man.
Conclusion: Therefore, Socrates is mortal.
These kinds of arguments are therefore evaluated in very different ways. An inductive argument does not give us certainty, but might be stronger or weaker in support of some conclusion. By contrast, deductive arguments are supposed to give us certainty if the premises are true, making the argument valid or invalid. Importantly, a deductive argument might be valid even when its conclusion is false. This is because logical validity is not testing the actual content of an argument, only its form. It tells us whether a conclusion must be true given the truth of its premises.
To show this, we can look at a deductive argument that has the same form as before, but which is clearly false.
Premise 1: All dogs are cats.
Premise 2: Spot is a dog.
Conclusion: Therefore, Spot is a cat.
This argument is valid because the conclusion really does follow from the premises. If all dogs were cats, then Spot being a dog would necessarily make her a cat. The problem is not with the form of this argument, but with the truth-value of its premises. We know that not all dogs are cats, which means we know that at least Premise 1 is wrong. Premise 2 and the conclusion might be wrong too, like if it turns out that Spot is actually a bird, but none of this actually affects the validity of the argument itself since the conclusion really does follow from the premises.
Likewise, we can also have an argument which is made up entirely of true statements, yet is logically invalid. Take this argument, for example:
Premise 1: All squares have four sides.
Premise 2: All squares are rectangles.
Conclusion: Therefore, all rectangles have four sides.
Each statement here taken on its own is true. However, the logic here doesn’t actually work. It does not follow from the fact that all squares have four sides that all rectangles must have four sides. Rather, we’d need to reason in the other direction, showing that we know all squares have four sides because all rectangles do, and squares are a kind of rectangle.
Validity and truth are therefore two separate concepts. In logic, we call a deductive argument that is both valid and uses true premises a “sound argument.” Our first argument about Socrates, for example. Otherwise an argument is “unsound” because it either reasons in an invalid way, or because at least one of its premises are false.
Logic, properly speaking, is focused on the study of the structure of arguments and the rules of inference. It studies the form of argument while looking past its particular content. In this way, we can look at a few different statements that might have the same by replacing certain parts with variables. For example, the statements “all men are mortal,” “all dogs are cats,” and “all squares are rectangles” have the same form of “all Xs are Ys.” We can similarly look at the structure of entire arguments this way.
For example, both of our examples of valid logical reasoning had arguments of this structure:
Premise 1: All M are P. (“All men are mortal” or “all dogs are cats”)
Premise 2: Some S are a M. (“Socrates is a man” or “Spot is a dog”)
Conclusion: Therefore, some S are a P. (“Socrates is mortal” or “Spot is a cat”)
We use “all” to indicate every member of the named group, while “some” indicates that we are asserting the existence of at least one member of this group, which is implicit in naming certain individuals like “Socrates” or “Spot.” All arguments that match this kind of logical format are valid.
By contrast, consider the structure of the invalid argument from before about squares and rectangles:
Premise 1: All M are P. (All squares have four sides)
Premise 2: All M are S. (All squares are rectangles)
Conclusion: Therefore, all S are P. (All rectangles have four sides)
Presented like this, the difference between this argument and the previous one becomes immediately clear. We will go into precisely what makes it invalid later, but for now it is enough that we can recognize its entire structure is different from what we saw before.
What then about Stefan Molyneux? How does he approach these basic ideas of logic? Let’s see.
This section is appropriately named “examples of bad deductive reasoning.” However, when Molyneux gets around to explaining why it is bad deductive reasoning, he completely misses the mark. His objections are directed at the truth of the premises and the conclusion, rather than whether the argument is valid.
However, when Molyneux analyzes this, he gets his terminology mixed up. He says “the only thing that could be provisionally valid is the proposition that Bob knows how to swim.” But validity is something we say about arguments, not propositions. Molyneux seems to have entirely confused ‘validity’ and ‘truth.’ But even that isn’t entirely clear, since he says that it is the only thing that might be true when Bob could, of course, both know how to swim and be a plumber, even though we know not all plumbers can swim. So it remains a complete mystery what Molyneux means by ‘valid.’ Chartier explores this in more detail.
So what is actually wrong with the first argument? Let’s figure it out by putting it in a more general notation, like we used before:
Premise 1: All P are M. (All plumbers can swim.)
Premise 2: Some S are M. (Bob knows how to swim.)
Conclusion: Therefore, some S are P. (Bob is a plumber)
Specifically, the formal fallacy committed by this argument is called the “undistributed middle.” Neither premise of this argument addresses all members of middle term, M. It is therefore impossible to use the middle term to guarantee a connection in the conclusion. (For the record, we use S to indicate the subject of the conclusion and P to indicate what is being predicated of subject.)
We will cover the different kinds of fallacies later, but for now it is enough to point out that Molyneux ignores explaining things like this entirely. For Molyneux here, logic and argument aren’t a matter of understanding the structure of the actual argument made, but simply asserting what he feels is obvious to be true. It has nothing to do with logic whatsoever.
This is made especially clear when we examine the second argument Molyneux presents, which he clearly (and mistakenly) believes to be the same kind of mistake as the first argument.
Premise 1: All M are P. (Kind people are socialists.)
Premise 2: Some S are M. (Bob is a kind person.)
Conclusion: Therefore, some S are P. (Bob is a socialist.)
Molyneux presents this as a common mistake, as an example of bad deductive reasoning that his study of “The Argument” has revealed to him to be faulty.
But this is a perfectly valid argument. If all kind people are socialists, and Bob is a kind person, then he must be a socialist. The conclusion is entailed by the premises.
Molyneux might avoid this if he meant to say that only some kind people are socialists. The ‘all’ in his argument was implicit if he is just definitively declaring “kind people are socialists,” especially given that his attacks on the first argument was about the truth or ‘provisional validity’ of statements that made inaccurate claims about all members of some group. If Molyneux meant ‘some,’ then this argument would be invalid, but in that case his critique of the first argument doesn’t apply to the second.
I find Molyneux’s errors here endlessly entertaining, and there are many more after this, but I will leave that to Chartier’s critique. Instead, I will end this article by putting away all this right-wing nonsense and instead focus on explaining other basic points of formal logic. Hopefully any readers will find this as interesting and useful as I do, but do keep in mind that from here on this paper is, for the most part, setting politics aside and instead is just giving a crash course in logic 101.
Basics of Categorical Logic
What is a Categorical Syllogism?
So far we have been examining “categorical syllogisms.”
A syllogism is a deductive argument made up of three propositions (i.e. statements which can be true or false). Two propositions are used as premises, and one proposition is presented as a conclusion derived from the premises.
A categorical syllogism is a syllogism made up of categorical propositions. These are propositions that come in one of four forms, depending on whether they are universal or particular and whether the are affirmative or negative. They are traditionally represented by the vowels A, E, I, and O, representing four moods.
A: All S are P. (Universal Affirmative)
E: No S are P. (Universal Negative)
I: Some S are P. (Particular Affirmative)
O: Some S are not P. (Particular Negative)
The Square of Opposition
The relationship between these four kinds of statements are often represented by the “square of opposition,” which works so long as we assume that we are not working with any empty sets (i.e. at least one member of the category exists).
You can see how the notations of A, E, I, and O are all being used here. For example, the A-form of “All S are P” has been reduced to “SaP.” This is also represented using Venn diagrams. The black areas represent that the area is empty (e.g. if “No S are P,” then the intersection between S and P in the diagram is black). The red areas represent areas which are not empty. I and O statements have “existential import” because they aren’t just saying anything about the category, but asserting the existence of something in that category, also represented here with an “x.” The white areas may or may not be empty.
Propositions are contradictory if they cannot both be true and cannot both be false. For example, if “all cats love lasagna” is true, then “some cats do not love lasagna” is false.
Propositions are contrary if they cannot both be true, but might both be false. If “all balloons are red” is true, then “no balloons are red” is false. But if some balloons are red and some are not red, then both statements can be false at the same time.
Propositions are subcontrary if they cannot both be false, but might both be true. If “some This is, again, assuming we are not dealing with any empty sets. So long as at least one member of this group exists, then there is at least one case where the predicate either applies or does not apply. The statements “some trees are evergreens” and “some trees are not evergreens” can both be true at the same time, but as long as trees exist they cannot both be false.
Propositions is a subaltern of another if it must be true whenever its superaltern is true. If “all S are P” is true, and we assume the existence of at least one member of S, then we know that “some S are P” is also true. We can say “some recipes use potatoes” and “some recipes do not use potatoes” are both true, but they cannot both be false so long as at least one recipe exists.
The examples I provide should also illustrate that the English statements that could be turned into a categorical syllogism do not need to match it verbatim. If I say “no beaches have sand,” I can treat this as a kind of “no S are P” statement, even though it uses the word “have” instead of “are.” If necessary, we could think of logically equivalent statements like “no beaches are sandy” or “no beaches are sand-havers.” These kinds of standardized formats help remove some of the ambiguity that might exist in more everyday speech.
We can also see from this relation why it’s important that, for all this to work, we need to assume are not dealing with any empty sets. If I say “all leprechauns wear green,” is this true or false? We might be tempted to say it’s true, since, as far as I know, leprechauns do not exist, meaning there are no cases of leprechauns who do not wear green. But we could not follow the square of opposition here to therefore declare the subaltern that “some leprechauns wear green,” since that would declare the existence of at least one leprechaun. Likewise, if we can say “all leprechauns wear green” is true by virtue of there being no leprechauns, it would be equally true to say “no leprechauns wear green.” That would give us two contrary statements which are true simultaneously, which the very idea of a “contrary” was meant to exclude.
As we saw before, the categorical syllogism uses three terms. There is the subject of the conclusion (S), what is being predicated of the subject in the conclusion (P), and the middle term being used to link the other two in the premises (M) while not appearing in the conclusion. It is a fallacy to use more than three terms in a categorical syllogism, often called the fallacy of four terms.
We call the premise which links M with P the major premise, while the premise which links M with S is the minor premise. Traditionally, the major premise is always given first, and the minor premise is given second. We can arrange each premise in two different ways. For example, the major premise might be “all M are P,” or it might say “all P are M.” This ordering is very important for determining whether the argument is valid or invalid. In total, we can order these in four different orders, which gives us for figures for the categorical syllogism:
Figure 1
Major Premise: M - P
Minor Premise: S - M
Figure 2
Major Premise: P - M
Minor Premise: S - M
Figure 3
Major Premise: M - P
Minor Premise: M - S
Figure 4
Major Premise: P - M
Minor Premise: M - S
We therefore have four different figures made up of two premises (the major and the minor), each of which will be one of four distinct kinds of statements (A, E, I, or O). Additionally, there are four kinds of statements the conclusion might be. This means we have a total of 256 possible categorical syllogisms (4 x 4 x 4 x4 = 256). We can use this shorthand to refer to the different kinds of statements.
Take our earlier example of a categorical syllogism:
Major Premise: All men are mortal.
Minor Premise: Socrates is a man. (Or, “Some individual named Socrates is a man.”)
Conclusion: Therefore, Socrates is mortal.
The subject of the conclusion is “Socrates.” The predicate of the conclusion is “mortal.” The middle term is “men.” We can therefore present this syllogism like so:
Major Premise: All M are P.
Minor Premise: Some S is M.
Conclusion: Therefore, some S is P.
As we can see, the major premise is an A-type proposition (All S are P), while the minor premise and the conclusion are both I-type propositions (Some S is P). We can also recognize that this is a Figure 1 syllogism (M-P, S-M). Or if we want to use a short code for this specific kind of syllogism, refer to the three moods that make it up and combine it with the argument’s figure to give us the an “AII-1” syllogism. The first letter refers to the major premise, the second to the minor premise, and the third to the conclusion. Each of the 256 kinds of categorical syllogisms can be expressed using this kind of code. For example, we could have a EIO-4 syllogism, a AAO-3 syllogism, etc.
However, not every categorical syllogism will be valid, as we have seen. In fact, of these 256 syllogisms, only 24 are valid. Of these 24, only 15 are unconditionally valid, while 9 are only conditionally valid if we assume no empty sets. Following medieval traditions, each one of these 24 syllogisms have been given mnemonic names.
For example, the AII-1 syllogism we saw before is “Darii” syllogism, since that name has the vowels A, I, and I. Using overlapping Venn diagrams, we can illustrate this logic like so:
I leave the other kinds of valid syllogism listed above as an exercise for any readers, if you want to test them yourself.
Every one of the remaining 232 categorical syllogisms will commit at least one fallacy that renders them invalid. To understand this, we should examine the rules that determine the validity of the syllogism.
We have already mentioned one possible error already with Molyneux’s example of Bob the swimming plumber: The fallacy of the undistributed middle.
A term is considered “distributed” if every the proposition refers to every member of a given class.
If I say “all candy is sweet,” then this distributes the first term “candy” but not the second “sweet.” I know something about all candy (that it is sweet), but I do not know something about all sweet things. All I know is that some sweet things are candy.
By contrast, if I say “no candy is bitter,” then that distributes both candy and bitter. I now know something about all candy (that it is not bitter) and about all bitter things (that it is not candy).
If I say “some candy is sour,” this doesn’t distribute either the first or second term. I don’t know anything about all candy or about all sour things. I just know some candy is sour, and I know some sour things are candy.
Finally, and a bit awkwardly, if I say “some candy is not chewy,” this does not distribute “candy,” but it does distribute “chewy.” This is because we now know that “All chewy things are not some candy.” There is a group (some candy) which all chewy things are excluded from. For example, some candy are bars of dark chocolate, which are not chewy. This is equivalent to saying, “all chewy things are not bars of dark chocolate.”
For a syllogism to be valid, the middle term needs to be distributed in one or both of the premises. If we don’t know something about all members of the middle term, then we can’t use it to arrive at our conclusion.
Besides this fallacy, there are a few other rules syllogisms must follow:
If P is distributed in the conclusion, it must be distributed in the major premise. Otherwise this is the fallacy of the illicit major.
Example: AEE-1
Major Premise: All stars are bright. (A: All M are P.)
Minor Premise: No lightbulbs are stars. (E: No S are M.)
Conclusion: Therefore, no lightbulbs are bright. (E: No S are P.)
If S is distributed in the conclusion, it must be distributed in the minor premise. Otherwise this is the fallacy of the illicit minor.
Example: AAA-4
Major Premise: All tigers are cats. (A: All P are M.)
Minor Premise: All cats are animals. (A: All M are S.)
Conclusion: Therefore, all animals are tigers. (A: All S are P.)
At least one of the premises must be affirmative (i.e. an A or an I proposition). Two negative premises is the fallacy of exclusive premises.
Example: EEE-3
Major Premise: No carrots are fruit. (E: No M are P.)
Minor Premise: No carrots are apples. (E: No M are S.)
Conclusion: Therefore, no apples are fruit. (E: No S are P.)
If one of the premises are negative, then the conclusion must be negative too. Otherwise this is the fallacy of the affirmative conclusion from a negative premise.
Example: EEA-2
Major Premise: No dollars are euros. (E: No P are M.)
Minor Premise: No yen are euros. (E: No S are M.)
Conclusion: Therefore, all yen are dollars. (A: All S are P.)
If both premises are affirmative, then the conclusion must be affirmative too. Otherwise this is the fallacy of the negative conclusion from affirmative premises.
Example: AAO-4
Major Premise: All virtues are dispositions to excellence. (A: All P are M.)
Minor Premise: All dispositions to excellence are good habits. (A: All M are S.)
Conclusion: Therefore, some good habits are not virtues. (O: Some S are not P.)
Additionally, if we allow for empty sets, then there is also a fallacy committed if two universal premises are used to arrive at a particular conclusion. But if we assume that at least one member of each term exists, then there is no fallacy in this. Hence the 9 conditionally valid syllogisms out of the 24.
All of these are formal fallacies. This means the argument has been made invalid by the very logical structure of the argument itself. This might be contrasted to informal fallacies, which is more about issues in arguments that arise in ordinary language. For example, the issue with an ad hominem (“to the person”) argument has nothing to do with structure of the argument itself, but the fact that the argument is being directed at the wrong thing (i.e. the person instead of the substance of the argument itself).
In my experience, the people that tend to be the most annoying about logic know very little about the above analysis of syllogisms, and have more or less just memorized a list of informal fallacies to yell out in (mostly online) debates that may or may not apply.
Credit to Existential Comics.
People like to associate logic and reason with debate because, optimistically, debate should be about people coming together to really understand and interrogate different topics. But in practice, debate is more a matter of rhetoric, not logic, which opens the door to sophistry. Randomly gesturing towards things used in logic, like the names of fallacies, may be good practice if you want to look good in front of an audience that doesn’t know any better. But if you really want to understand logic as a way to improve yourself and get a better grasp of truth, the actual formal study can help you here, but it can’t work just as something quick you can throw out to look good in front of strangers.
Propositional and Symbolic Logic
Categorical logic is an important part of classical logic, found in Aristotle, but logic of course extends far beyond this. Some syllogisms do not involve categorical propositions. For example:
If it is raining outside, then the ground is wet.
It is raining.
Therefore, the ground is wet.
This is an example of a modus ponens argument, also called “affirming the antecedent.” Unlike with categorical logic, this argument uses a conditional “if-then” proposition. This argument always takes this kind of form:
If P, then Q.
P.
Therefore, Q.
It is closely related to another similar argument, the modus tollens or “denying the consequent.”
If P, then Q.
Not Q.
Therefore, not P.
As readers here might remember, I reduced Friedrich Engels’ argument in his essay “On Authority” to a kind of modus tollens argument in my paper Read On Authority.
Premise 1: If the anarchists are correct, then socialists must reject all authority. (If P then Q.)
Premise 2: Socialists must not reject all authority. (Not Q.)
Certain forms of authority are inevitable given the conditions of production and will therefore be necessary in a future socialist society.
Workers must exercise authority in the present to abolish capitalism.
Conclusion: Therefore, the anarchists are not correct. (Not P.)
I objected to Engels by arguing that he is either (1) defining authority in such a way that anarchists would not require socialists to reject all authority, (2) he is defining authority in a way consistent with anarchists, but is then wrong about why he thinks socialists must not reject all authority, or (3) he is switching back and forth between definitions and is just equivocating terms.
We keep seeing examples for how the study of logic can be limited by ambiguity in phrases, even when we present things in more standardized formats compared to how we’d phrase things in ordinary language. To help get around this problem, modern logicians often present the ideas of logic in symbolic forms, not unlike mathematics, and which therefore also allows us to express propositions which would be far too confusing if expressed in ordinary language. Advances in this direction have been essential to developing things like computer programming, which needs to go through these kinds of complicated checks to function.
To help with this introduction, we can look at a few of the basic logical operators used in symbolic logic. Let’s say there are two basic propositions:
April drives her car.
Brandon walks outside.
We will assign each statement here the symbols A and B respectively. In this case, if I wanted to represent the proposition “April drives her car and Brandon walks outside,” I could do so like this: “A∧B”.
The symbol “∧” represents a conjunction, read here as “and.” More specifically, it means that the expression A∧B can only be true if both A and B are true. If one or both of these propositions are false, then A∧B is false.
By contrast, the symbol “∨” represents a disjunction, read as “or.” This means that the expression A∨B is true A is true, B is true, or both are true. A∨B is only false if both A and B are false.
Here is a quick list of a few symbols for reference:
Conjunction
Symbol: ∧ or · or &
Read As: “And”
Explanation: A∧B is true if and only if both A and B are true
Disjunction
Symbol: ∨
Read As: “Or”
Explanation: A∨B is true if A and B are not both false
Negation
Symbol: ¬ or ~ or !
Read As: “Not”
Explanation: ¬A is true if and only if A is false.
Material Conditional
Symbol: → or ⇒ or ⊃
Read As: “Implies” or “if the previous, then the following”
Explanation: If A→B is true, then it is not the case that A is true and B is false.
Material Equivalence
Symbol: ⇔ or ↔ or ≡
Read As: “If and only if” or “iff”
Explanation: A⇔B is true only if both A and B are false or both A and B are true.
Exclusive Disjunction
Symbol: ⊕ or crossed out versions of the material equivalence symbols
Read As: “Either… Or… But not Both”
Explanation: A⊕B works like a normal disjunction, except it is false when both A and B are true. So if A is true then B is false, and if B is true then A is false.
We can also see with this how these symbols can help remove some ambiguity in our more standard language. Take A⇔B for example. If I say, “April drives her car if and only if Brandon walks outside,” this sounds like I’m implying some kind of causal relationship between the two, as if April driving her car is causing Brandon to walk outside or something. But none of that is implied by these symbols though, which is only asserting something about the relation between the truth-values of propositions. If, just by complete coincidence, it is both true that April drives her car and Brandon walks outside, then A⇔B is true, despite one not causing the other.
This can be represented on a “truth table,” show how the different truth values of A and B (i.e. whether they are true or false) determines the value of the expression.
As this gets more complex, it is normal for parentheses to be used to contain one section together, noting it should be done first or that the operation outside of it is applying to everything within the parentheses, just as in mathematics.
With this truth table, we can see a proof that “If p then q” is materially equivalent to “not-p or q” since they have the same truth tables. On the left we have the different possible combination of truth-values for p and q. Either they are both true, one is true, or neither are true. We can then consider how these influence the truth-value of the different parts of the expression. (¬p∨q) is made up of ¬p. Whenever p is true, ¬p is false, and whenever p is false, then ¬p is true. Knowing this, we can tell whenever ¬p∨q is true or false, and we can see this perfectly matches whenever p→q is true or false.
Symbolic logic also allows us to use quantifiers, which brings us back to what we saw in categorical syllogisms. Consider these symbols:
Universal Quantification
Symbol: ∀
Read As: “For All” or “Given Any”
Explanation: ∀xP(x) means “for all x, x has the property of P.”
Existential Quantification
Symbol: ∃
Read As: “There exists”
Explanation: ∃xP(x) means “there exists at least one x that has the property P.”
With these symbols on hand, we can actually reconstruct the four types of categorical propositions we saw earlier.
A: All S are P.
∀x(S(x) → P(x))
For all x, if x has the property of being an S, then x has the property of being P.
E: No S are P.
∀x(S(x) → ¬P(x))
For all x, if x has the property of being an S, then x does not have the property of being a P.
I: Some S are P.
∃x(S(x) ∧ P(x))
There exists some x where x has the property of being an S and the property of being a P.
O: Some S are not P.
∃x(S(x) ∧ ¬P(x))
There does exist some x where x has the property of being an S and does not have the property of being a P.
Importantly, we can note that these formulations of the universal statements do not carry any existential import. In other words, it is not working on the assumption that we are not working with any empty sets. We therefore cannot deduce that “some S are P” from “all S are P.” In fact, the only thing that remains from the Square of Opposition is that contradictory statements. An A statement can not only be presented as “∀x(S(x)) → P(x)),” but as “¬∃x(S(x) ∧ ¬P(x)).”
With these tools, much more complex arguments can be expressed. To give one example I find cool, here is Kurt Gödel's ontological proof for the existence of God using symbolic logic.
If you wish, you can walk through this argument by understanding just a few more terms:
The diamond means “it is possible that…”
The box means “it is necessary that…”
P(x) means x is a positive property
G(x) means x has the property of being God or is God-like
“φ ess x” means φ is an essential property of x
E(x) means x exists necessarily
To get things started, the first axiom is stating that “If φ is a positive property, and if any x that is a φ must also be a ψ, then ψ is also a positive property.” In other words, if something is a positive property, and another property necessarily follows from it, then the property following from it is also positive.
More generally, the development of this kind of mathematical logic, especially found from Boole, Frege, and Russell, was an extremely significant development in the history of logic, presenting a rather radically different way at approaching logical problems compared to the more limited forms we saw before with categorical syllogism. Advances in modern mathematics, especially as seen in things like set theory, and computer science go hand-in-hand with these developments. Symbolic logic was especially important for Bertrand Russell and Alfred North Whitehead’s Principia Mathematica, attempted to demonstrate the logical foundations of mathematics, including this (in)famous “occasionally useful” proof that 1+1=2:
I believe that logic of this kind is fascinating in its own right, but this paper is not meant to be much more than an introduction to logic and an encouragement for others to show that it is worthy of study. The study of political philosophy and ethics is, of course, not rooted in logic in the same way mathematics is, but I believe the general practices and habits learned by studying symbolic logic are transferrable. Being able to analyze and interrogate arguments and grasping the underlying principles to things or understanding what is needed for actual proofs or being able to identify to identify weaknesses in arguments that might not otherwise be obvious. My own use of this within Read On Authority hopefully helps to serve as one small example.
If this is useful when dealing with as intelligent a figure as Engels, it is far more useful, sometimes even trivial, when dealing with right-wing pseudo-intellectuals like the Molyneuxs or Rands of the world. Their attempts to distort the foundations of logic in their general war on truth itself should also highlight why even a rudimentary understanding of these disciplines can help people to avoid their simple errors, or at least help stop their strategy of deliberately confusing concepts as a way to get better control of their audiences. But as we go beyond these figures into actually developing logic, math, and science beyond the rudimentary, the more we leave the conspiratorial and pseudo-intellectuals behind.
For anyone interested in getting a better understanding of sophistry and the corruption of rhetoric in politics, I think a good place to begin is Plato’s Gorgias. For those who do want to continue their study of logic, I will borrow one more thing from Chartier’s paper: his list of recommended introductions to logic for people interested in learning more.
Introductions to logic:
Irving M. Copi and Carl Cohen — Introduction to Logic, Fourteenth Edition (Pearson Education Ltd., 2014) [For those without a mathematical background]
Elliott Mendelson — Introduction to Mathematical Logic, Fourth Edition (Chapman and Hall, 1997) [My own introduction, but only for those who already have a mathematical background, i.e. are experienced with the practice of mathematical proof]
L.T.F. Gamut — Logic, Language and Meaning, Vols. 1 & 2 (University of Chicago Press, 1991) [Provides an introduction to logic from the perspective of its applications to formal semantics, culminating with possible worlds semantics and Montague grammar. For those familiar with the content of a mathematical logic textbook like Mendelson’s only Vol. 2 is necessary.]
Introductions to argumentation and rhetoric:
Anthony Weston — A Rulebook for Arguments, Fourth Edition (Hackett Publishing Co., 2009) [A concise instruction for undergraduate students across disciplines on how to write persuasively]
Stephen Toulmin, Richard Rieke, and Allan Janik — An Introduction to Reasoning, Second Edition (Macmillan Publishing Co., 1984) [A wide-ranging introduction to the study of argumentation]
This was a dense but rewarding read.