two notes on logic

- Logical consequence and logical consistency are inter-definable. An axiomatic theory T is consistent if there is no sentence P such that P and ¬P are both consequences of the axioms; P is a consequence of the axioms of a theory T if the axioms together with ¬P are inconsistent.

- While doing calculations, it may be helpful to keep in mind a picture of what is happening in the dual world. Here is how one can quickly recognize that logical equivalence is an associative binary operator. Note that p⇔q can be stated as (q⇒p)∧(p⇒q) which in turn can be stated as (¬q∨p)∧(¬p∨q). Hence the negation of p⇔q can be written as ¬((¬q∨p)∧(¬p∨q)) which in turn can be written as (¬¬q∧¬p)∨(¬¬p∧¬q). But the last statement is simply ¬pΔ¬q, namely the symmetric difference of ¬p and ¬q. So ¬(p⇔q) holds if and only if ¬pΔ¬q holds. In other words, ⇔ is dual to Δ. Since Δ plays the role of addition in the Boolean ring corresponding to the Boolean Algebra (∧,∨,¬,0,1) and since addition is associative, we can conclude that ⇔ is associative as well.