Introduction to Logic

In our daily lives, we are often told to be logical. However, it is rare for us to be told what logic is. As a result, many of us stumble through life with a vague sense of syllogism without formality.

Syllogisms

A syllogism (WP, nLab) is a foundational form of deductive proof. It operates according to the following schema:

  1. If P, then Q.
  2. If Q, then S.
  3. Therefore: If P, then S.

Here P, Q, and S are statements made out of words. This is the first principle of logic: we are oriented around words and rules which manipulate them, not the meanings of the words. As logicians, we recognize that natural usage of language does not always respect logical rules, which means that we cannot let P, Q, or S be arbitrary statements; instead, they must be constructed according to logical rules. We will revisit this later, but first let us consider syllogisms in Lojban, which have the following schema:

  1. {ganai broda gi brode}
  2. {ganai brode gi brodi}
  3. Therefore: {ganai broda gi brodi}

For now, the logical rules are easy to state: broda, brode, and brodi must be grammatical bridi with valsi drawn from a basis set. The basis set contains valsi like {ganai} and {gi} which are necessary to state the logical rules themselves; if we didn't choose a basis, then we wouldn't be able to define our logic! However, the basis set will not contain discursive connectives, so it may seem like a restricted way of using Lojban at first. We will see later how to define new valsi in terms of the basis set.

modus ponens

Modus ponens (WP, nLab) is the most common name for the following schema:

  1. Suppose: P.
  2. If P, then Q.
  3. Therefore: Q.

To be direct with the reader: "modus ponens" or "MP" is one of those few names that you must memorize in order to read English prose about logic; it occurs too frequently in Anglophone literature and is too fundamental to not be understood by readers. I apologize and wish it had a better common name.

In Lojban, there are several schema we might consider. Here is one classic schema:

  1. Suppose: {broda}
  2. {lo du'u broda cu nibli lo du'u brode}
  3. Therefore: {brode}

We will eventually build variations on this, but it turns out to be untenable for deep reasons. Instead, the schema used throughout la brismu is:

  1. Suppose: {broda}
  2. {ganai broda gi brode}
  3. Therefore: {brode}

Compare and contrast with the fully-formalized axiom statement, ax-mp. This schema is compatible with the syllogism schema. Both schemata have a bridi of the form {ganai broda gi brode}. Considering this more deeply leads to…

Categorical logic

Categorical logic (WP, nLab) is a way of studying logic by studying the properties of structures generated by systems with modus ponens, like the "if"/"then" of English or {ganai}/{gi} of Lojban. This lets us study the ontology of a logic (what a logic can construct) without doing metaphysics (what a logic thinks is real.)

To gently introduce the concept, we can arrange our structure as a graph. For here, a graph is

  • a set of abstract objects, known as vertices, along with
  • a set of connectors between objects, known as edges, such that
  • each edge connects from one vertex to another vertex, and
  • every vertex has an edge which connects it to itself.

Assign a vertex to each bridi. A path in a graph is a sequence of edges with each edge following the other. In our running example, a path from {broda} to {brodi} might consist of two edges, one from {broda} to {brode} and another from {brode} to {brodi}. We'll let {ganai broda gi brode} represent a path from the {broda} vertex to the {brode} vertex. An application of modus ponens to a path is merely a traversal along the path. A syllogism is a composition of paths; path (1) ends where path (2) begins, constructing path (3).

At this point, the reader is encouraged to let this concept settle. Categorical logic is important for understanding the overall direction of the database of theorems, for justifying some of our axioms, and for explaining the structure of Lojban in a mathematically satisfying fashion. It is not important for understanding individual portions of la brismu or carrying out the rules of logic. We will merely note that there is a category-theoretic justification to add {ge} to our basis set.

There are actually two structures worth studying here. One structure uses {ganai} and the other uses {go}. We may always weaken {go} to {ganai} in either direction; it is as if the edges of the graph for {go} are undirected, while the edges of the graph for {ganai} are directed. These structures are known as Loj and Core(Loj) respectively. The main reason to include {go} is…

Definitions

We don't want to add every valsi to the basis set. Ideally, we want to define some valsi in terms of others. Since this effectively extends the basis set with new valsi, it is known as extension by definitions. We will follow a sort of schema:

  1. Suppose: {broda}
  2. {go broda gi brode}
  3. Therefore: {brode}

This is merely modus ponens for {go}, and indeed it is a theorem, bi. However, let us be more specific:

  1. Suppose: {ko'a bu'a}
  2. {go ko'a bu'a gi brode}
  3. Therefore: {brode}

Then we might say that the unary selbri {bu'a} is defined in terms of {brode}. It is possible (and likely) for {ko'a} — which is really just a placeholder, or metasyntax, for some terbri — to appear in {brode}, but not required. The idea is that we may treat any occurrence of {bu'a} as equivalent to {brode} and vice versa, so if everything in {brode} is already defined in terms of the basis set, then {bu'a} is defined in terms of the basis set too.

Ironically, we cannot define {go} and {ge} fully in terms of themselves, and this is the best way to understand why they are in the basis set. However, once they are established, we can define {ga}, {gonai}, and many selbri.