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If α = E ¬ (β ), β must have length 3 we have shown to be impossible. Since α is not just a sentence symbol, α = E ¬ (β ) or α = E  (β, γ ) for some wffs β and γ where  = ∧, ∨, ↔ or →. Consider a shortest construction sequence and its last element. Suppose a wff of length 6 is possible and denote it by α. From the proof of lemma 1, each formula building operation adds 3 to the length of its constituents. However the only wffs that do not contain any connectives are made up of just one sentence symbol. Thus any wff of length 2 or 3 must not contain any connectives. By lemma 1 (from this document), any wff with connectives will be of length at least 4. All wffs must have at least one sentence symbol. Show that there are no wffs of length 2, 3, or 6, but that any other postive length is possible. By a simila similarr argumen argument, t, the number of left parenthesis, right parenthesis and connective symbols in (α ∧ β ), (α ∨ β ), (α → β ) and (α ↔ β ) are equal. lγ = lα + 3, rγ = rα + 3, cγ = cα + 3 = so lγ = rγ = cγ by the induction induction hypothesi hypothesis. Inducti Inductive ve step: step: Let γ = (¬α). Inducti Induction on hypothesi hypothesis: s: Let α and β be two wffs. Let l, r and c be, respecProof: We will use the inducti tively, the number of left parenthesis, right parenthesis and connective symbols in a wff. Lemma 1 The number of left parenthesis, right parenthesis and connective symbols are equal in a wff. The The Langu Language age of Senten Sententia tiall Logic Logic Do feel free to email me if you have any comments or to point out any errors (of which there will likely be plenty). It’d be great if this document helps anyone out in any form or fashion. I have posted it online as a possible resource for others also using the book. Its main purpose is to facilitate my own learning. Overview This document details my attempt to solve some of the problems in Herbert Enderton’s A Enderton’s A Mathematical Introduction to Logic (2nd Edition) . Some Solutions to Enderton’s Mathematical Introduction to Logic Kelvin Soh