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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?
The deduction theorem for first order logic is a fundamental theorem in mathematical logic that states that if a statement can be deduced from a set of premises in a formal system, then it can also be deduced from a larger set of premises that includes the original set. This theorem forms the basis for many proofs and reasoning processes in mathematics and computer science.
The deduction theorem works by allowing us to add additional premises to a set of premises and still be able to deduce the same conclusion. This means that if a statement can be proven from a smaller set of premises, it can also be proven from a larger set of premises that includes the smaller set.
The deduction theorem is significant in logic because it allows us to simplify and streamline the process of proving statements. Instead of having to start from scratch each time we want to prove a statement, we can build upon previously proven statements and expand our set of premises to reach new conclusions.
No, the deduction theorem can also be applied to higher order logics, such as second order logic. However, the theorem is most commonly used and discussed in the context of first order logic, as it is the simplest and most widely used logical system.
Yes, the deduction theorem can be applied to any formal system that follows the rules of first order logic. This includes mathematical and logical systems, as well as computer programming languages.