Truth value for objects that are not included in definitions

Question

Consider the statement "This triangle has radius 3" and the statement "This cat is a chihuahua". Both radius and chihuahua are terms defined for different kind of objects than the objects we are talking. Are the above staments true/false or we can't assign a truth value? The latter is what it troubles more because we can say "A cat is not a dog and therefore is not a chihuahua". Any ideas?

Answer 1

According to my point of view, the two statements have different "logical form".

We have to consider the well-known (since Aristotle) issue of the multiplicity of uses of "is".

From one side, we may translate "This triangle has radius 3" as follows:

"The radius of the triangle T = 3"

using equality to translate "is".

In this case, the proposed translation can be classified as meaningless because the object "the radius of..." is defined only for circles.

With "This cat is a chihuahua" instead, the "is" has a different meaning: "to be an element of".

The sentence means: "The cat C belongs to the breed of chihuahuas."

Answer 2

Short Answer

In mathematics, it might be meaningless depending on the rules of logic and the definitions involved, but that form of meaningless is a convention, not absolutist truth, and that same statement could become meaningful if definitions in mathematics and rules of logic that subtend it were changed. The modern linguistical approach to evaluating meaning is not just dependent upon logical, technical truth values, but a broader appeal to pragmatic function. In short, the second two statements are meaningful in natural language, and most logicians are capable of producing simple classical logic to demonstrate the formal truths of the statements.

Long Answer

You have chosen to classify your philosophical question as one of philosophy of language, and what you present is a question that goes to the heart of the nature of what consitutes meaning, and which theories of semantics govern it. I am going to address the broader issue as Mauro has spoken to the specific examples. You ask:

'0.4 is even' is meaningless?

The answer to your question depends on your metaphysical position in linguistics in meaning. The more orthodox answer, particularly by those brought up in formal logical curriculum is yes. It is a meaningless statement under generally accepted logical principals like the law of excluded middle, the law of identity, and so on. Even and odd are predication that are used with integers who by definition do not include (2/5). So, if you examine the philosophical literature, you will find many ontologists and epistemologists struggling with a nature of meaning based on truth-conditional semantics. It is standard fare for logicians and philosophers to start off with this model and its prominence in the philosophy of language is undeniable. A giant in the field (and there are many bright minds) is Alfred Tarski whose name you'll likely come across repeatedly along with men like Kurt Gödel and Willard V.O. Quine.

What is the nature of truth-conditional semantics? From WP:

Truth-conditional semantics is an approach to semantics of natural language that sees meaning (or at least the meaning of assertions) as being the same as, or reducible to, their truth conditions. This approach to semantics is principally associated with Donald Davidson, and attempts to carry out for the semantics of natural language what Tarski's semantic theory of truth achieves for the semantics of logic.

But such a position can be misleading, not least of which is because the nature of truth is itself contested. Among theories of truth are disquotationalism, coherence, and correspondence to name three. In fact, your question is actually a variant of a well known sentence in linguistics asked by Noam Chomsky. "Colorless green ideas sleep furiously." WP:

Colorless green ideas sleep furiously is a sentence composed by Noam Chomsky in his 1957 book Syntactic Structures as an example of a sentence that is grammatically correct, but semantically nonsensical. The sentence was originally used in his 1955 thesis The Logical Structure of Linguistic Theory and in his 1956 paper "Three Models for the Description of Language".1:116 Although the sentence is grammatically correct, no obvious understandable meaning can be derived from it, and thus it demonstrates the distinction between syntax and semantics.

So, under the truth-conditional notion of semantics, this as well as your sentence are both without meaning. But you ask the question, which likely suggests, how can a sentence whose parts of meaningful be completely devoid of meaning? Our philosophical intuition seems to be struggling with reconciling to ideas:

1. There is no meaning IN the example sentence.
   2. Every part IN the sentence is very meaningful.

I suspect you would agree with these statements. In order to put yourself at ease, you need merely accept a simple fact. Truth-conditional notions of semantics are INCOMPLETE. But if that is so, how else do words have meaning besides their place in your epistemological belief system? The recent answers to that question come from outside the circles of logicians and from actual linguists. As such, they have less concern about the rational utility of truth, and more about the empirical utility of communication. In linguistics, this discipline is known as pragmatics.

Let's take a simple example. Let's presume that a sentence is used for a reason other than a direct, literal truth. Perhaps there is a local group of rebellious mathematicians who have redefined their even to include any number that can be expressed as (2m)/(10^n):m,n in R. In this case, the sentence is true. In fact, you'd be hard-pressed to find a philosopher or mathematician who would claim that there is any logical ground to deny the acceptability of the alternative definition of even (we'll call them even1 and even2 from now on). Suddenly, that exact same sentence composed of the same words of the same character goes from having no meaning to have meaning. Odd isn't it? And yet it's the case. Mathematicians redefine axioms for theories all the time without which we wouldn't have non-Euclidian geometries.

In fact, let's expand the hypothetical context. As the movement grows to use even2 versus even1, if history and Thomas Kuhn have any truth to them, there will be a political struggle over which definition is orthodox. Perhaps both even1 and even2 will come to gain usage like non-Euclidian geometrical theorems. Perhaps not, and feelings will get hurt. The rallying cry around the movement to defend even2 becomes "0.4 is even!" Now, then sentence acquires a hortative flavor. It has become a rallying cry, and now has two meanings at two levels, one literal and technical (mathematical) and one as an exclamatory with what linguists call perlocutionary force. Now, when two young bucks look at each other and one says "0.4 is even!" and the other says "amen", it has another clear indicative meaning. Ironically, most defenders of even1 would accept that the second use of your example has meaning in rhetorical context, and is essentially synonymy for "The definition even2 is true.".

So now we are in a position where one language community believes the sentence is meaningless, and the other believes and uses both meanings. As much as the adherents of the first community believe the sentence has no meaning, a neutral linguist can't escape the EMPIRICAL fact that a second community uses it as such, and most linguists today are language descriptivists, not prescriptivists. So, no linguist worth her salt is going to claim your example is meaningless. They merely would accept it's meaningless within the mathematical system of the first community. In fact, most intelligent mathematicians would recognize that even1 is a convention, not absolutist truth.

Modern linguists whether they support conceptual metaphors or generative grammar or some hybrid theory accept that meaning is created at various levels: phonology, phoneme, word, sentence, text, and context. In our extended example, the context makes clear the sentence is meaningful. These days, this approach to semantics is called cognitive semantics and it is safe to say many logicians and mathematicians are unaware of its existence because of political pressures within departments.

Instead of truth-conditions as the sole and dominant criterion for meaningfulness, cognitive semanticists operate from a broader paradigm of meaning. From WP:

Cognitive semantics is part of the cognitive linguistics movement. Semantics is the study of linguistic meaning. Cognitive semantics holds that language is part of a more general human cognitive ability, and can therefore only describe the world as people conceive of it.1 It is implicit that different linguistic communities conceive of simple things and processes in the world differently (different cultures), not necessarily some difference between a person's conceptual world and the real world (wrong beliefs).

The main tenets of cognitive semantics are:

That grammar manifests a conception of the world held in a culture;
That knowledge of language is acquired and contextual;
That the ability to use language draws upon general cognitive resources and not a special language module.

So, is '0.4 is even' meaningless? The answer to that question is depends on which language community and definitions of meaning you use.