Are these propositions?

[jason1990]

A proposition is a statement that is either true or false. Are the following statements propositions?

1. "x = 2."

2. "He owns a blue shirt."

3. "My name is Jason."

4. "Today is Monday."

[djames]

Glad to see your name pop up on this board again. I'm interested to read where you're going with this.

Do we not need a tad more information here? That is, isn't a proposition a statement that is either true or false within some stated context? Without some bit of additional language, I'm not sure it can be determined whether these statements are either true or false. Perhaps one could construct (perhaps elaborate) setups for which each of these items can be both true and false.

I don't follow yet.

[bunny]

I think they are all capable of being true or false (although the way I remember it being defined was that statements are truth-bearers not propositions). So I would say yes.

[Philo]

[ QUOTE ]
A proposition is a statement that is either true or false. Are the following statements propositions?

1. "x = 2."

2. "He owns a blue shirt."

3. "My name is Jason."

4. "Today is Monday."

[/ QUOTE ]

Are you giving a special definition of "proposition" here that you don't intend to be the one that philosophers usually give?

In philosophy, standardly at least, there is a distinction between a statement (i.e., a declarative statement) and a proposition. A declarative statement is a sentence in some natural language that can be evaluated as either true or false, whereas a proposition is said to be that which is expressed by a declarative statement (or, alternatively, that which is given by the meaning of a declarative statement).

Since the examples you gave contain indexicals (at least the last three), they cannot stand on their own as declarative statements, at least by the standard definition in philosophy of a declarative statement. Whether or not the first one is a declarative statement will depend on whether or not we treat the variable 'x' as essentially indexical.

[jason1990]

I was just recalling from memory the definition of "proposition" that I learned in my freshman logic class. It was essentially the one presented here. You appear to be making a distinction between the statement itself and its meaning. I did not intend (nor even think about) this distinction.

But you seem to know a lot about this. Can you elaborate on this (maybe with some links to further reading):

[ QUOTE ]
Since the examples you gave contain indexicals (at least the last three), they cannot stand on their own as declarative statements, at least by the standard definition in philosophy of a declarative statement.

[/ QUOTE ]
You seem to be saying that 2, 3, and 4 are not "propositions" (according to the freshman definition above). Therefore, they cannot be used (in isolation, without further qualification) in a logical argument. Is that right? It makes sense for 2. But it feels like a stretch to say that 4 is not a declarative statement.

[borisp]

The truth value of (1) is a function of x. The truth value of (2) is a function of who "He" is. The truth value of (3) is a function of the speaker. And the truth value of (4) is a function of time.

(When I say "a function of" I essentially mean "depends precisely on." Forgive me, I do math.)

Plug in values for each of these arguments, and yes, they become propositions? Is this what a proposition means?

[Philo]

[ QUOTE ]
I was just recalling from memory the definition of "proposition" that I learned in my freshman logic class. It was essentially the one presented here. You appear to be making a distinction between the statement itself and its meaning. I did not intend (nor even think about) this distinction.

But you seem to know a lot about this. Can you elaborate on this (maybe with some links to further reading):

[ QUOTE ]
Since the examples you gave contain indexicals (at least the last three), they cannot stand on their own as declarative statements, at least by the standard definition in philosophy of a declarative statement.

[/ QUOTE ]
You seem to be saying that 2, 3, and 4 are not "propositions" (according to the freshman definition above). Therefore, they cannot be used (in isolation, without further qualification) in a logical argument. Is that right? It makes sense for 2. But it feels like a stretch to say that 4 is not a declarative statement.

[/ QUOTE ]

For the purposes of teaching logic philosophers typically use the terms "proposition" and "declarative statement" interchangeably (for example, the terms "propositional logic" and "sentential logic" are used interchangeably), although they are different. A declarative sentence is a linguistic item, while a proposition--the meaning expressed by a declarative sentence--is something more abstract.

In order to evaluate an argument in propositional logic we must evaluate the truth of the premises. To take one of your examples, we cannot determine whether the statement "He owns a blue shirt" is true or not until we know what the indexical "He" refers to in that very statement. That is why students are taught to replace the indexical elements in declarative sentences when evaluating arguments in propositional logic. So if "He" is replaced by "John Malkovich" we know now what proposition is being expressed by the sentence "He owns a blue shirt"--it is the same proposition expressed by the sentence "John Malkovich owns a blue shirt"-- and so we can evaluate whether or not it is true. To evaluate #4 we need to know on what day the statement is being made (e.g., it is false if uttered on Tuesday, but true if uttered on Monday).

Here is a discussion of propositions:

http://plato.stanford.edu/entries/propositions/

[jason1990]

Thanks for the elaboration and the link. I may have to look more closely at the link later, when I have the time. But a text search indicates that "indexical" does not appear on that page. You said,

[ QUOTE ]
To evaluate [determine the truth value of] #4 we need to know on what day the statement is being made

[/ QUOTE ]
and

[ QUOTE ]
students are taught to replace the indexical elements in declarative sentences when evaluating arguments in propositional logic.

[/ QUOTE ]
How do I replace the indexical, "today"? Am I supposed to find an alternative and "timeless" way to refer to the present day? For instance,

5. Geri Halliwell's 35th birthday is a Monday.

6. Charlize Theron's 32nd birthday is a Monday.

The truth value of 4 cannot be evaluated. But the truth value of 5 and 6, which represent 4 when said either yesterday or today, can be evaluated.

The origin, for me, of this line of thought is the following. There is an interpretation of probability theory which regards probability theory as an extension of logic. In this interpretation, probabilities are truth values. Probabilities 1 and 0 corresponds to true and false, as we use them in ordinary logic. Other probabilities correspond to "degrees of plausibility." In this interpretation, probabilities are meant to be assigned to "propositions," in the freshman logic sense. So can I extend your above quote to the logical interpretation of probability and say this:

In order to evaluate the probability of #4, we need to rewrite the sentence in a way which avoids the use of the indexical, "today".

[Philo]

You replace the indexical "today" simply by determining on what day the statement is being made. Arguments usually have a context in which they are presented that makes this possible, but not always. If you have an argument that leaves ambiguous what "today" refers to then there is no way to evaluate the argument.

Indexicals:

http://plato.stanford.edu/entries/indexicals/

[jason1990]

[ QUOTE ]
Glad to see your name pop up on this board again. I'm interested to read where you're going with this.

[/ QUOTE ]
I have recently been trying to learn more about the logical interpretation of probability. It is often regarded as a special case of Bayesian philosophy, but it is a little different. As I said before, in this interpretation, probability is an extension of propositional logic. Probabilities are assigned to propositions (or perhaps declarative statements) and represent degrees of plausibility. The mathematical laws of probability represent the rules of inference which we may apply to these degrees of plausibility.

This interpretation differs somewhat from pure Bayesian philosophy. In Bayesianism, prior probabilities represent subjective degrees of belief. They may vary arbitrarily from one person to another. But in the logical interpretation, it is believed that two rational persons with the same information should arrive at the same prior probabilities.

In his book, "Probability Theory: The Logic of Science," Jaynes sets out to lay down the rules which ought to be followed in order to translate a given set of (incomplete) information into prior probabilities. He describes it as building a "robot" which will reason according to these rules. The rules are derived from a set of "axioms" which are meant to be self-evident in a certain sense. Namely, they are supposed to be such that, if any person were to realize they were violating these axioms, they would wish to revise their thought processes to conform to them.

So, in this interpretation, a given set of information corresponds to a unique assignment of prior probabilities. Jaynes is trying to describe an "algorithm" for determining these probabilities from the information. In fact, at one point in the past, I was trying to describe this idea to chezlaw in somewhat loose and colorful language. That exchange is what prompted Sklansky to post his supposed counterexample to the "perfect probability machine." In reading about Sklansky's somewhat foggy views of the philosophy of probability, it seems clear to me that he leans strongly toward this logical interpretation. It is therefore somewhat bizarre that he would want to deny the existence of a method for determining probabilities from information. But that is not the point of this thread.

The OP comes from thinking about a scenario in which I might have incomplete information about the current day of the week. In that case, I might wonder, "What is the probability that today is Monday?" However, in the logical interpretation, I can only assign probabilities to propositions. As Philo points out, the sentence "today is Monday" contains an indexical. In order to convert the sentence into a stand-alone declarative statement, and hence assign to it a probability, I must replace the indexical. Philo says we replace it simply by determining on what day the statement is being made. Of course, that does not make much sense in the context of probability, since if I could do that, then I would not need probability. In other words, I must reword the sentence "today is Monday" in a way which does not use an indexical, yet I am not allowed to gather further information about the day on which the statement is being made. I must use only the incomplete information that I already have. If the incomplete information in my possession is not sufficient for me to identify "today" by some universal, stand-alone property, then it seems I am stuck, unable to assign a probability to this seemingly harmless statement.

Here is an example where there is not a problem. Suppose I am locked in a prison for a long time and I lose track of the day of the week. Suddenly, I am unexpectedly released. On that day, I wonder about the probability that "today is Monday." In this case, the information I have allows me to identify today as the day I was released from prison. So I could rephrase this as the probability that "the day jason1990 was released from prison is a Monday." I have removed the indexical using only the partial information on which I want to base my probability. This rephrased sentence stands alone as a declarative statement and I can now assign to it a probability. It may be that, unbeknownst to me, today is actually Monday, August 6, 2007. But it would not be right to translate "today is Monday" as "Monday, August 6, 2007 is Monday," since this latter statement has probability 1 according to anyone, and clearly 1 is not the probability that corresponds to my information.

I will leave it to anyone who is interested to come up with their own example where there is a problem. I have one in mind, but I do not want to rehash old discussions that many may be burned out on.