Logical Entailment

Definition

Logical entailment refers to the relationship between statements where one statement necessarily follows from another. If a set of premises logically entails a conclusion, then if the premises are true, the conclusion must also be true. This concept is crucial in understanding validity in logical arguments and is often tested using truth tables to determine if a conclusion is a necessary outcome of its premises.

5 Must Know Facts For Your Next Test

1. Logical entailment is often represented with symbols such as '⊨', indicating that one statement entails another.
2. Truth tables are a systematic way to test logical entailment by comparing all possible truth values of the premises and the conclusion.
3. If every row in a truth table where the premises are true also shows the conclusion as true, the premises logically entail the conclusion.
4. In propositional logic, logical entailment can also indicate that certain propositions cannot both be false at the same time.
5. Understanding logical entailment helps in constructing sound arguments and identifying invalid reasoning in discussions.

Review Questions

• How can truth tables be used to establish logical entailment between statements?
  • Truth tables can be utilized to establish logical entailment by listing all possible combinations of truth values for the premises and the conclusion. If, in every scenario where the premises are true, the conclusion is also true, then we can say that the premises logically entail the conclusion. This method provides a clear visual representation of how different truth values interact and helps identify valid arguments.
• Discuss how logical entailment relates to the concepts of validity and soundness in logical reasoning.
  • Logical entailment is central to understanding validity, as a valid argument requires that if its premises are true, then its conclusion must also be true. Soundness extends this concept by adding that an argument must not only be valid but also have true premises. Therefore, while logical entailment ensures that conclusions follow from premises, soundness guarantees that those conclusions are based on true statements, making them reliable.
• Evaluate a scenario where an argument is valid but not sound; how does this illustrate the concept of logical entailment?
  • Consider an argument where the premises state that 'All cats are reptiles' (which is false) and 'Whiskers is a cat' (which is true), leading to the conclusion 'Whiskers is a reptile.' This argument is valid because the structure ensures that if the premises were true, the conclusion would logically follow. However, since one premise is false, it demonstrates that while logical entailment connects premises to conclusions, it does not guarantee soundness. This highlights the importance of examining both validity and truth in reasoning.

Related terms

Validity: A property of arguments where if the premises are true, the conclusion must also be true.

Soundness: A property of arguments that are both valid and have all true premises.

Truth Table: A mathematical table used to determine the truth value of logical expressions based on the truth values of their components.