Deduction theorem for first order logic

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SUMMARY

The deduction theorem for first-order logic (FOL) states that if a closed formula P entails Q (P ⊢ Q), then one can infer that P implies Q (⊢ P → Q). This holds true under the condition that P is closed. The discussion emphasizes the importance of the rules of inference applied when deriving implications, specifically when transitioning from T ⊢ S to ⊢ T → S. Understanding these nuances is crucial for correctly applying the deduction theorem in logical proofs.

PREREQUISITES
  • Understanding of first-order logic (FOL)
  • Familiarity with closed formulas in logic
  • Knowledge of rules of inference
  • Basic concepts of logical entailment
NEXT STEPS
  • Study the formal definition of the deduction theorem in first-order logic
  • Explore the implications of closed formulas in logical reasoning
  • Research various rules of inference and their applications in FOL
  • Examine examples of logical entailment and implications in mathematical proofs
USEFUL FOR

Logicians, mathematicians, philosophy students, and anyone interested in the foundations of logical reasoning and formal proofs.

Jeroslaw
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If I have P l- Q in FOL and P is closed, can I infer l- P -> Q. IIRC, this is valid as long as P is closed, but my memory is a little hazy. Is that how it works?
 
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Let me elaborate on my question. Say that we have T l- S. In order to infer l- T -> S, what restrictions must be placed on the formulas? How does the deduction theorem for first order logic depend upon the rules of inference that are allowed?
 

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