@EliahKagan That is, in ∀x ∀y Gxy
, the variables x
and y
are both bound, which is to say that they are bound in the whole sentence. But x
is free (and y
still bound) in ∀y Gxy
. And both x
and y
are free in Gxy
.
That is an almost exact restatement of what I just previously said... but I think putting it this way helps to clarify that free or bound variables are free or bound in some particular sentence, and a variable that is free in one sentence may be (for example, and this is extremely common) bound in a large sentence that contains it.
@Zanna The problem with that, formally, is that with F
as a unary predicate and x
as a variable of quantification, the sentence Fx
will be true or false for particular specific values of x
, yet Fx
itself does not have a truth value.
But in terms of the time question... one way to look at something like "Smith is away" is that its meaning stays the same across time but its truth value changes because the world changes. I think that's the view you're taking, and it's reasonable. But another way to approach it is to say that the meaning is what changes.
So perhaps "Smith" is a sometimes adequate, sometimes inadequate way to say something like "Smith at 04:17 Z December 3, 2015". Or maybe the information about time should go in the predicate ("is away") instead of in the object. Or maybe it should be a separate argument.
People sometimes say, "I'm not the same person I was ten years ago."
I think this is also related--though perhaps not in an obvious and decisive way--to the question of whether the past and future exist.
I strongly believe the past and future exist; that is, as I understand
eternalism (which is not necessarily the same as all other eternalists understand it), I am eternalist.
I am taken to understand that presentism used to be a popular philosophical view, but that special relativity has made it less popular--due to the seeming inconsistencies between presentism and the relativity of simultaneity.
I am interested in your thoughts about this, which I suspect may be considerably more developed or informed than mine.
@Zanna In standard logic, no sentence is both true and false. (I am not saying you are doing this... but I am saying that the claim that a sentence has a truth value is associated with the idea that it has one particular truth value, true or false, and not the other.)
The appearance that a sentence is both true and false, in standard logic, has to be resolved either by admitting defeat in the sense of recognizing that one's premises are inconsistent and cannot reasonably be adopted together, or by figuring out where one has gone wrong. Viewing a sentence as both true and false because it is true at one time but false in another is a problem, because it allows you to prove something of the form:
One of the reasons that's no good is that, if your system can prove that, then your system cannot meaningfully distinguish what is true and what is not ever, even to the slightest extent.
Not all logics have this, btw. But standard logic, which I've been presenting and which is typically used as a foundation for mathematics (whether or not set theory is used, which it usually but not always is) does have the law of the excluded middle. Some other logics that are not standard logic do also have it.
Suppose Smith is away and Smith is not away. Since Smith is away, it is surely true that Smith is away or I am the pope. (After all, Smith is away.) But Smith is not away. Since Smith is away or I am the pope, but Smith is not away, I am the pope.
@EliahKagan * not saying that you are saying this
This does not mean that standard logic cannot be used to describe a changing world. It can, and there are multiple ways to do it.
This issue with the principle of explosion is something I intended to go into, btw -- it is not motivated by the discussion of time, I just happened to be reminded of it by that.
So, one convention is to talk about open and closed sentences and recognize both as sentences.
Another convention, which I prefer, though of course it breaks quite drastically with what a sentence is in ordinary language, is to say that a sentence has no free variables (though variables may be free in parts of it, of course), and to call that which would be a sentence if its free variables were bound a formula.
That is, the other convention, which I prefer but have not adopted for our recent conversations, is to say that sentences are what some people call closed sentences, and formulas are what some people call sentences (open or closed).
Then all sentences are formulas but not all formulas are sentences, and in particular, ∀x ∀y Gxy
is a sentence (and thus also a formula) but that ∀y Gxy
and Gxy
are formulas but not sentences. Gxy
is then regarded to be an atomic formula but not, in the strictest sense, an atomic sentence.
I'm comfortable with either convention so long as I know which one I'm using.