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 <p>It's common to apply <code><span class='Function'>∧</span><span class='Modifier'>´</span></code> or <code><span class='Function'>∨</span><span class='Modifier'>´</span></code> 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 <code><span class='Function'>&gt;⌈</span><span class='Modifier'>´</span><span class='Bracket'>⟨⟩</span></code> yields <code><span class='Number'>¯∞</span></code> instead of <code><span class='Number'>0</span></code>—a bug waiting to happen. Always using <code><span class='Number'>0</span></code> as a left argument to <code><span class='Function'>⌈</span><span class='Modifier'>´</span></code> fixes this problem but requires more work from the programmer, making errors more likely.</p>
 <p>It is easy to prove that the bilinear extensions have the identity elements we want. Of course <code><span class='Number'>1</span><span class='Function'>∧</span><span class='Value'>x</span></code> is <code><span class='Number'>1</span><span class='Function'>×</span><span class='Value'>x</span></code>, or <code><span class='Value'>x</span></code>, and <code><span class='Number'>0</span><span class='Function'>∨</span><span class='Value'>x</span></code> is <code><span class='Number'>0</span><span class='Function'>×</span><span class='Modifier2'>⌾</span><span class='Function'>¬</span><span class='Value'>x</span></code>, or <code><span class='Function'>¬</span><span class='Number'>1</span><span class='Function'>׬</span><span class='Value'>x</span></code>, giving <code><span class='Function'>¬¬</span><span class='Value'>x</span></code> or <code><span class='Value'>x</span></code> again. Both functions are commutative, so these identities are double-sided.</p>
 <p>Other logical identities do not necessarily hold. For example, in boolean logic And distributes over Or and vice-versa: <code><span class='Value'>a</span><span class='Function'>∧</span><span class='Value'>b</span><span class='Function'>∨</span><span class='Value'>c</span> <span class='Gets'>←→</span> <span class='Paren'>(</span><span class='Value'>a</span><span class='Function'>∧</span><span class='Value'>b</span><span class='Paren'>)</span><span class='Function'>∨</span><span class='Paren'>(</span><span class='Value'>a</span><span class='Function'>∧</span><span class='Value'>c</span><span class='Paren'>)</span></code>. But substituting <code><span class='Function'>×</span></code> for <code><span class='Function'>∧</span></code> and <code><span class='Function'>+-×</span></code> for <code><span class='Function'>∨</span></code> we find that the left hand side is <code><span class='Paren'>(</span><span class='Value'>a</span><span class='Function'>×</span><span class='Value'>b</span><span class='Paren'>)</span><span class='Function'>+</span><span class='Paren'>(</span><span class='Value'>a</span><span class='Function'>×</span><span class='Value'>c</span><span class='Paren'>)</span><span class='Function'>+</span><span class='Paren'>(</span><span class='Value'>a</span><span class='Function'>×</span><span class='Value'>b</span><span class='Function'>×</span><span class='Value'>c</span><span class='Paren'>)</span></code> while the right gives <code><span class='Paren'>(</span><span class='Value'>a</span><span class='Function'>×</span><span class='Value'>b</span><span class='Paren'>)</span><span class='Function'>+</span><span class='Paren'>(</span><span class='Value'>a</span><span class='Function'>×</span><span class='Value'>c</span><span class='Paren'>)</span><span class='Function'>+</span><span class='Paren'>(</span><span class='Value'>a</span><span class='Function'>×</span><span class='Value'>b</span><span class='Function'>×</span><span class='Value'>a</span><span class='Function'>×</span><span class='Value'>c</span><span class='Paren'>)</span></code>. These are equivalent for arbitrary <code><span class='Value'>b</span></code> and <code><span class='Value'>c</span></code> only if <code><span class='Value'>a</span><span class='Function'>=</span><span class='Value'>a</span><span class='Function'>×</span><span class='Value'>a</span></code>, that is, <code><span class='Value'>a</span></code> is 0 or 1. In terms of probabilities the difference when <code><span class='Value'>a</span></code> 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 <code><span class='Function'>∧</span></code> are independent, but the two arguments to <code><span class='Function'>∨</span></code>, <code><span class='Value'>a</span><span class='Function'>∧</span><span class='Value'>b</span></code> and <code><span class='Value'>a</span><span class='Function'>∧</span><span class='Value'>c</span></code>, are not. The relationship between these arguments means that logical equivalences no longer apply.</p>
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