Logical consequence

Definition

Logical consequence refers to a relationship between statements where one statement necessarily follows from another. In essence, if a set of premises is true, then the conclusion drawn from those premises must also be true, given the structure of logical reasoning. This concept is foundational in understanding interpretations and models, as it establishes how truths propagate within a logical system.

5 Must Know Facts For Your Next Test

1. Logical consequence can be expressed formally through deduction, where conclusions are derived using rules of inference.
2. The symbol used to denote logical consequence is usually '⊨', indicating that a conclusion holds in every model that satisfies the premises.
3. Understanding logical consequence is crucial for constructing valid arguments and proofs in mathematics and philosophy.
4. In first-order logic, logical consequence involves quantifiers and variables, expanding its complexity compared to propositional logic.
5. The completeness theorem states that if something is logically valid (a conclusion follows from premises), there exists a proof for it within a formal system.

Review Questions

• How does the concept of logical consequence relate to the evaluation of arguments in formal logic?
  • Logical consequence is fundamental in evaluating arguments because it determines whether conclusions necessarily follow from given premises. When analyzing an argument, if the premises logically lead to the conclusion, then we say that the conclusion is a logical consequence of those premises. This evaluation is critical in ensuring that arguments are valid and sound within formal logic.
• Discuss how interpretations influence the understanding of logical consequence in different models.
  • Interpretations provide meaning to the symbols used in logical systems, which directly influences how we determine whether a conclusion is a logical consequence of certain premises. Different interpretations can lead to different truth values for the same statements, which may affect whether the conclusion holds under that interpretation. Therefore, understanding various interpretations helps clarify when a logical consequence is true or false across different models.
• Evaluate the implications of logical consequence on proof theory and its role in establishing mathematical truths.
  • Logical consequence plays a critical role in proof theory as it establishes a foundation for what can be considered mathematically true. By ensuring that conclusions logically follow from established premises through sound reasoning, proof theory relies on this principle to build rigorous mathematical frameworks. Evaluating logical consequences allows mathematicians to construct proofs that validate theories, highlighting its significance in establishing coherent and reliable mathematical truths.

Related terms

Model: A model is a mathematical structure that represents a set of objects and their relationships, used to determine the truth values of statements in logic.

Interpretation: An interpretation assigns meaning to the symbols and formulas within a logical language, allowing for the evaluation of truth in different contexts.

Validity: Validity refers to an argument where if the premises are true, the conclusion must also be true; it is closely related to the idea of logical consequence.