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Types of Logical Fallacies and How to Recognize Them
Discrete Mathematics · Chapter 1: Logic and Mathematical ProofSection 1.2 · lesson 2 of 8712 min · free with a free account
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In this lesson
The specific ways an argument fails to earn its conclusion.
- Distinguish formal fallacies from informal ones
- Recognize the 3 formal fallacies beside the valid patterns they imitate
- Recognize fallacies of relevance and of insufficient evidence
- Examine a confident claim instead of simply accepting it
Transcript
Everything said in the video, section by section (1,657 words). Read along, or search it.
Logic and Mathematical Proof
Types of Logical Fallacies
Section one point one separated an argument's validity from the truth of its conclusion. This section names the specific ways an argument can fail to earn its conclusion. The goal is practical. A confident-sounding claim in a bug report, a code review, or a policy debate should be examined, rather than simply accepted. We'll cover three formal fallacies, beside the valid patterns they imitate, and then seven informal ones.
The Opening Problem
An Incident Channel at 2:47 a.m.
It's two forty-seven in the morning, and an on-call engineer named Priya Nair is paged. Response latency on a mid-size logistics platform has roughly tripled. The incident channel fills with theories before anyone has pulled a single log. Over the next two hours, six colleagues each offer an explanation. However confidently stated, the theories collapse the moment someone asks a single follow-up question. A fallacy is an argument that fails to give adequate support for its conclusion, whether or not that conclusion happens to be true. Fallacies fall into two families. A formal fallacy breaks the argument's structure. Replace every noun with a different one and the flaw remains, because it lives in the pattern. An informal fallacy leaves the structure alone, and breaks something else. The evidence is too thin, the appeal is irrelevant, or the argument quietly assumes what it's supposed to prove.
Formal Fallacy 1
Affirming the Consequent
A formal fallacy looks enough like a valid pattern to be mistaken for one. That's exactly what makes it dangerous. The first is affirming the consequent. A conditional statement, if condition then result, guarantees the result whenever the condition holds. But it doesn't promise that the condition is the only way to reach that result. Affirming the consequent runs the conditional in reverse. It sees the result, and concludes the condition must have caused it. Here's the broken argument. If a deployment introduces a memory leak, memory usage rises steadily afterward. Memory rose steadily after Tuesday's deployment. Therefore, Tuesday's deployment introduced a leak. The first premise describes one route to rising memory. It never says it's the only one. An unrelated nightly reporting job could have started that evening. A legitimate cache could have grown during a traffic spike. A monitoring tool could have begun sampling more often. Any one of those fits both premises and leaves the deployment blameless. So the conclusion doesn't follow with certainty.
Formal Fallacy 2
Denying the Antecedent
A closely related error runs the same conditional in the opposite direction. Denying the antecedent observes that the condition did not occur, and concludes that the result did not occur either. Here's the broken argument. If a service exposes an unauthenticated API endpoint, it's vulnerable to unauthorized data access. This service does not expose an unauthenticated endpoint. Therefore, this service is not vulnerable. Again, the first premise names one route to vulnerability, and says nothing about whether it's the only one. Unauthorized access can also come from a leaked credential, from an authenticated endpoint that never checks who owns the resource, or from a session token that never expires. Ruling out one cause doesn't clear a system of every cause of the same effect. A conditional promises what happens if the condition is met. It promises nothing about what happens if it isn't.
Formal Fallacy 3
The Undistributed Middle
The third formal fallacy concerns claims about whole categories. Two categorical claims can connect two categories, but only if the connecting category, called the middle term, is fully pinned down by at least one premise. Here's the broken argument. All relational databases store data in tables. All spreadsheet applications store data in tables. Therefore, all spreadsheet applications are relational databases. The middle term is things that store data in tables. Neither premise pins down its full membership. Each premise only places one category somewhere inside a broad class, which also includes comma-separated files and static web tables. A spreadsheet that stores values in flat, unindexed grids, with no enforced schema, makes the conclusion false while both premises stay true. And that confirms the argument was never valid, however much the two kinds of software resemble each other.
Table 1.2.1
Each Formal Fallacy Imitates a Valid Pattern
This table puts each formal fallacy beside the valid pattern it imitates. Modus ponens says, if the condition holds, the result follows. The condition holds. So the result follows. That's valid. Affirming the consequent swaps the second step. The result holds, so the condition holds. It fails, because other conditions can produce the same result. Modus tollens says, if the condition holds, the result follows. The result did not occur. So the condition did not hold. Also valid. Denying the antecedent says instead, the condition did not hold, so the result did not occur. It fails, because other paths lead to the same result. And a valid syllogism says, all M are P, and S is an M, so S is P. The undistributed middle only shows two categories sharing a property, which doesn't relate them. In every case, the difference lies entirely in which direction the inference runs, or whether the middle term is pinned down.
Informal Fallacies of Relevance (1)
Attacking the Wrong Target
Informal fallacies are different. The sentence can look perfectly well formed, while its content fails to support the conclusion. The first group shares a common thread. Each redirects attention away from the claim actually in dispute, toward something that feels persuasive but is beside the point. An ad hominem argument attacks the person, rather than the claim. Even accurate personal criticism says nothing about whether the claim is true. In the incident channel, someone replies to a junior engineer. You only joined the team three months ago, so the caching layer obviously isn't the problem. Tenure has no bearing on whether the caching layer caused the outage. The reply swaps a fact about the speaker for a fact about the system. A straw man replaces the actual claim with a more extreme version, then attacks that. Marcus asks for integration tests before shipping one new feature. Diego replies, so you want to freeze all shipping until we reach one hundred percent coverage? Refuting the exaggerated version proves nothing about the narrower request actually on the table.
Informal Fallacies of Relevance (2)
False Choices and Borrowed Credibility
A false dichotomy presents two options as if they were the only possibilities. Either we require two-factor authentication on every internal API by Friday, or we accept that a breach is inevitable. But there are reasonable middle options. Rate-limiting suspicious requests, isolating internal APIs from the public internet, or rolling out two-factor authentication to the highest-risk services first. An appeal to authority treats a claim as true because a credible source holds it. An appeal to popularity treats it as true because many people act as if it is. A major streaming company runs its whole backend on microservices. Most large engineering organizations use them. So this five-person startup should restructure into microservices. Both premises may be accurate, and neither addresses the actual question. Do microservices suit a five-person team's capacity? What a well-resourced organization does is evidence about organizations in that position. Not about a team in a very different one.
Insufficient or Circular Evidence
Evidence That Only Looks Relevant
The last three informal fallacies fail for a different reason. They offer evidence that looks relevant, but isn't sufficient, or isn't independent of the conclusion. Circular reasoning, also called begging the question, uses the conclusion as a premise, in different words. Functions must stay under twenty lines, because functions over twenty lines violate the style guide. Nothing explains why twenty lines is the right cutoff, rather than fifteen or thirty. A hasty generalization draws a broad conclusion from a sample that's too small or too narrow. A new static-analysis tool correctly flagged the two files it was tested on, so it will catch every bug in a million-line codebase. Two files license a claim about those two files only. And a slippery slope claims one step will inevitably lead, through a chain of worse consequences, to an extreme outcome. If one engineer skips code review for an urgent hotfix, soon everyone will skip it, and the codebase collapses. Each link in that chain is asserted, not demonstrated.
Figure 1.2.1
10 Fallacies, 2 Families
This figure organizes all ten fallacies into their two families. On the left are the three formal fallacies. They fail because of structure alone, whatever the subject matter. The informal fallacies fail because of content, relevance, or insufficient evidence, even when the sentence structure looks fine. In the middle are the four that redirect attention. On the right are the three that offer evidence that's circular, too thin, or an unproven chain. Naming the family a flawed argument belongs to is often the fastest way to say, precisely, what has gone wrong with it.
How the Incident Was Resolved
Independent Evidence, Not Confidence
So how was Priya's incident actually resolved? None of the early theories survived scrutiny. What worked was a different kind of reasoning. Priya cross-referenced the exact timestamp of the latency spike against the deployment log, the batch-job schedule, and the traffic dashboard. Only one candidate cause lined up within the same two-minute window as the spike. A change to a database connection-pool setting, shipped in Tuesday's deployment, and unrelated to the memory-leak theory. It worked because each candidate was checked against independent evidence. Not accepted because a colleague sounded confident, was senior, or offered the first plausible story. The valid pattern behind that reasoning, and the machinery for verifying it, is what the next few sections build, in symbolic form.
Takeaways
Name the Failure Precisely
To summarize. Formal fallacies fail because of structure alone. Informal fallacies fail because of content, relevance, or insufficient evidence. Affirming the consequent and denying the antecedent both make the same mistake. They treat one sufficient path to a result as the only path. The undistributed middle shows that two categories sharing a property aren't thereby related. Ad hominem, straw man, false dichotomy, and appeals to authority or popularity redirect attention. Circular reasoning, hasty generalization, and slippery slopes offer evidence that's insufficient, or assumes its conclusion. Here's something to think about. Build an argument that affirms the consequent, but whose conclusion happens to be true. Does the structure become acceptable because the conclusion turned out right? And why is relying on it still a poor habit, even when it works out?
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