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Validity and Logical Consequence
Argument Validity
An argument is valid when it is impossible for the premises to be true and the conclusion false. Validity is a formal property that depends on the structure of the argument, not on the content of the propositions.
Characteristics of Validity:
- Validity is a syntactic property
- A valid argument preserves truth
- Validity does not guarantee the truth of the premises
- An argument can be valid with false premises
Logical Consequence
A proposition φ is a logical consequence of a set of premises Γ if and only if in every interpretation in which all formulas of Γ are true, φ is also true.
Notation: Γ ⊨ φ
Proof Methods
There are several methods to prove the validity of an argument:
1. Truth Tables
A table is constructed with all possible combinations of truth values. The argument is valid if there is no row where the premises are true and the conclusion false.
2. Natural Deduction
Inference rules are applied to derive the conclusion from the premises through a series of logically valid steps.
3. Venn Diagrams
For arguments with categorical structure, the relationships between sets are represented using Venn diagrams to verify validity.
Invalid Arguments
An argument is invalid when there exists at least one interpretation (or row in the truth table) in which all premises are true but the conclusion is false. This interpretation is called a counterexample.
Relationship with Other Concepts
- Sound argument: valid and with true premises
- Correct argument: valid with true conclusion
- Tautology: valid formula without premises (always true)