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Working P.D. Magnus. forallX: an Introduction to Formal Logic (p. 182, exercise B. 4). To solve it, I tried that sentence is a theorem:

$ \def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}} \def\Ae#1{\qquad\mathbf{\forall E} \: #1 \\} \def\R#1{\qquad\mathbf{R} \: #1 \\} \def\ii#1{\qquad\mathbf{\to I} \: #1 \\} \def\ne#1{\qquad\mathbf{\neg E} \: #1 \\} \def\Ee#1{\qquad\mathbf{\exists E} \: #1 \\} \def\IP#1{\qquad\mathbf{IP} \: #1 \\} \def\ei#1{\qquad\mathbf{\exists I} \: #1 \\} \def\ie#1{\qquad\mathbf{\to E} \: #1 \\} \def\bi#1{\qquad\mathbf{\leftrightarrow I} \: #1 \\} \def\oi#1{\qquad\mathbf{\lor I} \: #1 \\} $ $ \fitch{\, }{ \fitch{1.\, L}{ \fitch{2.\, N \to N}{ 3.\, L \R{1} }\\ 4.\, (N \to N) \to L \ii{2-3} }\\ \fitch{5.\, (N \to N) \to L}{ \fitch{6.\, N}{ 7.\, N \R{6} }\\ 8.\,N \to N \ii{6-7} 9.\, L \ie{11,5} }\\ 10.\,(L \leftrightarrow ((N \to N) \to L) \bi{1-4,5-9} 11.\, (L \leftrightarrow ((N \to N) \to L) \lor H \oi{10} } $

Is the usage of $\mathbf{\leftrightarrow I}$ a good approach in this case?

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A good approach? Sure! What else could you have used? What is your worry here?

It is evident (as you saw) that you'll need to prove the given wff is a tautology by proving the first disjunct. The proof you then give is the neatest and most direct option, establishing a biconditional by separately proving each direction by a Conditional Proof ($\to$I).

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  • $\begingroup$ @F. Zer Since Peter asked, "what else could you have used?" I'll answer that outside the scope of the text, one might use equational logic. We might want to keep straight what is a meta-variable and what is not one. Using 'c' as the classical conditional, 'e' as equivalence, and 'a' as disjunction, some equations are cXX = 1, c1X = X, eXX = 1, and a1X=1. So aeLccNNLH = aeLc1LH = aeLLH = a1H = 1. Thus, since the formula always takes on the value 1 in our model, it's not contingent since contigent formulas always have more than one possible value. $\endgroup$ Apr 1, 2020 at 23:41
  • $\begingroup$ The OP was obviously asking about providing a proof in Magnus's Fitch-style natural deduction system. My remark "what else could you have used" was equally obvious a remark about a efficient proof-strategy in that ND system. So I fail to see the relevance of this. $\endgroup$ Apr 2, 2020 at 6:42

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