Logical Module

This post first appeared on Postlib.com.

Logical Interface

An updated, more general, abstraction of the concept originally introduced in the section below.

A logical module defined. (More abstract, more fully general, newer, definition.)

1. A set of symbols within some lexicon L.
2. A grammar system (rules of formation for wff) for L.

A logical interface defined.

Given two languages L₁, L₂ sharing the same proof system S :

1. A module m strongly interfaces between L₁, L₂ whenever:

i. Each symbol in L₁ is in L₂.

ii. Each symbol in L₂ is in L₁.

iii. The grammar system defining wff in m is in L₁.

iv. The grammar system defining wff in m is in L₂.

2. A module m weakly interfaces between L₁, L₂ whenever:

i. Each symbol in L₁ is in L₂.

ii. The grammar system defining wff in m is in L₁.

iii. The grammar system defining wff in m is in L₂.

Example Formulation

We assume a logical module or grammatical fragment suitable for first and higher-order implementations. More precisely, I define a [specific] module m per the following:

1. A negation operator {¬}.
2. A set of conceptual variables {λ1, …, λn}.
3. A belief operator {●}.
4. A set of temporal operators {t1, …, tn}.
5. A time indexing operator {@}.
6. The following grammatical rules (where wff is any well-formed formula in the language of implementation):

i. Where ¬wff is a well-formed formula.

ii. Where ●( λa , λb ) is a well-formed formula.

iii. Where ●( λa ) is a well-formed formula.

iv. where λa, λb range over conceptual variables.

v. Where wff @ ta is a well-formed formula where ta ranges over temporal operators.

A module m is implemented by a language L whenever:

1. The syntactic marks negation operator, conceptual variables, belief operator, temporal operators, and time indexing operator are in L's vocabulary.

2. The grammatical rules above are consistent with and part of L's grammar.

I define plug and play with respect to a language L as an attribute of a module whenever it, the module, can satisfy the two conditions (immediately preceding this one) with respect to L.

Tertiary bits (no pun):

A module m interfaces whenever it is implemented by two languages L₁, L₂.

Usually we think of this as a morphism (a kind of relation) between two languages. I’d prefer to focus on the module here.

Potential Applications of the Above

1. Carnap's Linguistic Frameworks.
2. Hegel's Dialectical Method and Science of Logic.
3. A conceptual variable denotes a logic.