what is the difference between axiom and assumptions
Answers
Explanation:
The difference is mostly one of scope. Axioms are assumptions global to any deduction of the logic, whereas a mere assumption is local to some part of a deduction and needs to be discharged. Only deductions with all non-axiomatic assumptions discharged are called proofs. What discharging an assumption does, is make the conclusion of a deduction independent of the the truth of the assumption. Axioms on the other hand do not need to be discharged at all.
But that doesn’t really change anything. It’s a mere syntactic convenience, since we don’t need to everywhere introduce the same assumptions in the deduction. That this is so, becomes evident, by considering a logic, in which we dropped the axioms, but used it as an ordinary assumption in deductions.
Then you would notice, since we need to discharge assumptions, that, whereas the deduction with axioms allows you to draw a conclusion Conc , without them we would then prove (Ax1∧⋯∧Axn)⟹Conc .
The meaning is exactly the same, since conclusions are only guaranteed to be true if the axioms are in fact true. So basically axioms just save us some typing, by leaving the assumptions needed implicit in the expression of the conclusion. They are ‘factored out’ of the deductions and left implicit in the syntax.
Answer:
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Explanation:
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