Premises

Definition

In logic and proof theory, premises are statements or propositions that provide the foundational support for an argument or inference. They serve as the starting point in constructing logical proofs, establishing the validity of conclusions drawn from them. Understanding how premises function is essential for evaluating arguments across various proof systems, as they dictate the structure and validity of the conclusions derived from them.

5 Must Know Facts For Your Next Test

1. Premises can be either axioms, which are accepted as true without proof, or derived statements that are established through other premises.
2. In natural deduction systems, premises are often introduced at the beginning of a proof and can be discharged through inference rules.
3. Tableau methods utilize premises by systematically exploring their implications and contradictions to determine the satisfiability of arguments.
4. The validity of an argument depends on the truth of its premises; if the premises are true and the argument is valid, then the conclusion must also be true.
5. Different proof systems handle premises differently, with some allowing for assumptions that can be discharged while others require explicit statements to be proven.

Review Questions

• How do premises influence the structure and outcome of logical arguments?
  • Premises are crucial to logical arguments as they provide the foundation upon which conclusions are built. The structure of an argument often follows a pattern where the premises lead logically to a conclusion. If the premises are sound and logically connected, they help ensure that the conclusion is valid. Thus, a strong understanding of premises allows for better evaluation and construction of effective arguments.
• Compare how premises are treated in natural deduction versus tableau methods.
  • In natural deduction, premises are introduced at the start and can be used to derive conclusions through a series of inference rules, with some rules allowing for premises to be discharged. Conversely, tableau methods treat premises by systematically breaking them down to explore their implications; they evaluate whether a set of premises can lead to a contradiction. While both methods rely on premises to establish arguments, they approach their usage differently within their respective frameworks.
• Evaluate the role of premises in establishing soundness and completeness in proof systems.
  • Premises play a vital role in ensuring both soundness and completeness within proof systems. Soundness means that if all premises are true, then the conclusions derived must also be true; this upholds the integrity of logical reasoning. Completeness ensures that if a conclusion is true, there is some set of premises from which it can be derived. Therefore, careful formulation and assessment of premises are essential in maintaining the robustness of any proof system.

Related terms

Conclusion: The statement or proposition that follows logically from the premises in an argument, representing what is being argued or proven.

Inference: The process of deriving logical conclusions from premises, often using rules of logic to connect the two.

Logical Form: The abstract structure of an argument that highlights its logical relationships, often used to analyze the validity of premises and conclusions.