"Has a value of" vs. "has the value of"

Question

The basic problem

The following types of expressions are ubiquitous, especially (but not only) in scientific and technical literature; note the indefinite article in the construction "...has/with a value of...":

At period 1, inflation has a value of about 1.021.
The electric field within the capacitor has a value of 170 N/C
The energy charge quotient has a value of unity (or, 1.00) when only ATP is present and a value of zero when only AMP is present.
The shipments of cattle were 84,205 head, with a value of $5,473,325.

In fact, it seems that all quantifiable properties obey the same relevant rule:

The Commodore PET was also released in 1977 with a price of $800.
The front of the train has a speed of 23 m/s.

The indefinite article is apparently at least allowable even when the relevant property refers to something previously mentioned, like in the second appearance of speed in the following example:

The train is moving with a speed of 5 mph. It then stops, and then slowly speeds up, until it is again moving with a speed of 5 mph.

(I don't have links to published versions of exactly these sentences, but I hope the following two examples come close enough: look for then accelerates back to a velocity of 25 m/s here, and for again with a speed of 250 r.p.m. here, the latter in the paragraph right below Fig. 2. See also Some analyses that (probably) don't work, B, below.)

The basic question is, why are we not using the definite article here? After all, in all these examples, the "of"-phrase would seem to be providing a further specification, a narrowing down, of the property in question. In other words, it would seem to be a standard prepositional phrase, which normally entails the definite article.

A fill-in-the-blank exercise

To help illustrate this last point, let's try an exercise. Fill in the blank ( __ ) in the following:

"The train is moving with a certain speed."
"What speed?"
"___ speed of 5 mph."

What did you put in the blank, "the," "a," or nothing? Surely, you put "the"?

I think I can show that the reason you put "the" is not that the speed is being mentioned for the second time---see Some analyses that (probably) don't work, A., below. The reason thus must be something else, and I think it is that "5 mph" provides a "narrowing down of possibilities to a single one," a "specifying," of the "speed."

And yet, just as surely, we would say,

The train is moving with a speed of 5 mph

even though here, too, it would seem that "5 mph" serves to "narrow down" the speed to a single value.

So, why the indefinite article? Why doesn't it matter that the numerical value narrows down the possibilities to just one? Why doesn't it matter if the relevant noun refers to something previously mentioned?

Categorical properties

The rule even seems to apply to the cases of categorical (as opposed to quantifiable) properties:

Forty-five percent of the population has a blood type of O.
He has a major in social work.
Wilde looked at the copy with an expression of surprise.

I am tempted to (semantically) analyze these examples as follows: they all involve a determinable property---i.e. a property that can get more specific---being more precisely characterized. In the case of quantifiable properties (inflation, speed, price, charge,...), we make them more specific by giving a numerical value. In the case of categorical properties, we restrict them by specifying the category: so a blood type can be A, B, AB or O; an expression can be one of surprise, fear, love, etc. And all of this talk of making things "more specific" would suggest the definite article, in contradiction to what is actually used.

Apparent counterexamples to the rule

Based on the preceding, we would expect the property of having an emotion (which can be more precisely characterized as happiness, sadness, boredom, surprise, anger, delight, ...) should require the indefinite article as well. And yet, instead we have

God eternally has the emotion of compassion.
For example, consider what happens when the reader has the emotion of surprise.

And to make things really confusing, consider the property called having a property, which we further characterize by specifying the kind of property. Here there seems to be no pattern at all. You might hope that some of the usual rules would explain the particular choices of the articles in the sentences below (e.g. the property was---or was not---previously mentioned). However, if you look at the full texts, it will be apparent that such an explanation doesn't really work here:

The class of all spoons has the property of not being member of itself.
There is a thing which has a property of being the only writer of Waverly and of being Scotch.
As a set of points space has the property of containing points.
The construction of womanhood has a property of Otherness.

Here and then one encounters counterexamples to the paradigmatic cases discussed at the beginning. In the following, the first and the third boldfaced article are the counterexamples, while the second one is the (usual) indefinite article.

A vehicle departing the roadway at the mean speed of 49.3 mph subjected to an effective friction of 0.7 due to braking would need to travel 30 ft before it slowed by 10 mph. If this vehicle was encroaching at the mean departure angle of 16.9 degrees...

You may think that it is significant that we have mean speed. But, no...

Elite human athletes run 100-m races in about 10 s, at a mean speed of 10 m/s.
Ahead of the fleet lay a journey of some 420 miles, scheduled to last for thirty-five hours at a mean speed of twelve knots.
By now the Exeter and the Graf Spee were approaching each other at a mean speed of forty miles an hour.

Additionally, it seems that with the weight of used to be more acceptable in the past (see here), but for many decades already, with a weight of is preferred (see here).

The question summarized: All the examples above would seem to be clear cases of a prepositional phrase telling us "which one," i.e. "which value" of the many possible ones, and so we would expect the definite article in all of them. But instead, we usually use the indefinite article instead. Why? And why don't we use the indefinite articles in the case of having an emotion and in half the cases of having a property?

It could be that this an example where English hasn't yet really worked out what the rule should be. It could be that matters here are simply illogical (see here and here for some comments on why some usages of the English articles---including the zero article---may well be simply illogical). But are they illogical in the particular cases presented here? Or is there, after all, some rule, some pattern behind all of this?

Update 1 (with thanks to Edwin Ashworth): Sometimes when we more precisely characterize a determinable property, we do it by saying that it is the same as something else. In this case, we do use the definite article:

It has the color of deep rich caramel.
One copper coin has the value of a measure of rice.

I think the reason is this: when, to our ear, it seems that the "of"-phrase is coming from a transformation of a possessive phrase, the urge to use the definite article is simply too great. So we say It has the value of a Spanish milled dollar because it feels like it is a rephrasing of It has a Spanish milled dollar's value.

On the other hand, we don't have the same urge to say I have five grains of gold, with the value of $13, because we are not inclined to hear this as a rephrasing of a possessive phrase. And this for the simple reason that there simply is no corresponding possessive phrase. Note that with a $13's value doesn't sound right at all; it would instead have to be with a $13 value, which is not a possessive phrase. So the urge to use the definite article is not as great here, and, indeed, we rather have I have five grains of gold, with a value of $13, consistent with the previous examples. (Of course, a "decreased urge" to use the definite article is one thing; what is the actual reason why we use the indefinite article is another...)

Some analyses that (probably) don't work:

A. It has been suggested that the reason we put "the" in the fill-in-the-blank exercise (the second subsection of the text; see above) is that it is the second mention of the speed, i.e. because it is a reference to something previously mentioned. But if it were so, then this "the" would be obligatory, and it isn't:

"The train moving with a certain speed."
"What speed?"
"A certain speed. The point is, the speed is increasing."

Here the second-to-last sentence is used by the speaker to let his correspondent know that the precise value of the speed doesn't matter. (If it did matter, the speaker's reply would begin with the definite article, e.g. The speed of 5 mph.) But even so, in the sentence that follows, it is obligatory to say the speed and not a speed, because this really is a reference to something previously mentioned. Since in the second-to-last sentence the definite article is not obligatory, I conclude that when we do put it, we don't put it because it is referring to something previously mentioned (if that were the reason we put it, it would be obligatory). Instead, I think the reason is that "5 mph" provides a "narrowing down of possibilities to a single one."

B. It has been suggested that the reason we put the indefinite article in The train is moving with a speed of 5 mph is that this is the first reference to that speed. But this analysis does not work. Consider the following:

The train is moving with a speed of 5 mph. It then stops, and then slowly speeds up, until it is again moving with a speed of 5 mph.

Note that, at the end of the second sentence, it is again the indefinite article, even though we are referring to a previously mentioned speed (a point emphasized by the appearance of again). From this example, I conclude that the whole business of previous mention is irrelevant to what article is to be used here, although, of course, I still don't understand the grammatical reason why it is irrelevant, or why "5 mph" doesn't count as a "narrowing down of possibilities to a single one," like it (I think) does in the fill-in-the-blank exercise (see sub-subsection A, just above).

(I don't have links to published versions of exactly these sentences, but I hope the following two links come close enough: then accelerates back to a velocity of 25m/s, and again with a speed of 250 r.p.m., the latter in the paragraph right below Fig. 2.)

Answer 1

When the value is subject to change, then has a value of N is correct, it implies at different time or circumstances, the value may be different. Over time, the train will move at lots of different speeds. The value of gold changes depending on the market. The value read from a capacitor changes depending on charge time and voltage.

He has a major in social work. This implies he could have other majors in different subjects.

It has the color of rich deep caramel. This implies the color property does not change. Whatever "it" is, we wouldn't expect it to be blue tomorrow or black yesterday.

Using a or the implies variability in the value. There will be many examples, as you have cited above, that appear inconsistent with this. I'd put it down to most writers not being philosophers or logicians, and they write what sounds appropriate, using a or the almost interchangeably.

Answer 2

First, I note that my response to the fill-in-the-blank exercise is not what the OP expected. For me, the most natural answer to "What speed?" is "A speed of 5 mph." I don't know if this is a Germanism (I think my native German works that way), my déformation professionnelle as a mathematician, or just how English really works.

My first intuition for the general phenomenon is that "a ... of ..." is shorthand for "a ... that has been measured/determined to be ...". My second intuition is that this actually makes sense because in most cases we are dealing with approximations rather than exact values, and approximations are not determined in a way that would justify the definite article. (This also justifies my fill-in-the-blanks response.)

We can talk of the speed of the train, even though it (a) is variable and (b) can't be measured with infinite precision. However, the speed of the train is not "the" speed of 5 mph. It is actually one of many speeds (e.g. 4.9814264 mph) that fall into the class of speeds to which we refer as (approximately) 5 mph.

Obviously the examples involving precise dollar values cannot be explained this way. The explanation here is probably that the reasons given are valid for most other types of values and the choice of article is determined grammatically rather than semantically.

Whereas speed and value have a strong bias toward the indefinite article, with price we usually have a choice between indefinite and definite, and amount even has a bias toward the definite article.

Technical terms of mathematics such as sum are special cases. In principle they seem to behave much like price, with a choice between a and the. However, both are best avoided as they can lead to ambiguity or at least irritation:

1. Having determined a sum of 4,291 Euros, ...
2. Having determined the sum of 1,381 Euros and 2,910 Euros, ...
3. Having determined the sum of 1,381 Euros and 2,910 Euros as 4,291 Euros, ...
4. Having determined the sum as 4,291 Euros, ...
5. Having determined the sum of 4,291 Euros, ...

The problem with 5 is that it's not a priori clear whether it's parallel to 1 and 4 or to 2. Even 1 itself is not completely unambiguous and could be interpreted as parallel to "a sum of ... and ...". In other contexts the workaround of using as is not possible.

(Note: Prescriptivist extremism has caused a bias towards treating obviously uncountable measurements such as 5 mph as countable and applying "fewer" to them instead of the natural "less", just because they involve a number that happens to be approximated by an integer. Maybe this question is somehow related.)

Answer 3

At period 1, inflation has a value of about 1.021.

1.021 isn't the value of inflation. Inflation is invaluable. You can't buy it.

When an ampere meter displays a value of 1.000, it isn't how much the ampere meter itself is worth.

God has the emotion of compassion.

The wording is terrible, but, for what it's worth, the emotion is God's, not compassion's, hence the definite article. Ditto the other example.

I hope this helps, even though I realize it's kind of confusing. I hope someone can furnish a better explanation.

Answer 4

Based on the sentences here, I thought of several cases:

1. In accordance with Edwin's explanation, sentences of the form "...the (same) value (as that) of X" almost require the definite article.

However, this case is insufficient, since it doesn't deal with the counterexamples you provided. This leads me to some other cases:

2. "...the (previously mentioned/otherwise salient) value of X" -- almost requires the definite article

You mentioned yourself that this follows from natural English rules, e.g. "we experimentally verified the (expected) value of 0.5."; "the (aforementioned) feeling of cynicism was common among younger individuals"

3. "...the (fixed) value of" -- suggests the definite article

Ricky had an interesting idea here; I'm thinking of the first couple of counterexamples, although I think this case applies to all of them. For example, God can have "a fixed emotion of compassion", but more probably has "the fixed emotion of compassion". As far as the second counterexample goes, the (previously mentioned by you) emotion of surprise is being fixed in this hypothetical situation with an arbitrary reader.

4. Not otherwise covered above -- probably requires the indefinite article

So I doubt the first three cases are comprehensive in the least -- I'm working off the weak heuristic of "if I can justify using the definite article beyond just the value being some number/category/etc, it's probably valid to use the definite article". That is, it's insufficient to say that having a given value is enough to say that it warrants the definite article; it has to be implicitly or explicitly qualified or described in some way ("the [adjective] value of"). Even something as trivial as "the specific value of" would suffice, although if it's not self evident in the sentence what the qualifying adjective is, it's probably best to spell it out.

Answer 5

'A' would be preferable when giving exact value, like in numericals. The other would support relative measure, a case of which would be stating equality, as you mentioned. You may notice it's in the examples. Reason? I'll come to that.

Now for the question related to "_ speed of 5 mph." We use 'the' because that article is also used for referring to things which have already been stated. Clearly, we are talking about the same speed that was asked in the question. But for a sentence like 'The train has a speed of 5mph', we are telling about the speed for the first time. You'll notice the same rule applied here - The train has a speed of 5mph. The speed clearly conveys that the train is not so fast. That may seem a bit odd, but is grammatically appropriate.

'A' would be supporting numericals, when the thing is defined for the first time. 'The' would be used when stating the value in relation to something. 'The value', the same value of xyz that was said before.

Hope this helps. Thanks

Answer 6

The indefinite article 'a[n]' is simply the determiner 'one'. Some determiners are indefinite (one, two, some, many ...) and some are definite (all, every) [because they specify the universal, which is unique]. Indefinite noun phrases represent new information to be added to the memory, while definite noun phrases represent information that should already be in the memory (and not be redundantly added). Any indefinite noun phrase can be made definite by marking it with the definite article 'the'. Note that the determiner 'one' is omitted following 'the', because singularity is also marked by lack of a plural noun. Also note that the determiner 'some' is omitted following 'the', because it has no information about quantity other than '>0', which is already conveyed by 'the'.

As to your question, an indefinite noun phrase might be preferable, but the same sentences are also giving a value to be stored, so the value is in some way 'already in memory'; thus, it is OK to use the definite article.

Answer 7

Though issues at stake have been discussed many times over, but the sincerity and meticulousness with which the the question has been raised demands admiration and a detailed discussion.

HOW TO USE, WHEN TO USE AND WHICH ARTICLE BEFORE uncountable ABSTRACT NOUNS

Articles are actually precision tools for limiting, quantifying and determining countable concrete nouns; they are a pointer to the noun and brings in a kind of unique accuracy. While (a/an) are indefinite, 'the' can be used with nouns of either number for one either specifically in discussion or about to be in discussion.

For the present, we would like to subdivide nouns into 3 categories namely Concrete, Abstract and Ambiguous. Concrete has physical existance, abstract conceptual and ambiguous can be viewed as either abstract or concrete:

*Beauty is truth, truth beauty.

**She is a beauty.

Abstract uncountable nouns in general do not take article.

*Power corrupts and absolute power corrupts absolutely.

However, for qualified abstract nouns either followed by a defining relative clause or 'of + NP' we place a 'the' before the abstract noun. The rule of article with countable abstract noun (ambiguous) are same as before concrete count nouns.

SPEED or, for that matter, other words like value/emotion/type etc. are abstract noncount nouns concentrating on special instances or special aspects of the given idea and can be ranked as an ambiguous noun of our earlier prescription. When we tell "... at a speed of" we are in a sense upholding the concept of qualified abstract noun but the problem remains– why are we using 'a' instead of ' 'the'. But there are instances when we use 'a' before qualified abstract nouns:

• She lived a long life and died an abnormal death.

So it is established that 'speed' is qualified by a given numerical, it becomes an ambiguous noun and, more to it, combines a generecic concept which has to be made peculiar or account specific. Hence "... A rate of..."

However, the emotion referred to God is unique to Him alone. Hence 'the'. In the gap-filling-exercise, I an more inclined to put one "A" there and see, it works fine under the impression that generic concepts are made situation specific; it may change any moment.

Great masters of grammar writes that use of 'a/an' with non count abstract noun is somewhat like the use of 'the'.

Answer 8

One should not necessarily ascribe a given "value" to be the worth of a specific quantity. We can conclude by stating that when one variable quantity [given a title and definition; be it amperage, mass, length etc] is calculated, determined, or observed to be x units , then we say that this quantity has a (indefinite, varible) value of x units; whereas when one invariable or constant quantity [e.g the speed of light; the SI definition of the kilogramme; Planck's number, etc] is calculated, determined, or observed to be x units , then we say that this quantity has the (definite, invarible, constant) value of x units.

In categorical properties, such as the blood typing example, the attribute might linked to the presence or absence of an object of some specificity: e.g. 5% of the population possess red blood cells that express the A-antigen on their membranes (...has the A blood type). The presence of a certain entity is then characteristic of a property type and this may be the direct attribute of that class by definition. The same can be said for emotions which have distinct characteristics [such as palor, pupil dilation etc] used in combinations to define a specific emotion. The absence of entities can also result in indirect or non-specific attributes.

If the observable entity is directly associated with an attribute, then it is from here that the article type is inherited; the absence of an entity may result in a different article type e.g. The leaves have an orange colour: their reduced exposure to sunlight and photosynthesis rendered the chloroplast inactive, and the lack of chlorophyll confers the lack of the green colour of the leaves. Note that leaves may appear a different observable colour and so an indefinite article may be used.

Answer 9

Compare:

1. The front of the train has a speed of 23 m/s

2. The front of the train has the speed of 23 m/s

If you ask #2, the reader will understand that there are a very narrow set of possible values. It would make the reader think this is a problem with, for example, a binary choice, where we're comparing Train A with speed 23 m/s and Train B with speed 27 m/s. Reason: "The" is used to refer to something previously mentioned.

Note that the following is acceptable and won't be misunderstood:

3. The front of the train has speed 23 m/s.

In the discussion, you asked

Why would we say, "Forty-five percent of the population has a blood type of O?" After all, if anything can be said to be fixed, it is a person's blood type.

Same thing. If you use "the," the reader will take that as an indication that you are analyzing a situation where only two (or three) blood types are possible. (That assumption would have been established in the set-up of the problem, in earlier sentences in the article or homework or exam problem.)

If you want to discuss "He has an emotion of approval" I suggest writing a separate question.