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Propositional Logic – A Primer

7. Logical Fallacies

Book cover: Ideas That Matter: The Concepts That Shape the 21st Century by A. C. Grayling

With the tools and methods illustrated in the previous sections, you can formally prove the validity and invalidity of some basic arguments expressed in propositional logic. In addition, you are also now able to identify reasonably quickly some formal logical fallacies. These are errors in reasoning that arise from violations of the inference rules of propositional logic covered previously.

Keep in mind that there are many informal errors of reasoning that are sometimes labeled incorrectly as 'logical fallacies'. These are more typically cognitive biases and include such errors as loss aversion, anchoring bias and confirmation bias.

There are four basic logical fallacies identified using propositional logic. Each of these formal fallacies is included under the general umbrella term: 'non-sequiter' (from Latin ‘it does not follow’). In different ways, the invalid conclusion does not follow logically from the stated premises. These four basic logical errors are as follows.

Affirming the Consequent

Form:

If P then Q

Q

Conclusion: P

Example:
If it is raining, then the ground is wet. The ground is wet. Therefore, it is raining. In fact, a pipe burst.
Explanation:
This fallacy mistakenly assumes that if the consequent of a conditional statement is true, the antecedent must also be true. However, there may be other reasons for the consequent to be true.

Denying the Antecedent

Form:

If P then Q

Not P

Conclusion: Not Q

Example:
If it is raining, then the ground is wet. It is not raining. Therefore, the ground is not wet. In fact, a pipe burst.
Explanation:
This fallacy incorrectly concludes that if the antecedent of a conditional statement is false, the consequent must also be false. However, there may be other reasons for the consequent to be true.

Affirming a Disjunct

Form:

P or Q

P

Conclusion: Not Q

Example:
Sue is working or Sue is at home. Sue is working. Therefore, Sue is not at home. In fact, Sue is working at home.
Explanation:
This fallacy incorrectly concludes that if one of the disjuncts of a disjunction is true, the other disjunct must be false. However, there may be instances where both disjuncts are true together.

Denying a Conjunct

Form:

Not P and Q

Not P

Conclusion: Q

Example:
It is not the case that both Sue is at work and Sue is at home. Sue is not at work. Therefore, Sue is at home. In fact, Sue is at a park.
Explanation:
This fallacy incorrectly concludes that if one of the conjuncts of a conjunction is false, the other conjunct must be true. However, there may be instances where both conjuncts are false together.

You are now in a position to identify some basic logical errors using your knowledge of propositional logic. See how many of these errors you can spot in your readings and in conversations with others.

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