Today I Learned

Some of the things I've learned every day since Oct 10, 2016

130: Consequentia Mirabilis

Consequentia Mirabilis, also known as Clavius’ Law (after the German mathematician) is the principal in classical logic

$(\neg X \rightarrow X) \rightarrow X$.

That is, it’s the principle that if a proposition’s negation implies the proposition, then the proposition must be true.

Its use is similar in form to that reductio ad absurdum. To use reductio ad absurdum, one assumes that a proposition is false, sees that a contradiction follows, and from this concludes that the proposition must be true. To use consequentia mirabilis, by contrast, one assumes that a proposition is false, sees that the proposition itself follows, and from this concludes that the proposition must be true.

126: Principle of Explosion

In logic systems such as classical logic, the principle of explosion is the principle that from a contradiction any proposition can be inferred to be true.

To show this, suppose we’ve arrived at a contradictory statement, $P \wedge \neg P$. Let $Q$ be any proposition we want to prove. By conjunction elimination, we have $P$, and then by disjunction introduction we have

$P \vee Q$.

However, since we also have $\neg P$, it follows that $Q$ is true.

This principle suggests an interesting alternative attitude towards contradiction in systems that exhibit this behavior: contradiction is not undesirable because of any ‘intuitive’ nonsensicalness, but simply because a set of assumptions leading to contradiction gives us no information.

113: One Generalization of AND, OR, NOT to Fuzzy Logic

In fuzzy logic, where the set $\{ 0, 1 \}$ of possible values is replaced with the range $[ 0, 1]$, generalizations of the basic AND, OR, and NOT operators is needed which can handle input from the new domain. The only requirement of such generalizations is that they agree with their more specific counterparts when given input from $\{ 0, 1 \}$. However, there are choices which are more intuitive and advantageous than others.

One such choice is defining

• AND(a, b) = a*b
• OR(a, b) = a + b – a*b
• NOT(a) = 1 – a

The formulas for AND and NOT are intuitive enough, and even though the formula for OR may not be, it’s easily derived from those of the other 2 using the equivalence

OR(a, b) = NOT(AND(NOT(a), NOT(b))).