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Definition: Logical Arguments

A logical argument consists of some (say $n\ge 1$) given propositions $p_1,\ldots,p_n$, called premises, and a proposition $q$, called the conclusion.

There are two kinds of logical arguments, valid arguments, and fallacies:

• A logical argument is valid if and only if the conclusion is true whenever all the premises are simultaneously true, formally $$\text{ if }([[p_1]]_I=1\text{ and },\ldots,\text{ and }[[p_n]]_I=1),\text{ then } [[q]]_I=1$$ for all interpretations $I$.
• A fallacy is an invalid argument: The conclusion is false, while all the premises are true, formally $$([[p_1]]_I=1\text{ and },\ldots, \text{ and }[[p_n]]_I=1), \text{ and }[[q]]_I=0$$ for all interpretations $I$.

Mentioned in:

Chapters: 1 2
Examples: 3
Explanations: 4 5
Lemmas: 6 7 8 9 10 11 12 13 14
Proofs: 15 16 17 18 19 20 21 22 23 24

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References

Bibliography

1. Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016
2. Kohar, Richard: "Basic Discrete Mathematics, Logic, Set Theory & Probability", World Scientific, 2016