This essay is a response to a question posed by Prof. Patrick Grim on page 20 of his excellent book The Incomplete Universe. I am eager to learn how Prof. Grim deals with this problem.
The context of the question is a discussion of alternatives to solving the Liar's Paradox problems by making a clear distinction between sentences and propositions. With this distinction made, we agree that sentences say nothing and make no assertions. The propositions they stand for, on the other hand, do make assertions and therefore we should be able to assign truth values to them. (At least that is the way I understand it.)
The question as stated by Prof. Grim is the following: "[A] propositional approach is in some ways too easy a reply to the Liar and its kin, at least with regard to some of the questions that many who have taken on the Liar have tried to answer. Surely it is not enough merely to say that sentence (4) or (6) or (7) or (8) doesn't express a proposition, for we will also want to know why. What is it about such sentences that keeps them from expressing a proposition or deprives them of a proposition to express? A propositional account alone offers no theory of why certain sentences will supposedly fail to express propositions..."
My short answer to Prof. Grim's question is that such sentences are incomplete. i.e. they are not well defined.
Let me explain by using example (6). (6) appears on page 7 as follows:
(6) is not a member of set A. . . . . . . . . . . . (6)
I maintain that (6) is not only not-well-defined, but that it cannot be well-defined. The problem is that the universe does not contain enough material, nor time, in order to completely define the terms in (6). Let's try anyway.
The grammatical construction 'is not a member of' is well-defined since it is a meaningful mathematical expression. 'set A' is also well-defined in the context of the introduction to (6) so we needn't dwell on it.
That leaves us with only the '(6)' to define.
The '(6)' appearing on the right identifies the string '(6) is not a member of set A.' which serves as the definition of '(6)'.
By substituting the definition for the symbol, we get the more refined statement of (6) as,
((6) is not a member of set A.) is not a member of set A.
Again, we have the term '(6)' appearing in the statement, and we again substitute the definition in order to refine the statement with the goal of completely defining it:
(((6) is not a member of set A.) is not a member of set A.) is not a member of set A.
The pesky '(6)' persists. If we continue to replace it with its definition, we can see that we will run out of toner, or time, or fundamental particles to use in forming the expression. The definition of (6) will remain incomplete despite our best efforts.
It seems to me that the fundamental distinction between propositions and sentences is that the former are mental constructs and the latter are physical constructs. This distinction is easily made if Cartesian Dualism is accepted, which I do, and which I realize is no longer in fashion among modern philosophers.
Using this distinction, propositions are mental constructs such as ideas or concepts, whereas sentences are physical manifestations such as utterances, inscriptions, or encoded bit strings. Therefore, propositions require a mind and sentences require a physical world.
Now, what candidates do we have for a mind? Let's use our grammar to identify three. The most certain candidate comes from the first person singular: I. I have (or am) a mind and I know that I can and do harbor ideas and concepts. Some few of these can be expressed in sentences that I consider to be fairly accurate expressions of the idea or concept. For example, I harbor the idea that I have sensations that seem to come from interactions between what appears to be my body and some specific structure in an external world. I can express that idea in a sentence inscribed as follows: I see a tree.
The second candidate, and a little less certain (to me), is the second person singular: You. You are no doubt reading this sentence now (now being defined as the time at which you happen to be reading this essay.) I have some amount of confidence that you will agree with, and identify with the previous paragraph with the 'I' referring to yourself, the reader, rather than with me, the writer.
The third candidate, and less certain (to both you and me), is some third person or persons. There are strong indications that there are some 7 billion other minds much like mine (or yours), and these are all situated on this small insignificant planet among the vastness of space and its contents. There may be untold more billions of extra-terrestrial minds. And there is the cogent suggestion of Gregory Bateson that there may be a comprehensive mind involved in the establishment and operation of the physical universe itself. Any or all of these may harbor ideas and concepts.
Sentences are much simpler by comparison. Utterances are short-lived and unless they are heard, or recorded, they disappear as soon as they end. Inscriptions last much longer, and can reach more listeners/readers than utterances, but they are still limited by time and space. Encoded bit strings can last longer and travel further than utterances or inscriptions. They can, and have, escaped the limits of our planet. The odds of them being "heard" or "read" by minds beyond Earth, however, are small, the claims for prayers notwithstanding.
With this distinction made, the claim that sentences carry no truth value makes sense. What does make sense, however is that sentences can be seen as the carriers of concepts or ideas from one mind to another (or the same mind later in time). They are simply the links in a communication network with minds as the nodes and physical media as the links.
There are two troublesome points in such a communication network: the encoding of an idea or concept into a sentence, and the decoding of a sentence into an idea or concept. If the encoding and decoding are even possible, and the receiving mind forms a coherent concept that matches the original concept as a result, the claim can be made that an understanding has been achieved. In other words, an idea or concept has successfully been transferred from one mind to another.
In the unfortunate case that Cartesian Dualism is denied, then there is trouble in defining just what a mind is. If it is nothing more than patterns in physical parts of the brain, then it seems to be a little murky as to how to make the distinction between propositions and sentences. Sentences are still easy, but the patterns of brain states comprising ideas or concepts would seem to be not different in kind from the patterns of various other physical constituents which make up sentences.
This is but one of many reasons why I urge philosophers to give Cartesian Dualism a second chance and a new life.Please send me an email with your comments.
©2016 Paul R. Martin, All rights reserved.