Proving (∀x.ϕ) →(∃x.ϕ) using Natural Deduction?

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SUMMARY

The proof of the statement ⊢ (∀x.ϕ) →(∃x.ϕ) can be established using a natural deduction system. The process involves assuming the antecedent (∀x.ϕ) and applying the rule of universal instantiation to derive a specific instance of ϕ. Following this, the rule of existential generalization allows the transition from the instance to the existential quantifier (∃x.ϕ). Finally, the application of conditionalization completes the proof.

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  • Understanding of natural deduction systems
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Hello, I have no idea how to solve this proof and would actually appreciate your help. I cannot use soundness or completeness.

⊢ (∀x.ϕ) →(∃x.ϕ)


Thanks
 
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badwolf23 said:
Hello, I have no idea how to solve this proof and would actually appreciate your help. I cannot use soundness or completeness.

⊢ (∀x.ϕ) →(∃x.ϕ)Thanks

It will depend on the particular deductive apparatus. You might be using a set of axioms together with a set of rules of inference, or you might be using a system of natural deduction, etc. Let's suppose it's a natural deduction system.

Assume your antecedent. Often you will have a rule of universal instantiation that allows you to go from the assumed antecedent to an instance, then a rule that let's you go from the instance to the existential generalization of the instance (which is the consequent). Then an application of conditionalization gets you home. Does that help?
 
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