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Premises or Givens:

  • $∃x(A(x) → B(x))$
  • $∀x (B(x) → K(x))$

To Prove:

  • $∃x(A(x) → K(x))$

My Solution:

  1. $A(z) → B(z)$ From premise and Existential instantiation $x$ for $z$

  2. $B(z) → K(z)$ From premise and Universal instantiation $x$ for $z$

  3. $A(z) → K(z)$ Transitivity of 1,2

  4. $∃x(A(x) → K(x))$ From Existential generalization (Substitute $z$ for $x$)

OR

I was thinking about assuming $A(z)$ and then using Modus Ponens to get $B(z)$ and then further $K(z)$, then using the deduction theorem on $A(z)$ and $K(z)$, and then using Existential Generalization on that statement by substituting $x$ for $z$.

Can someone suggest which way would be more effective?

Is there any other effective way to solve it?

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Swaraj
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1 Answers1

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Your solution is a good one, and that's how I'd prove it also. I don't believe a more straightforward or simpler version is possible.

Draconis
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