## 2.2 Propositional Logic

Propositional logic is that part of mathematical logic that deals with the
composition of formulas. The composition starts with basic formulas that are
considered as "black boxes": we are interested in their truth value ("true
or false?") but not in what properties they actually describe. Various
operations then allow to combine simpler formulas to more complex ones.

**Definition 3 (Logical Connective)**
A *(logical) * *connective (Junktor)* is a syntactic operator that
combines formulas to a new formula.

In more detail, the formulas of propositional logic are constructed from the
connectives `F', `T', `~', ` /\ ', ` \/ ', ` => ', ` <=> ' as
follows:

**Proposition 1 (Formulas of Propositional Logic)**
- The following
*logical constants (Logische Konstanten)* are
formulas^{2}:
** ***False*- "false" ("falsch")
F

** ***True*- "true" ("wahr")
T

- If
`A` is a formula, then the following is a formula:
** ***Negation*- "not
`A`" ("nicht `A`")
~`A`

- If
`A` and `B` are formulas, then the following are formulas
** ***Conjunction*- "
`A` and
`B`" ("`A` und `B`")
(`A` /\ `B`)

** ***Disjunction*- "
`A` or
`B`" ("`A` oder `B`")
(`A` \/ `B`)

** ***Implication*- "
`A` implies
`B`" ("`A` impliziert `B`")
(`A` => `B`)

** ***Equivalence*- "
`A` is
equivalent to `B`" ("`A` ist äquivalent zu `B`")
(`A` <=> `B`)

We omit the parentheses if `A` and `B` are unambiguous (usually
`x` is free in `A`).

Above propositions provide the basis for a hierarchical construction of
formulas.

**Example**
The syntactic structure of the formula
T /\ F => F

is ambiguous; it may be understood as (T /\ F) => F or as T /\ (F => F).

In the following sections, we will discuss the meanings of these formulas.

Author: Wolfgang Schreiner

Last Modification: October 4, 1999