Everyone loves all lovers, Romeo loves Juliet $\vdash$ I love you (Natural deduction proof)

Yes, all the shortest proofs in natural deduction consist of 10 steps (included the 2 assumptions), because you have to apply 3 elimination rules for the quantifiers and 1 elimination rule for the implication, for 2 times.

For instance, if you translate the first assumption as $\forall x \forall y \forall z(Lxy \to Lzx)$ (which is logically equivalent to $\forall x (\exists y Lxy \to \forall y Lyx)$), the shortest proof in natural deduction is the following:

  1. $\forall x \forall y \forall z(Lxy \to Lzx)$ assumption
  2. $Lrj$ assumption
  3. $ \forall y \forall z(Lry \to Lzr)$ $ \ \forall_E$ from 1
  4. $ \forall z(Lrj \to Lzr)$ $\ \forall_E$ from 3
  5. $Lrj \to Lur$ $\ \forall_E$ from 4
  6. $Lur$ $\ \to_E$ from 5, 2
  7. $ \forall y \forall z(Luy \to Lzu)$ $\ \forall_E$ from 1
  8. $ \forall z(Lur \to Lzu)$ $\ \forall_E$ from 7
  9. $Lur \to Liu$ $\ \forall_E$ from 8
  10. $Liu$ $\ \to_E$ from 9, 6

Tags:

Logic

Predicate Logic

First Order Logic

Natural Deduction

Formal Proofs

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