L-Concepts, its Terms and Definitions

Kris Caballero • Jul 19, 2024 • 0 comments • Logic

Taken and learned from well-known philosopher Rudolf Carnap, from his book Introduction to Symbolic Logic and its Applications, let's talk about what he means introducing L-Concepts. Granted, it's on the very first chapter of his book, but very little has been talked about and discussed online, so we here at KCU Network of Mathematics decided we should make these terms out!

L-True

Statements are considered L-True if and only if its conclusion is tautological. Despite their truth values, its content is exclusive from empirical facts. It's "logically true," very much.

"Either I am thirsty or not."

"2 + 2 = 4*"

* This statement, while true, is not tautological, despite it is a universally accepted fact that two added by two equates to four.

Even if the statement isn't tautological, it still falls under the guise of an L-true statement.

L-False

The opposite of an L-True statement: These statements are logically false and whose conclusions are contradictory. Its depiction and meaning being conveyed is rather self-evident, and relatively doesn't speak of, nor often cares about, the reality of the world—"logically false," basically.

"It is hot and not hot outside."

"Paris is a city located in Canada."

Such examples ought to prove its invalidity, given that it's not possible for a weather pattern to be both hot and cold (not hot); It is also obvious that the beautiful city of Paris is located in France and not Canada.

L-Determinant

Statements that are purely L-True or L-False are considered L-Determinant. What this means is the statements have been (re)marked by its value and the 'L-class' to which it belongs (L-class meaning, L-True or L-False). Because such statements have been summed up as such, we say, or as Carnap says, those statements are L-Determinant—a term to secure its truth and false value for having "determined" its value, conclusion etc.

Example statements above are considered L-Determinant, as we know of its value and its conclusion from whence we have deducted—the statement itself.

L-Indeterminant

A statement is considered L-Indeterminant when its value is neither L-true nor L-false, and their conclusions are summed up as contigent. This means that empirical evidence and/or state of the world is wholly dependent on the validity of its truth or falsity of such statement(s). Because of that, logical rules, analysis and its truth value isn't enough to tell or prove such statement is true nor false.

"The sun is has risen up."

"The Earth is perfectly spherical."

These examples, while it holds some truth, need further investigation. Someone who loses track of time being indoors and was told that the sun has risen up may need to step outside for once, or witness sunlight beaming through their windows to believe it. For the second example, yes the Earth is spherical but to say it's a "perfect" sphere begs a deeper insight to secure such truth. Geophysicists say the Earth bulges at the Equator, crushing such claim that our planet is "perfectly" spherical in shape. Another sample of a statement that while true, needs more information.

L-Implication

Statements are L-implicated when its second statement follows and logically implies its derivation from the first. In other words, the truth of one statement, known as the antecedent, proves and secures its truth of the following statement known as the consequent.

"If it's hot outside, then I will go swimming."
"It's hot outside."

"Therefore, I will go swimming."

This example, the premise, constructed as a statement using an implication method, talks about the weather where if the temperature is hot enough to induce sweating, then swimming will be an option to cool off. The premise is followed by an antecedent knowing that the temperature outside is high. The consequent then follows, and concludes, that in order to cool off from such high temperature, swimming is then considered.

Statements CANNOT be L-implied if its statements do not follow nor derive from one another. It is non-sensical to consider statements like, "If pizza tastes like wood, then Mickey Mouse is the governor of Florida" as L-implicated. Premise(s) and its antecedent and consequent must follow and relate to each other.

L-Equivalence

Statements that are L-equivalent are ones that have the same truth, or false, value in every possible way. Such statements regardless of its content are "logically equivalent" if they follow the same result under its range of truth or false value.

"I will wear a raincoat if and only if it's raining outside."

Such example follows its equivalance, or "biconditional," given its statement talking about going outside and adapting to the natural environment. Even if it's not raining, making the second statement false, one could wear a raincoat anyway to keep warm; One could wear another kind of jacket, or go casual with shirt and jeans, with an umbrella if it's raining outside. Regardless, all possibilities prove each of their truth and false value, making this L-equivalent.

And there you have it! Examples of these 'Carnapian' terms may or may not be in use today, considering the decades of advancement to modern standards, but it's a great thing to look back and learn what they meant as they may come to good use again—who knows? It's exciting to discover terms old and new as means to witness the growth of Logic during its tenure back then, much like wondering why the early movies were black and white and not in color.

We hope you learned something new! Feel free to contact us, if you have any questions or would like to add to this article teaching about the terms we've discussed here!


(NOTE: This marks as our very first, official tutorial post here on KCU Network of Mathematics!)