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1:00 PM
@Zanna That starts tomorrow?
 
1:26 PM
@Zanna Another way say what validity means is this:
A valid argument is one in which the conclusion follows from the premises, without requiring any additional knowledge. That is, nothing other than the information in the premises is needed in order to infer the conclusion deductively.
A valid sentence is a sentence whose truth is guaranteed, without requiring any additional knowledge. Just as one can verify the validity of an argument by inspection, so too can one verify the validity of a valid sentence by inspection. One need not go out into the world to see whether or not the sentence's claims correspond to facts of the world.
A valid schema is a pattern for valid sentences. A valid schema need not match (and never does match) every valid sentence, but every sentence that a valid schema does match is valid, and its validity an be discerned merely by noticing that the schema matches it.
(It is also meaningful to say of an open formula--that is, a formula that has one or more free variables and that is therefore not a closed sentence--that it is valid. But I haven't covered that. Once the other meanings of validity are clear, IMO that one becomes intuitive. I don't mean that one need not define what it means for a formula to be valid, merely that thus far that has been a deliberate omission.)
The problem with those characterizations of validity is that, of course, one does have to know things about the world to know if an argument, sentence, or schema is valid. One has to know enough about its meaning to know its logical structure and the meaning of logical words and phrases like "not," "only if," and "there exists." (Or one could enumerate all the valid forms one recognizes and check it against them. I actually think that does constitute knowing something about its meaning.)
Still, I think those characterizations of validity of arguments, sentences, and schemata are, while not answering all important philosophical questions, both correct and intuitively useful.
@EliahKagan (And knowing the meaning of anything in a sentence, even logical words and phrases, comes from knowledge about the world.)
 
 
3 hours later…
4:18 PM
@EliahKagan it actually started today
@EliahKagan looks so nice!
I haven't read anything here yet today
 
5:10 PM
@EliahKagan Sorry, that "can verify by inspection" claim is not accurate. In my effort to write something one could sink one's teeth into, I wrote something that is false (and disastrously, obviously so, in the sense that its falsehood is a major result in formal logic that I am acquainted with...).
In full generality, validity is not so easy to determine. I'm going afk for a while but later I'll try to replace it with something of similar intuitive value but that is not incorrect. For now, though, I'll note that it is accurate for the valid forms I have shown thus far as examples, which are truth-functional validity and monadic quantificational validity. (Monadic quantification theory is quantification theory limited to monadic predicates, i.e., unary predicates, predicates of arity $1$.)
@Zanna ^^ That might be good, since one of the things I said today was horrendously wrong.
@EliahKagan I mean, it is accurate for the valid forms I have shown thus far, which are forms of truth-functional validity and monadic quantificational validity. All other truth-functional and monadic quantificational validities are also possible to determine by inspection.
 
5:40 PM
@EliahKagan :D
 
6:06 PM
@Zanna I'm back, for a bit anyway. The issue is in the meaning of "by inspection." This phrase means, "You can efficiently look and see whether it is the case," and that's not in general true of validity.
What I should say instead is that, if something is valid, then you can eventually discover that it is valid, and that no outside information (besides the meaning of logical words and phrases) is needed to do so.
So a valid argument is one where, based only on deductive logic and the information contained in the premises (if any), one can eventually satisfy oneself that if the premises are true then the conclusion is true. And a valid sentence is one where, based only on deductive logic, one can eventually satisfy oneself that it is true. That is, you can prove it without premises, so even an argument with no premises that concludes it is valid.
 
I think it is much less obvious than it might appear. I kind of see why you were talking about the meaning of words.
@EliahKagan since I'm vegan my teeth don't expect much and are easily satisfied
Wow I'm talking super unclearly
I'll go to sleep
 
@Zanna What you're saying seems clear.
@Zanna I think you're responding to "obviously so" in that message.
In polyadic quantification theory (polyadic means predicates of of $> 1$ are permitted), which is also called just quantification theory, there are proof procedures for validity and inconsistency. Given anything valid, there is a procedure guaranteed to eventually find a proof of it. Given anything invalid, there are procedures that guaranteed to eventually find a proof of its negation.
(Note that any proof procedure for validity can be converted into a proof procedure for inconsistency, and vice versa, since a sentence is valid iff its negation is inconsistent, and a sentence is inconsistent iff its negation is valid.)
However, there is no decision procedure for validity in polyadic quantification. There are ways of trying to prove validity that never give incorrect results, and that, if what you're trying to prove the validity of is valid, will always eventually finish.
In contrast, a decision procedure would finish and tell you either way whether or not something is valid. There are such procedures for truth-functional logic (truth tables are one, though there are more efficient ones) and monadic quantification theory. There is no such procedure for polyadic quantification theory.
Note that invalidity should not be confused with inconsistency. Valid means "true under all interpretations." Inconsistent means "false under all interpretations." Consistency just means "true under some interpretation."
What I mean when I say that my false claim, involving a wrong and unfortunate use of the term "by inspection," was obviously wrong, I mean something like how it is obviously wrong if someone points to a box and says, "This box is a closed physical system in which total energy is not conserved." It's not that the law of conservation of energy is obvious, but rather that the claim is obviously wrong in light of that law.
 
6:56 PM
Alternatively, if I misunderstood you and you meant that the exact meaning of validity is less obvious that it might appear, then I agree. I did not mean to suggest that precisely what validity does mean is obvious. On the one hand, I don't want to discourage you by overcomplicating validity--in actual use, the concept is typically clear enough, and I suspect you either understand it or almost do.
Furthermore, what is valid in a particular formal logic is clear-cut: it is those things whose validity that logic can ascertain, in accordance with formal rules. There are various systems of rules for determining validity in standard logic (some of which, in modified form, also have uses in other logics).
The method I showed above in my examples--more precisely, I showed a partially informal analogue of it and did so incompletely--is natural deduction. I am referring to what I said around there. I used and cited some rules of natural deduction, such as universal instantiation and disjunctive syllogism.
However, figuring out exactly what, in general, comprises validity, is a very deep question and a matter of considerable philosophical dispute, at least if one regards the validity, in the fully general sense, to mean logical truth. (I do regard validity, in the fully general sense meaningful outside any specific formal system, to mean logical truth. Identifying validity with logical truth is fairly common. But I don't think everyone does.)
The above messages would be more useful if they had more links, but I didn't want to include links (other than to earlier chat messages for context) except when fairly confident the linked resource would be useful--and I'm going afk again soon.
Anyway, good night!
@EliahKagan Anyway, what I said in those messages other than the use of the phrase "by inspection," is correct (unless I've missed other errors...). And I think what I said there, other than my use of "by inspection," is still useful. Validity can be verified, and can be verified procedurally, but this is not always immediate. And there is no sure-fire procedure for verifying invalidity.
@EliahKagan * at least if one regards validity (I did not mean to say "the")
 

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