Deduction theorem for first order logic

AI Thread Summary
In first-order logic (FOL), if P entails Q and P is closed, it is valid to infer P implies Q. To infer T implies S from T entails S, certain restrictions on the formulas must be considered, particularly regarding their closure. The deduction theorem's validity is contingent on the specific rules of inference employed in the logical system. Understanding these dependencies is crucial for correctly applying the theorem in various contexts. Overall, the deduction theorem in FOL is influenced by the closure of formulas and the inference rules in use.
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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