I have found refurbished laptops to have no more problems than new laptops, and they cost less (or are better for the same price).
In suggesting refurbished laptops, perhaps I've just made it so there are even more things to choose from. :)
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4:05 PM
@Zanna I do think it's likely that buying a refurbished laptop would contribute less to harming the environment, and also that it would contribute less to the exploitation and trafficking of workers including in ways that are not mediated by environmental harm. However, I expect that the degree of difference is extremely small. Your mentioning this, and also your concerns that the machine might not work well, motivate me to point out the distinction between refurbished and used.
A refurbished laptop will typically have been used for a short time and sent back to its manufacturer during an initial warranty period, serviced, and tested.
The number of refurbished laptops relative to the total number of laptops probably can't change much based on demand for refurbished laptops, so buying a refurbished laptop has a much more direct effect on how many new laptops are purchased than buying a used laptop (which also causes more new laptops to be purchased, but to a lesser extent).
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5:02 PM
There is something I can mention on the conceptual (i.e., not the stuff about identity elements) line of thought... but I don't want to make it so you're staying up to discuss it, since as I understand you that's not something you wish to do. I don't know if that would interest you. It can wait; please feel free to say no. (Also, if you don't reply, I'll assume you don't wish to hear it at this time.)
This is closely related to the question of which unary predicates and, more generally, sentences, have a set that correspond to them.
You might want--and early set theorists did hope for--a situation where, for every expressible property about an object x, you could have a set of all and only those x for which that sentence is true.
Unfortunately this is not possible--a system that is powerful enough to guarantee this is so powerful it can prove contradictions. The usual way to show this is with Russell's paradox. You may have heard of Russell's paradox. (Have you?)
Even if not, I could state it now and I believe it would make sense... but I would be interested in laying some of the groundwork both for that and more broadly for considerations about which sets exist, by first examining the general issue of which sentences/formulas can reasonably be said to be either true or false.
(Regarding the other line of thought, which currently is on identity elements, I should perhaps not have phrased what I said earlier as an offer of another hint -- instead, there is some general advice about uniqueness proofs that I can give, whenever we pick that up, which I believe will enable you to do the proof.)
However, if you like, I would also be interested say something now about which sentences can meaningfully be said to be true or false.
In particular, I believe you can come up with and very possibly have already come up with earlier, then discarded as not readily relevant some sentences--at least with what is widely regarded to be a sentence in a natural language, but even with what I've been talking about so far as sentences in the formal language of FOL--which cannot meaningfully said to be true or false.
That sentence isn't an example of one that's neither true nor false... though it is poorly written and I'm sorry. :)
That's what I would say. I am not trying to represent that view as something that people have to accept to do or understand logic.
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5:32 PM
However, there's one form of indexicality that is pretty universally regarded as sufficient to prevent a sentence--in the way I've been using that term lately--from having a truth value: when its truth value could be affected by object is supplied in place of a variable of quantification that is not quantified over.
This is often explained by analogy to the indexicality represented by pronouns in ordinary language, e.g., "She is an astronomer."
I do think pronouns are like variables, but in terms of whether or not something has a truth value, I think there are some subtle issues. I am not confident in the position that "She is an astronomer" doesn't have a truth value on the grounds that I don't know who "She" is. In the sentence, "Alice is an astronomer," at least as I use it in this example, I don't know who Alice is either.
That's not to say that it has a truth value, but rather than the presence or absence of a pronoun does not determine if a sentence may be ascribed a truth value in the way that a presence or absence of unquantified-over variables in decisive in first-order logic.
5:38 PM
I am not sure which style is preferable; in our recent conversations, I've been deliberately adopting a different convention from the one I usually use.
When speaking formally, some people--including me in our recent conversations--use "sentence" to mean the same thing as "formula."
A closed sentence is a sentence that is not open: any variables of quantification that appear in it are quantified over.
A variable that is quantified over is said to be bound (it is bound to the quantifier that introduces it) and a variable that is not quantified over is said to be free (it is not bound to any quantifier). If you're using a system that has constants (i.e., permanent names), those are not variables and are neither bound nor free, even though syntactically it is as though they are free.
That is a closed sentence, because for every occurrence of a variable in the sentence, the quantifier that quantifies over it is also present in that sentence.
That may seem questionable since I haven't said what G means, but consider, first, that we often have primitives whose meanings are not defined (like "∈" in a set theory) and we do still think we say things that are true or false, and second, we use the assumption that sentences like that are either true or false all the time when we take sentences like this one to be true:
But for this to work, all the closed sentences that are well-formed in the language of our system--that is, that we can actually express symbolically in it--must be regarded to be true or false. We might not know how to discern which. It might not even be possible to discern which. But we assume they are true or false.
5:55 PM
@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.
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.
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.
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).
@EliahKagan I should say: what it meant, in saying that I was not motivated by the discussion of time in bringing up the principle of explosion, is that I was not attempting to address most of what you're saying there. So if you're wondering why nothing I have said really seems like it addressed it... that's why. :)
Sentences about observations made in a particular reference frame, that do not mention the reference frame, are indexical, and saying they are not really true or false (in the same way that
Fx is not really true or false because x is a free variable) is one possible approach to them... but I am not saying that's necessarily the only approach.
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