# A calculus of the absurd

#### 21.2 Logics

##### 21.2.1 Foundational things

These are the meta-definitions for any given logic (that is, they are kind of a “template” for defining a specific logic). Once we answer these questions, we have a logic!

• Definition 21.2.1 An interpretation assigns a value to specified free symbols in a formula.

• Definition 21.2.2 An interpretation is suitable for a formula if it binds all free variables.

• Definition 21.2.3 Given a formula $$F$$ (or a set of formulas), a model $$M$$ is an interpretation for which $$F$$ is true.

In formulas, this is either

\begin{align} & \mathcal {A}(F) = 1 \\ & \mathcal {A} \vDash F \end{align}

• Definition 21.2.4 Logical consequence If for every interpretation whenever $$F$$ is true, $$G$$ is too, then we write

\begin{align} F \vDash G \end{align}

• Definition 21.2.5 Logical equivalence ???

\begin{align} & F \vDash G \\ & M \vDash G \end{align}

• Definition 21.2.6 Tautology notation If $$F$$ is a tautology (informally it is always true) then we write

$$\vDash F$$

That is, $$F$$ cannot be derived

• Definition 21.2.7 Unsatisfiability If $$F$$ is unsatisfiable we write

$$F \vDash \bot$$

• Theorem 21.2.1 Relation between tautology and unsatisfiability We can prove that

$$\big ( \vDash F \big ) \iff \big (\lnot F \vDash \bot \big )$$

This apparently follows from what this means.

• Theorem 21.2.2 Normal forms The statements of the theorem is that we can always write any formula in a normal form.

Consider some formula in propositional logic, for which we have a truth table. We then consider the $$1$$ entries. For these rows, we might have something in the form

$$\begin{pmatrix} A & B & C & F \\ 0 & 1 & 0 & 1 \end {pmatrix}$$

Then we add a term which is one if and only if $$A = 0, B = 1, C = 0$$. For example, in this case we would have something that looks a bit like

$$(\lnot A \land B \land \lnot C) \lor ....$$

For conjunctive normal form, this is a little more complex, but we could do something like

$$(A \lor B \lor C) \land (...) \land ... \land (...)$$

then instead of making terms one when the conditions trigger, we make terms zero when the conditions are not triggered.