Logical Fallacies as Bayesian Informational Shortcuts (7 of 9)

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Central Problem

Are formal logical fallacies mere errors, fit only for the conceptual junkyard, or do they provide any informational value? Traditional logic, since Aristotle, has condemned fallacies like denying the antecedent (DA) and affirming the consequent (AC) as worthless rubbish to be eradicated. Cognitive science experiments have used these fallacies to demonstrate human irrationality — that people systematically make reasoning errors.

However, this “wasteful” or “ungreen policy” toward logical fallacies may miss their potential contribution to information processes. If intelligent people consistently commit these fallacies despite training, perhaps the fallacies serve some epistemic function. The chapter asks: can we develop a “greener” interpretation that shows these fallacies provide quick and dirty, though riskier, ways to extract useful information from available resources?

The challenge connects to two standard explanations for why people accept fallacies as valid: (1) confusing “if” with “if and only if” (biconditional reading), or (2) confusing necessity with probability. Both explanations may be compatible through Bayes’ theorem.

Main Thesis

Floridi argues that formal logical fallacies — particularly affirming the consequent (AC) and denying the antecedent (DA) — are not mere mistakes proving human irrationality, but informational shortcuts that provide quick and dirty ways of extracting useful information. They are degraded versions of Bayes’ theorem, stripped of some probabilities.

The Bayesian Analysis:

For AC (A → B, B ∴ A):

• If there are no false positives (P(B|Ac) = 0), then P(A|B) = 1 and the reasoning degrades to a valid biconditional: A ↔ B, B ∴ A
• If there are some false positives (P(B|Ac) > 0), then P(A|B) < 1, and the formula resembles the AC fallacy

For DA (A → B, ¬A ∴ ¬B):

• If there are no false positives (P(B|Ac) = 0), then P(Bc|Ac) = 1 and the formula degrades to: A ↔ B, ¬A ∴ ¬B
• If there are some false positives, the formula resembles the DA fallacy

The Key Insight: The less the probabilities count (the fewer false positives/negatives), the closer these fallacies become to logically valid reasoning. They are “Bayesian ‘quick and dirty’ informational shortcuts” — when we use them, we bet that the improbable cases are negligible.

Conditions for increased information gain:

1. Soundness: A is true and A → B valid
2. Relevance: A and B are relevantly related (not independent)
3. Constraints: Events are mutually exclusive with defined sample space

Historical Context

The chapter situates itself against a long tradition of denouncing formal fallacies. Aristotle’s De Sophisticis Elenchis first condemned these reasoning patterns, arguing that people mistakenly suppose “the relation of consequence is convertible” — believing that because rain makes ground wet, wet ground implies rain.

Since the 1960s, cognitive science experiments (notably the Wason selection task, 1966) have systematically demonstrated that humans commit these fallacies, allegedly proving human irrationality. This research tradition, surveyed by Stein (1996), concluded that “cognitive science can and should play a role in determining whether or not humans are rational.”

However, semantic and social interpretations of the Wason selection task (Griggs and Cox 1982) challenged the reductionist tendency to equate rational thinking with logical reasoning. More recent Bayesian analyses of informal fallacies (Hahn and Oaksford 2006) have shown that argumentum ad ignorantiam, petitio principii, and slippery slope arguments “match structurally arguments which are widely accepted” — suggesting the problem lies in content, not form.

The connection between AC and abduction (inference to the best explanation) has become common in AI, where “backward modus ponens” is identified with abduction when properly constrained.

Philosophical Lineage

flowchart TD
    Aristotle --> Traditional
    Traditional --> Frege
    Bayes --> Floridi
    Wason --> CogSci
    CogSci --> Stein
    Hahn --> Floridi
    Oaksford --> Floridi
    Josephson --> Floridi
    Peirce --> Josephson

    class Aristotle,Bayes,Frege,Wason,CogSci,Stein,Hahn,Oaksford,Josephson,Peirce,Floridi,Traditional internal-link;

Key Thinkers

Thinker	Dates	Movement	Main Work	Core Concept
Aristotle	384-322 BCE	Ancient Philosophy	De Sophisticis Elenchis	Fallacies as convertibility error
Bayes	1701-1761	Mathematics	Essay towards solving a problem	Posterior probability theorem
Wason	1924-2003	Cognitive Science	Selection Task (1966)	Testing logical reasoning
Stein	—	Philosophy of Mind	Without Good Reason (1996)	Human irrationality thesis
Josephson	—	Artificial Intelligence	Abduction research	AC as inference to best explanation

Key Concepts

Concept	Definition	Related to
Formal Logical Fallacy (FLF)	Deductively invalid argument with appearance of validity; problem is morphological, not semantic	Logic, Aristotle
Affirming the Consequent (AC)	A → B, B ∴ A — invalid inference that affirms the consequent to conclude the antecedent	Logic, Bayes
Denying the Antecedent (DA)	A → B, ¬A ∴ ¬B — invalid inference that denies the antecedent to conclude negation of consequent	Logic, Bayes
Bayes’ Theorem	P(A|B) = P(B|A)P(A) / [P(B|A)P(A) + P(B|Ac)P(Ac)] — calculates posterior probability	Bayes, Probability
Informational Shortcut	Quick and dirty but riskier way to extract information; trades logical validity for speed	Floridi, Epistemology
False Positives	Cases where B occurs without A (P(B|Ac) > 0); their absence validates biconditional reading	Bayes, Logic
Inference to Best Explanation (IBE)	Abductive reasoning; AC read as IBE effort (retrodiction)	Peirce, Josephson
Greener Logic	Approach that recycles fallacies for informational value rather than discarding as waste	Floridi, Epistemology
Degraded Biconditional	When fallacy approximates valid A ↔ B because probabilities approach certainty	Floridi, Logic
Ontological Thrift	Trusting fewer items (being cautious) by relying on “logically greener” reasoning	Floridi, Epistemology

Authors Comparison

Theme	Floridi	Aristotle	Stein
Value of fallacies	Informational shortcuts, recyclable	Worthless errors to eradicate	Proof of human irrationality
Human rationality	Rationality regained through Bayesian lens	Rational when avoiding fallacies	Empirically undermined
Proper framework	Information-gathering and gain	Dialectical and argumentative strength	Cognitive science testing
Treatment	Greener policy — recycle	Ungreen policy — discard	Diagnose as systematic error
Role of probability	Central — degraded Bayes	Not considered	Background assumption

Influences & Connections

• Predecessors: Floridi ← revises ← Aristotle’s condemnation of fallacies
• Mathematical basis: Floridi ← applies ← Bayes’ theorem to logic
• Contemporaries: Floridi ↔ converges with ↔ Hahn, Oaksford on Bayesian informal fallacies
• AI connection: Josephson → shows → AC as abduction/IBE
• Opposing views: Stein ← claims → human irrationality from fallacy experiments

Summary Formulas

• Floridi: Formal logical fallacies (DA and AC) are degraded versions of Bayes’ theorem stripped of probabilities; they function as informational shortcuts that trade logical validity for quick information extraction from available resources.

• Aristotle: Fallacies arise because people mistakenly suppose the relation of consequence is convertible — believing that if A implies B, then B implies A.

• Bayes: The posterior probability P(A|B) depends on prior probability P(A) and likelihood P(B|A); when false positives approach zero, Bayesian updating approaches logical certainty.

• Josephson: Affirming the consequent (backward modus ponens) approximates inference to the best explanation when properly constrained; it is “smart to the degree that it implements IBE.”

Timeline

Year	Event
c. 350 BCE	Aristotle condemns fallacies in De Sophisticis Elenchis
1763	Bayes’ theorem published posthumously
1966	Wason introduces selection task experiment
1979	Marcus and Rips study participants’ judgements of MP, MT, DA, AC
1982	Griggs and Cox challenge Wason task interpretation
1996	Stein publishes Without Good Reason
2006	Hahn and Oaksford provide Bayesian analysis of informal fallacies
2008	Chater and Oaksford show cognitive science moving beyond mathematical logic

Notable Quotes

“Logical fallacies are not mere mistakes, of no value, but informational shortcuts that can be epistemically fruitful, if carefully managed.” — Floridi

“AC and DA are very powerful informational tools — inferentially, they are the equivalent of shoot first and ask questions later — developed by embedded and embodied agents to cope, as quickly and successfully as possible, with a hostile environment.” — Floridi

“The refutation which depends upon the consequent arises because people suppose that the relation of consequence is convertible.” — Aristotle

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Warning

This annotation was normalised using a large language model and may contain inaccuracies. These texts serve as preliminary study resources rather than exhaustive references.