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  "body": "\n## Statement\n\n| ( | *P* | → | ( | *Q* | → | *R* | )) | → | (( | *P* | → | *Q* | ) | → | ( | *P* | → | *R* | )) |\n|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n|  | 1 | **2** |  | 3 | 4 | 5 |  | **6** |  | 7 | 8 | 9 |  | **10** |  | 11 | 12 | 13 |  |\n\nIn propositional logic, Frege's theorem refers to this [[tautology|05006AJ7DZ]]:\n\n$$\\big(P\\to(Q\\to R)\\big)\\to\\big((P\\to Q)\\to(P\\to R)\\big)$$\n\nThe theorem already holds in one of the weakest logics imaginable, the constructive implicational calculus. The proof under the Brouwer–Heyting–Kolmogorov interpretation reads $f \\mapsto g\\mapsto p\\mapsto (f(p)\\circ g)(p)$. In words: \"Let *f* denote a reason that *P* implies that *Q* implies *R*. And let *g* denote a reason that *P* implies *Q*. Then given a *f*, then given a *g*, then given a reason *p* for *P*, we know both that *Q* holds by *g* and that *Q* implies *R* holds by *f*. So *R* holds.\"\n\nThe [[truth table|05RSK4Q657]] to the right gives a semantic proof. For all possible assignments of *false* () or *true* () to *P*, *Q*, and *R* (columns 1, 3, 5), each subformula is evaluated according to the rules for [[material conditional|0NMG7W7Q11]], the result being shown below its main operator. Column 6 shows that the whole formula evaluates to *true* in every case, i.e. that it is a tautology. In fact, its antecedent (column 2) and its consequent (column 10) are even equivalent.\n",
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