In the propositional logic, let the string $(a_1, a_2, \cdots, a_n)$ be WFF. And exist natural numbers $i< j\in \{1,2 \cdots, n\}$ search that string $(a_i,a_{i+1}, \cdots , a_j )$ is also WFF. Prove that if $(b_1, b_2, \cdots, b_m)$ is WFF, then $(a_1, a_2, a_{i-1}, b_1, b_2, \cdots, b_m, a_{j+1}\cdots, a_n)$ is also WFF.

In my logical book, substitution is defined only for proposition letter. But I think this is true. How can we prove this?

  • $\begingroup$ You are asked to subst into a formula a sub-formula with another formula. $\endgroup$ – Mauro ALLEGRANZA May 9 at 11:20
  • $\begingroup$ @MauroALLEGRANZA Yes! Thanks! Do you know the proof or material with this topic? $\endgroup$ – amoogae May 9 at 11:49
  • $\begingroup$ Do you have a link to a link to a copy of the book or the chapter where this exercise appears? $\endgroup$ – prime4567 May 9 at 11:50
  • $\begingroup$ @prime4567 Sorry. I made this problem. And the book that I read is written in my native.. $\endgroup$ – amoogae May 9 at 12:16
  • $\begingroup$ IMO the proof is by induction on the compelxity of the formula, considering the specifications for WFF. For simplicity, consider only the connectives : $\lnot, \lor$. $\endgroup$ – Mauro ALLEGRANZA May 9 at 12:37

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