From 673de1c627a2046123e1693e9e6508399a028508 Mon Sep 17 00:00:00 2001 From: Marshall Lochbaum Date: Sat, 3 Oct 2020 16:38:35 -0400 Subject: Write files directly from md.bqn, not gendocs (removes trailing empty lines) --- docs/doc/logic.html | 1 - 1 file changed, 1 deletion(-) (limited to 'docs/doc/logic.html') diff --git a/docs/doc/logic.html b/docs/doc/logic.html index 0ede46e7..40aa5906 100644 --- a/docs/doc/logic.html +++ b/docs/doc/logic.html @@ -44,4 +44,3 @@

It's common to apply ∧´ or ∨´ to a list (checking whether all elements are true and whether any are true, respectively), and so it's important for extensions to And and Or to share their identity element. Minimum and Maximum do match And and Or when restricted to booleans, but they have different identity elements. It would be dangerous to use Maximum to check whether any element of a list is true because >⌈´⟨⟩ yields ¯∞ instead of 0—a bug waiting to happen. Always using 0 as a left argument to ⌈´ fixes this problem but requires more work from the programmer, making errors more likely.

It is easy to prove that the bilinear extensions have the identity elements we want. Of course 1∧x is 1×x, or x, and 0∨x is 0×⌾¬x, or ¬1׬x, giving ¬¬x or x again. Both functions are commutative, so these identities are double-sided.

Other logical identities do not necessarily hold. For example, in boolean logic And distributes over Or and vice-versa: a∧b∨c ←→ (a∧b)∨(a∧c). But substituting × for ∧ and +-× for ∨ we find that the left hand side is (a×b)+(a×c)+(a×b×c) while the right gives (a×b)+(a×c)+(a×b×a×c). These are equivalent for arbitrary b and c only if a=a×a, that is, a is 0 or 1. In terms of probabilities the difference when a is not boolean is caused by failure of independence. On the left hand side, the two arguments of every logical function are independent. On the right hand side, each pair of arguments to ∧ are independent, but the two arguments to ∨, a∧b and a∧c, are not. The relationship between these arguments means that logical equivalences no longer apply.

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