What are "words"?

Question

I am reading Classical Mathematical Logic by Richard L. Epstein, page $3$:

B. Types

When we reason together, we assume that words will continue to be used in the same way. That assumption is so embedded in our use of language that it's hard to think of a word except as a type, that is, as a representative of inscriptions that look the same and utterances that sound the same.

...We will assume that throughout any particalar discussion equiform words will have the same properties of interest to logic....Briefly, a word is a type.

I don't really understand what a word is. The author says it is "a representative of inscriptions that look the same and utterances that sound the same." Does this mean that a word is the collection (not set?) of all equiform inscriptions and utterances that have been, are, will be, will never be, written or uttered? If I write: car, car, did I just pluck out two inscriptions from an infinite collection (cars, cars, cars...)? Is each "cars" distinguishable while it is in the collection? Meaning, does the collection actually look like ($\text{cars}_{\text{That will be used by Ovi on 7/23/2016}}$, $\text{cars}_{\text{That will be used by Ovi on 7/25/2016}}$, ...) or does each inscription become distinguishable only after it has been plucked out from the collection? This interpretation sounds a little bit platonic, which is the reason why I think it is probably wrong; the author had distinguished himself from platonics on a previous page.

When I think of a word, I think that the word is inscription or utterance itself. This inscription or utterance is a representative of the meaning of the word.

a type probably means a collection of properties (as in programming languages) — reuns, Jul 23 '16 at 19:31
It seems very difficult to know what the author has in mind without reading more of this book. — Qiaochu Yuan, Jul 23 '16 at 19:31
@QiaochuYuan You think I should just keep reading ahead and not worry about it too much? This is only page 3, and pages 1 and 2 were just spent on defining a proposition as "a written or uttered declarative sentence used in such a way that it is true or false, but not both". Then he proceeds to give an alternative view of propositions (platonic view), and then page 3. — Ovi, Jul 23 '16 at 19:34
@user1952009 Is your definition consistent to the definition in the answer of the linked question? I think they are slightly different, but I am just trying to see if I understood that definition correctly — Ovi, Jul 23 '16 at 19:35
I dunno, you might also give up on the book entirely and switch to something less confusing. — Qiaochu Yuan, Jul 23 '16 at 19:36
@QiaochuYuan Do you have any suggestions? Just from the first few pages I like it a lot, because it is much more rigorous than my Discrete Math textbook, it assumes no background knowledge (therefore it's all self contained). — Ovi, Jul 23 '16 at 19:39
@MauroALLEGRANZA Is there any difference between "word" and "meaning of the word"? — Ovi, Jul 23 '16 at 21:24
@MauroALLEGRANZA Sorry never mind, I should read the source before asking the question — Ovi, Jul 23 '16 at 21:26
I see no reason for voting to close this question: the OP is trying to understand a mathematical text, has done some work on understanding it and is asking for help. "I don't have the book" or "I think there are better books" is not a good reason to discount the question. — Rob Arthan, Jul 23 '16 at 23:01
@Ovi: See http://math.stackexchange.com/a/1684208 for my suggestions on intuitive yet reasonably rigorous introductions to logic. I didn't vote to close, but I do vote to stop reading that book. — user21820, Jul 25 '16 at 01:25
@user21820 Thanks "Introduction to Logic" seems very clear and intuitive. I am planning to read this and my own book at the same time, because I still like the level of rigor in "Classical Mathematical Logic". Is this book really that bad? — Ovi, Jul 25 '16 at 04:09
@Ovi: From the fragments that have been quoted so far, it has not proven itself rigorous at all. The fact that its intended meaning is not clear to you is not in general a good sign. In my experience I find that most texts that talk a lot of philosophy have nothing concrete to say and hence to be a waste of time. In fact, see http://math.stackexchange.com/a/1867336 for an example of a natural language statement that appears to be used in such a way that it is true or false but not both, which nevertheless cannot be a proposition without giving rise to contradiction. — user21820, Jul 25 '16 at 04:53

score 3 · Accepted Answer · answered Jul 24 '16 at 09:04

Epstein is exposing an anti-platonist point of view; see page 2:

There are other views of what propositions are. Some say that what is true or false is not the sentence, but the meaning or thought expressed by the sentence. Thus ‘Ralph is a dog’ is not a proposition; it expresses one, the very same one expressed by ‘Ralph is a domestic canine’.

Platonists take this one step further. A platonist, as I use the term, is someone who believes that there are abstract objects not perceptible to our senses that exist independently of us. Such objects can be perceived by us only through our intellect. The independence and timeless existence of such objects account for objectivity in logic and mathematics. In particular, propositions are abstract objects, and a proposition is true or is false, though not both, independently of our even knowing of its existence.

Thus, it is clear that for Epstein, propositions are not abstract objects, but "concrete" ones :

A proposition is a written or uttered declarative sentence used in such a way that it is true or false, but not both.

Where, in turn, a sentence is

a written (or uttered) concatenations of inscriptions (or sounds).

Consider for simplicity the written case:

dog, dog, Dog,

are all written concatenations of inscriptions.

If we consider them as "concrete" objects, they are (slightly) different: thus, have we to consider them as different words ?

Epstein's answer is: NO, because we have to consider

a word as a type, that is, as a representative of inscriptions that look the same and utterances that sound the same [i.e. that are equiform].

Words are types. We will assume that throughout any particular discussion equiform words will have the same properties of interest to logic [in particular, same meaning]. We therefore identify them and treat them [i.e. the singular utterances, i.e. the tokens] as the same word.

Briefly, a word is a type.

score 1 · Answer 2 · answered Jul 24 '16 at 13:32

A "word" is many things. Consider this example:

It is what it is.

How many words are in that sentence? In one sense, there are five words: the first "it", the first "is", the "what", the second "it", and the second "is". So in that sense there are two "it"s, two "is"s, and one "what". In another sense, there are three words: "it", "is", and "what". In terminology that dates back to Peirce, words in the first sense are called "tokens" and words in the second sense are called "types". So every copy of "it" is a different token of the same word type.

This is not really a mathematical topic - many mathematicians get by without ever thinking about this distinction - but it is of interest in philosophy. See Types and Tokens in the Stanford Encyclopedia of Philosophy for a much more thorough account. In that context, they also distinguish between "tokens" and "occurrences" of words, so that there are at least three concepts instead of two.

In general, by the way, you can ignore this distinction when learning introductory logic, and you are very unlikely to miss any mathematical issues. Logic texts of a more philosophical character sometimes spend time on the distinction, but texts of a more mathematical character may not mention it at all. It is not one of the most pressing things to learn about mathematical logic for a new student.

What are "words"?

2 Answers2