Lesson 6: Reason: Logic and Its Limits

When we say "that is not logical" or "be reasonable", we appeal to reason: the use of structured thinking to move from what we accept to what follows from it. Reason is central to mathematics, philosophy, law and science. Yet TOK asks a sharper question: how far can reason take us, and where does it fail?

Three patterns of reasoning

Deduction moves from general premises to a specific conclusion that is *guaranteed* if the premises are true.

• All mammals breathe air. Whales are mammals. Therefore whales breathe air.

A deductive argument is valid if the conclusion follows necessarily from the premises, and sound if it is valid *and* its premises are true. Validity concerns form; soundness concerns form and truth. "All birds can fly; penguins are birds; therefore penguins can fly" is valid but unsound, because the first premise is false.

Induction moves from specific observations to a general claim that is *probable*, not guaranteed.

• Every swan I have seen is white; therefore all swans are white.

Europeans believed this until black swans were found in Australia. Induction drives science (we generalise from experiments), but it can never offer certainty. The philosopher David Hume pointed out that the assumption "the future will resemble the past" cannot itself be proved without circular reasoning: the problem of induction.

Abduction (inference to the best explanation) starts from an observation and proposes the most plausible explanation. A doctor sees a fever, cough and a particular scan, and infers pneumonia. Detectives, scientists and historians often reason this way. The risk: the best explanation we can think of may not be the true one.

Formal logic and its power

Logic gives us tools to check argument structure. Two valid forms:

• Modus ponens: If P then Q. P. Therefore Q.
• Modus tollens: If P then Q. Not Q. Therefore not P.

And two classic invalid forms that people often mistake for valid ones:

• Affirming the consequent: If P then Q. Q. Therefore P. (If it rained, the street is wet. The street is wet. Therefore it rained. Not necessarily: a hose could have been used.)
• Denying the antecedent: If P then Q. Not P. Therefore not Q.

Fallacies in everyday reasoning

• Ad hominem: attacking the person rather than the argument.
• Straw man: misrepresenting an opponent's view to refute it more easily.
• False dilemma: presenting two options when more exist.
• Appeal to authority: accepting a claim because someone prominent said it, without checking relevance or evidence. (Expert testimony is legitimate when the expert is in the relevant field and there is broad agreement; the fallacy is the unexamined appeal.)
• Slippery slope: claiming without evidence that a small step will lead to extreme consequences.
• Circular reasoning: assuming what you set out to prove.
• Post hoc ergo propter hoc: assuming that because B followed A, A caused B.

Recognising fallacies is useful, but TOK cautions against merely labelling arguments. Ask whether the reasoning is weak, and also why it is persuasive.

The limits of reason

Cognitive biases. Psychologists have documented systematic patterns in human reasoning. Confirmation bias leads us to seek and favour information that supports what we already believe. The anchoring effect makes an initial number influence later estimates. The availability heuristic makes vivid events (plane crashes) seem more probable than common ones (car accidents). Daniel Kahneman and Amos Tversky's research on heuristics showed that intuitive judgement can be fast and efficient but error-prone.

Bounded rationality. Herbert Simon argued that real decision-makers have limited information, time and cognitive capacity, so they "satisfice" (accept good-enough solutions) rather than optimise.

Rationalisation. Reason can be used after the fact to justify decisions made on other grounds. People sometimes build logical-looking arguments for conclusions they were already attached to.

Formal limits. In mathematics, Kurt Gödel's incompleteness theorems (1931) showed that in any consistent formal system rich enough to express basic arithmetic, there are true statements that cannot be proved within the system. This limits the dream that all mathematical truth could be captured by a single complete set of axioms. (Be careful: the theorems apply to specific kinds of formal systems; they are often over-generalised in popular writing.)

Reason across the areas of knowledge

• Mathematics: proof by deduction gives certainty relative to axioms.
• Natural sciences: combines deduction (predictions from theory) and induction (generalising from data), with a role for abduction in forming hypotheses.
• History: reasoning about evidence to infer the most plausible account; counterfactuals ("what if?") are reasoning tools but speculative.
• The arts: reasoned interpretation and critique, though artists may deliberately challenge logic.
• Ethics: reasoning about moral principles; disagreement may arise from different starting premises, not just faulty logic.

A case study: the Monty Hall problem

You face three doors: behind one is a car, behind the others goats. You choose door 1. The host, who knows what is behind each door, opens door 3 to reveal a goat and offers you the chance to switch to door 2. Should you? Many people (including many mathematicians when it was popularised in 1990) feel it makes no difference: two doors remain, so 50:50. Careful reasoning shows that switching wins with probability 2/3. Your first choice had a 1/3 chance of being right, and the host's action does not change that. The case illustrates that intuition can mislead us even when we feel logical, and that reasoning can correct intuition, but only if we slow down and examine our assumptions.

import random
def play(switch, n=100000):
    wins = 0
    for _ in range(n):
        car, pick = random.randrange(3), random.randrange(3)
        opened = next(d for d in range(3) if d != pick and d != car)
        if switch: pick = next(d for d in range(3) if d != pick and d != opened)
        wins += pick == car
    return wins / n
print(play(False), play(True))     # about 0.33 and 0.67

Possible knowledge questions

• To what extent can reason alone justify knowledge claims?
• How do different areas of knowledge rely on different forms of reasoning, and what does this imply for their certainty?
• To what extent do cognitive biases undermine our confidence in reasoned conclusions?
• Is a valid argument always a reason to accept its conclusion?

Case file: the Challenger decision

On the eve of the Space Shuttle *Challenger* launch in January 1986, engineers at a contractor expressed concern that cold temperatures could reduce the flexibility of rubber O-ring seals. Managers asked them to "prove" the launch was unsafe rather than prove it was safe, and the launch went ahead; the shuttle broke apart 73 seconds after liftoff. Later investigations, including the physicist Richard Feynman's demonstration with ice water, highlighted failures of reasoning and organisational culture. Features relevant to TOK: the burden of proof was reversed; normalisation of deviance meant earlier minor failures were treated as acceptable; the data presented did not make the relevant pattern visible; and pressures of schedule and reputation influenced judgement. The case shows that reasoning is never done in isolation from institutions and incentives.

Two sides of a question

"Is human reasoning fundamentally flawed?"

• Flawed: bias, framing and base-rate neglect appear across many experiments and professions.
• Capable: humans created mathematics, scientific methods and legal systems that correct individual errors. Reasoning can be improved by training, by external tools such as checklists, and by groups that include dissenting voices.

A nuanced view: individual reasoning is fallible, but *collective, institutional reasoning* can compensate, provided its structures are designed to reward challenge.

Worked model answer

*Practice task: evaluate the argument "All scientists are biased, so scientific claims cannot be trusted."* The argument is deductively invalid as stated, because the conclusion does not follow from the premise: bias in individuals does not imply that the community's output is untrustworthy if its methods reduce the effect of individual bias. The premise is also an overgeneralisation: bias varies in degree and kind. The stronger claim that would follow is that *some* conclusions may be influenced by bias, and that is why scientific institutions use blinding, replication and peer review. Evidence for the argument's weakness comes from cases where individual bias was detected and corrected, such as the discovery of errors in early claims about N-rays. However, the argument has a grain of truth: conflicts of interest and publication incentives can systematically skew results, as seen in industry-sponsored research on tobacco. The reasonable conclusion is to calibrate trust to the transparency and independence of the process.

Discussion question with guidance

• *Is it irrational to trust experts, since that is accepting a conclusion without reasoning it yourself?* No, because reasoning about *who* to trust is itself reasoning: you can ask about track record, independence and methods. The rational stance is to calibrate trust to evidence about reliability.