2:24 AM
@Zanna I've recently reread our whole logic and set theory conversation, minus whatever parts I inadvertently missed on this rereading, which I think is little, and which hopefully is very little. :)
I've noted some corrections and clarifications to things I have said--mostly clarifications, fortunately, though less fortunately than if everything I said had originally been fully clear. I'm not sure if I should go into them now, or hold off. In particular, so far, I've written 100% of the TeX in this room. So it would feel excessive to write more at this time... even though that's sort of an unreasonable attitude, since it would only be a 0% increase. :)
3 hours later…
6:07 AM
6:20 AM
@EliahKagan apart from the most recent part on 17th January, which for some reason seemed very dense (more accurately I seemed very dense while trying to read it), I have also reread all of this talk about logic and set theory now!
I have not put as much effort as I would have liked to into doing that, but I am accepting that I've put in all the effort that is likely to be available for it
I have some work which is starting properly this weekend coming. From then I will be super busy until 10th February. It's possible I'll start reading the logic book after that
@Kulfy So-called continental breakfast is popular in the UK. What you mention there is something people actually eat and expect to be available for a regular breakfast here
If you sleep in a hotel in the UK, the breakfast options will usually be continental breakfast or full English breakfast.
The breakfast is typically served on a buffet with juices, coffee, tea, croissants, maybe pain au chocolate, bread for toasting, butter, Marmite, jams, processed cereals, muesli, granola, milk, cream, bacon, sausages, hash browns, fried mushrooms, baked beans (haricot beans cooked in a sauce of tomato paste, sugar, salt and cornflour), scrambled or fried eggs, and possibly more obscure items from the "full English" such as black pudding
6:46 AM
I can't really imagine any British people I know getting up and going to a bakery before breakfast to get something fresh. But I have this image of Italian and French and Spanish people going out to bakeries and buying some delicious bread and eating that for breakfast
My understanding is that a small breakfast is eaten on the continent because dinner is the main meal of the day there, but I have no idea if that is actually true. It's just what I've read in cookbooks and travel books and stuff
my parents were working class (I think they have graduated (literally) to being middle class now) and we ate very simple food growing up. I guess everyone feels what they ate growing up is typical of the wider culture they belong to. We would eat sugary processed cereal with milk for breakfast. Very occasionally at the weekend my mum would make some kind of full English breakfast with fried meat things, eggs, fried bread. The main meal at the weekend was lunch, which my parents call "dinner"
The afternoon/evening meal they call "tea". I think this reflects that in their families it was a small meal. Just tea, bread sandwiches or toast, maybe a piece of cake (my mum's mum would serve bread and jam, poached egg or cheese on toast, something like that)
I started working in a health food shop when I was 19 and from then I started eating muesli (a mixture of whole cereal flakes with dry fruits and nuts) or dry fruit and oats porridge (a dish that originated in Scotland), with yogurt and fresh fruit for breakfast. That habit stuck with me. When I lived/travelled in Brazil and when I lived/travelled in China, I maintained it despite the traditional breakfasts in those countries being something else.
I refined my oat/yogurt/nuts/fruit breakfast over many years and I would very rarely eat other things because if I ate something else I would find it unsatisfying. It's only coming to India that has finally enabled me to break the habit of eating almost the same routine breakfast every day
@Kulfy if you feel like elaborating on your thoughts about this, I would be very interested. I'm sorry that I disputed the part about the royal family which was probably discouraging. Even if I happen to be right that it wasn't necessarily a royal priority to expand BE, I assume it may have been a priority anyway...
yesterday I went to the Chennai book fair and I was tempted by some history books, but in the end I only bought some books of short stories, partly because my friend promised to lend me his history books soon
I stopped studying history at the options in the last part of secondary school because I found the history we were studying frustratingly repetitive and boring. We had a peek into the syllabus for the GCSEs (exams at age 16 which we study for for the last two years of secondary school), and we were going to study WWII yet again. I studied geography and religious studies instead (geography and English were my favourite subjects at the time)
1 hour later…
8:30 AM
2 hours later…
10:24 AM
@Zanna Regarding the part from the 17th, can you identify the range of messages you mean? We're in quite different time zones, and the chat transcript that shows dates is in a different time zone from both of us (UTC), and it doesn't show messages about logic and set theory on that day. It may be obvious which messages you mean but for some reason I am not sure about it.
10:42 AM
For easier reading, and also for some conceptual reasons, I've uniformly put three spaces between the column labels, i.e., between successive schemata in the first row of the table. Also, I've aligned every "
T" and "⊥" directly beneath the major connective in the schemata it applies to. You don't have to do these things (though you can).
Much nicer truth tables can be written in TeX and this will work in chat... but I don't think it's easier to do that, and I don't actually know how to line things up the way I want. So I'll keep using plain text for truth tables, but otherwise use TeX.
$\neg (p \wedge q)$ is equivalent to $\neg p \vee \neg q$ because they have the same truth value under all interpretations of their sentence letters $p$ and $q$. Same situation with $\neg (p \vee q)$ and $\neg p \wedge \neg q$. Having the same truth value under all interpretations is what it means for truth-functional schemata to be equivalent.
What I was recommending to do there was to examine the relationship between equivalence of truth-functional schemata and $\leftrightarrow$ (i.e., the biconditional, i.e., "if and only if "). Currently that truth table has columns for $p$, $q$, $\neg (p \wedge q)$, $\neg p \vee \neg q$, $\neg (p \vee q)$, and $\neg p \wedge \neg q$.
So I was recommending to add two additional columns, one for each of those biconditionals, which are: $$\neg (p \wedge q) \leftrightarrow (\neg p \vee \neg q)$$ $$\neg (p \vee q) \leftrightarrow (\neg p \wedge \neg q)$$
@Zanna That part was very dense. It would be better to cover it as a back-and-forth conversation about set-builder notation, in which each of us would probably write terms in various forms of set-builder notation. My hope was that using TeX would facilitate such a conversation. If we do that, then those messages could serve as a summary or restatement of a less dense discussion.
@EliahKagan It seems to me that you did largely understand what was going on with empty conjunctions and empty disjunctions, except that I derailed your understanding there. I've mentioned this before but I'd like to touch on it again.
I think you replied to that message but for some reason I cannot find that reply, and while I recall reading the reply, I don't recall its details, except something about how you thought it might make more sense when you reread it. So I hope it's not too annoying if what I'm saying here doesn't account for some of what you've already told me.
I'm not sure what I was thinking there--or what I thought you were expressing--but indeed, as you said, if there are no things then any existentially quantified sentence (i.e., sentence that starts with an existential quantifier) will be false. This is analogous to how, if there are no things, any universally quantified sentence is true.
Since an empty universe of discourse is a finite universe of discourse, it is in principle possible to express any universally quantified sentence as a conjunction of sentences about each thing that exists, and likewise to express any existentially quantified sentence as a disjunction of sentences about each thing that exists.
One thing that is needed, in order to do so, is a way to name each of the objects in the universe of discourse. But this is trivially satisfied in any empty universe: there are no objects, so we have names for them all. (The theory's signature necessarily has no constants. These constants are sufficient! :) The result of doing so would, for universal quantification, be a conjunction of zero sentences and would, for existential quantification, be a disjunction of zero sentences.
Thus, if we choose to permit empty conjunctions and disjunctions, then their semantics must correspond to that of universal and existential quantification, respectively, over an empty universe of discourse. That is, if we permit the empty conjunction then it is true, and if we permit the empty disjunction then it is false.
That's what I had meant to convey before. Please note, when I said that there was more I could say about empty conjunctions and disjunctions, I wasn't referring to this, but rather to other stuff that I have not said, having to do with practically (and arguably also conceptually) why the connectives $\wedge$ and $\vee$ (as well as $\neg$, $\rightarrow$, and $\leftrightarrow$) are used.
Okay. I'll proceed, unless you'd prefer I not do so, and you can read what I'm saying when convenient.
(I hope also that you will reply about that at some point after you return, though there is no hurry whatsoever.)
I also wished to clarify what I was saying about one theory being an extension of another. The context was that NBG is a conservative extension of ZFC. Furthermore, what I said contained a major misstatement. So I'll correct that and then clarify from there.
@EliahKagan I did not mean to say that a theory $T'$ is an extension of a theory $T$ only if they have the same language and signature, even though that is what I did say. Sorry! That is of course not true. Besides saying the wrong thing in that message, I realize I did not explain this concept well at all, nor to adequate depth.
When $T'$ is an extension (or supertheory) of $T$, the language of $T'$ recognizes everything that the language of $T$ recognizes, and perhaps more.
Likewise, regarding the signature: every constant, function symbol, and predicate of $T$ is a constant, function symbol, and predicate of $T'$ respectively. (Also, their arities--and, if applicable, sorts--are unchanged. All the ways they can be used in $T$ are ways they can be used in $T'$.)
Likewise, regarding what can be proved: every theorem of $T$ (that is, every sentence in the language of $T$ that can be proved in $T$) is a theorem of $T'$.
The converses need not hold, of course; if $T'$ has a richer language, more constants, more function symbols, more predicates, more theorems, or any combination thereof, that doesn't keep it from being an extension of $T$. (So I meant to say something like, "That is, every formula in the language of ZFC is a formula in my system, and likewise the signature of my system may add to the signature of ZFC but does not take anything away.")
(Recall that, by "my system," that was the bobcats example, which turned out to be equivalent to -- and a metaphor for -- NBG.)
When $T'$ is an extension (also called a supertheory) of $T$, we say $T$ is a restriction (also called a subtheory) of $T'$.
An extension $T'$ of $T$ is said to be conservative when every theorem of $T'$ that can be expressed in the language of $T$ (i.e., every sentence that can be proved in $T'$ and that is also a sentence in the language of $T$) is a theorem of $T$. That is, a conservative extension may have theorems that a theory it extends does not have. But if it does, none of those theorems can even be stated in that original theory.
Note that this is a different notion of extension than the notion of it in the axiom of extensionality, i.e., than the notion of the extension of a formula or predicate. The extension of a formula or predicate is the things that satisfy it; regarded singularly, it is the class of those things; in a set theory where it can be proved that there is a set of all and only those things, it is the set of those things.
But the other meaning of an extension of something is a thing that does what it does but may also go beyond it. For example, given appropriate constructions of the integers and reals (that is, constructions where $\mathbb{Z} \subseteq \mathbb{R}$), let's call the less-than relation on the integers $<_\mathbb{Z}$ and the less-than relation on the reals $<_\mathbb{R}$. Then $<_\mathbb{R}$ is an extension of $<_\mathbb{Z}$ and $<_\mathbb{Z}$ is a restriction of $<_\mathbb{R}$.
That is, $<_\mathbb{R}$ is $<_\mathbb{Z}$ extended to the reals, and $<_\mathbb{Z}$ is $<_\mathbb{R}$ restricted to the integers. (Extensions are not in general unique, of course. However, in this case, there's only one $<$ that can be defined on $\mathbb{R}$ with the usual properties.)
It is in this sense, conceptually, that applies when one says that one formal theory is an extension (or restriction) of another.
@EliahKagan I meant to say, "Though strictly speaking you only stated that 0 is a left identity of addition and that 1 is a left identity of multiplication." (This was in reference to how $0 + x = x$ and $1 \times x = x$.)
@EliahKagan On the second line, I meant to say $\forall S\, \forall T\, (T \supsetneq S \leftrightarrow S \subsetneq T)$. That is, the first occurrence of "$\subsetneq$" on that line should instead be "$\supsetneq$".
@EliahKagan I mean that what I had just said was the clarification I had in mind regarding empty conjunctions and disjunctions, not that whatever we do later in that vein will be the clarification.
@EliahKagan In hindsight I think it was not all that helpful for me to talk about extensions of theories at that point, or perhaps at all. However, the two different broad meanings of "extension" is something I should have covered explicitly long before. :)
@EliahKagan All the forms of set-builder notation sort of pop out intuitively, without one even noticing, as one tries to express various sets in set-builder notation. Especially as you've not used TeX much before, you might start by expressing something like "the set of just $a$, $b$, and $c$" in the set list notation (i.e., a comma-separated listed enclosed in curly braces).
12:25 PM
12:31 PM
@EliahKagan A schema that is true under all interpretations (i.e., always satisfied) is said to be valid or a tautology. A schema that is false under all interpretations (i.e., never satisfied) is said to be inconsistent or a contradiction. Schemata that are neither valid nor inconsistent are said to be contingent. When a schema is not valid, it is said to be non-valid or invalid.
For example, the schema $p$ is not valid. That doesn't mean there's anything wrong with it; it does mean that it is a mistake to assume that a sentence is true merely because it can be written in that form. In contrast, sentences that can be written in the form of a valid schema are all true, and true by virtue of their form.
Even this is not really praise of those sentences; those sentences, effectively, because they are logical truths, do not actually convey any information (or, if you prefer: they do not convey any information other than the information that can be inferred from the meaning of their truth-functional sentential connectives).
@Zanna You may be able to get them to render on your phone. I don't know if Firefox for Android supports any extensions that facilitate the user of user scripts. MathJax does in general work in Firefox on Android, at least for me. I recall my phone is significantly older and worse than yours. MathJax in Firefox on my phone often impacts performance quite badly, but I wouldn't make a prediction for your phone based on that.
@EliahKagan Also, you can right-click on rendered TeX and click Show Math As and then click TeX Commands.
But what might be most useful is to open the chat in a private browsing window, in addition to having it open in a regular window.
By default, extensions are disabled in private browsing mode, so your user script manager is disabled by default in private browsing mode, so ChatJax++ doesn't run, so TeX doesn't render in chat.
I should perhaps mention that, while I am in the habit of using
$\neg$, $\wedge$, and $\vee$ for "$\neg$", "$\wedge$", and "$\vee$", you may prefer the commands $\lnot$, $\land$, and $\lor$, which also render to "$\lnot$", "$\land$", and "$\lor$".
@Zanna In this same sense of "extension" one might say that when $S \subseteq T$ (i.e., when $S$ is a subset of $T$), $S$ is a restriction of $T$ and $T$ is an extension of $S$. However, this is best avoided, since it invites confusion between the two meanings of "extension."
But note that, because we construct relations as their graphs and even tend to think of them as their graphs whenever convenient (in contrast, say, to real numbers, which we construct as sets, but which we don't tend to think of as any particular set), the way you would express that $<_\mathbb{R}$ is an extension of $<_\mathbb{Z}$ is by saying $<_\mathbb{Z} \subseteq <_\mathbb{R}$.
12:50 PM
When translating sentences from a natural language like English into a formal language of some logic or other, there are often choices to be made. You might translate "Jones is ill or Smith is away" into first-order logic as a disjunction of atomic sentences; you might write it as $Ij \vee As$ or $\mathrm{Ill}(\mathrm{jones}) \vee \mathrm{Away}(\mathrm{smith})$. Or if what you need to do does not require that level of granularity then you might just translate it truth-functionally as $p \vee q$.
@EliahKagan Likewise you might need less granularity than any of those ways, and translate the whole thing as just $p$. One reason you might do that is if "Jones is ill or Smith is away" always appears by itself everywhere you're interested in it. Then it makes sense to translate it opaquely as just $p$.
@Zanna I was going to suggest, and had even begun to suggest, that you write some things in set notation (using TeX). Since you are low on energy today, though, it occurs to me that you may not wish to do that today, even if you are interested eventually to do it.
@Zanna From that description it sounds like the issue is not yet being comfortable with TeX and that the solution may be to start small and build up. As you now seem to be doing... please proceed...
@EliahKagan For example, suppose you know that if Jones is ill or Smith is away, then Cassidy will be called in. Furthermore, you know Jones is ill or Smith is away. You wish to infer that Cassidy will be called in. If you represent "Jones is ill" as $p$, "Smith is away" as $q$, and Cassidy will be called in as $s$, you can write your argument as: $$(p \vee q) \rightarrow r$$ $$p \vee q$$ $$\therefore r$$ (The symbol "$therefore$" means "therefore.")
@EliahKagan (Those should be aligned on the left, with "$r$" starting in the same place as the first symbols of the preceding lines and thus with "$\therefore$" to the left of them... but I have not managed that.)
1:21 PM
I am pretending that I wrote $therefore$ instead of $\therefore$ on purpose to demonstrate that it is no big deal to write wrong TeX in chat and then not manage to fix it in time, and that I hope you do not regard that as something to be feared. Of course, I did not do it on purpose and I am somewhat frustrated by this error of mine, but I am still pretending that I did it to demonstrate this soothing truth, which I do believe it demonstrates.
1:27 PM
1:40 PM
Yes. Except that an argument cannot be true or false. A argument is valid if its premises follow from its conclusion, and otherwise invalid. An argument that is valid and whose premises are all true is said to be sound. The conclusion of a sound argument is always true, but not all arguments with true conclusions are sound; not all arguments with true conclusions are even valid.
I should also say that I am talking about deductive arguments here. Deductive arguments are not the only arguments that are reasonable to accept; it is also often reasonable to accept inductive arguments, which are based on empirical evidence. An inductive argument that ought to be believed because the premises provide good evidence for the conclusion is said to be cogent.
Thus validity in deductive reasoning takes a place analogous to the place taken by cogency in inductive reasoning. In a valid argument, the premises are said to entail the conclusion.
You may already be quite familiar with all this; if so, I hope it is not too annoying. But I think it is important to state it explicitly.
@Zanna However, I think you have actually noticed something deep and important: given any deductive argument, it is possible to write a compound sentence that is (a) valid if and only if the argument is valid and (b) of a form that directly resembles the argument, so that either may be mechanically translated into the other. This is the connection between the meaning of validity as it applies to deductive arguments and the meaning of validity as it applies to sentences and schemata.
Oh. Cool. I am very glad I am mentioning it then. This is both important in its own right and deeply related to why, in practice, we use the particular truth-functional sentential connectives we use.
@EliahKagan This is the argument's corresponding conditional (so called because it is a conditional and it corresponds to the argument). Recall that in a conditional $p \rightarrow q$, we call $p$ the antecedent and $q$ the consequent. A deductive argument's corresponding conditional has, as its antecedent, the conjunction of all the argument's premises, and has, as its consequent, the conclusion of the argument.
Cool. You will notice that validity of an argument is invariant under the operations of replacing any number of premises by their conjunction and of replacing any premises that are conjunctions with their conjuncts. In particular, the immediately preceding argument is valid if and only if this argument is valid: $$S_1 \wedge S_2 \wedge \ldots \wedge S_n$$ $$\therefore T$$
Yes. When an argument is valid we say we are permitted to infer its conclusion from its premises. Such an inference is called a valid inference. (If you formalize logic via formal rules of inference, all the inferences they bless had better be valid inferences, or your rules of inference are... not good.) You can also have a situation where you can infer a premise from the other premises. When that happens, the validity of the original argument is invariant under removal of that premise.
Often, though not always, there is more than once choice of premise to remove. As a really simple example, note first that this argument is valid: $$p$$ $$\therefore p$$ So this argument is also valid: $$p$$ $$p$$ $$\therefore p$$ You can remove either premise $p$.
As a less trivial but still simple example, consider first this valid argument, which does not have any premise that may be removed: $$p$$ $$q$$ $$\therefore p \wedge q$$ But now consider this other, also valid argument, in which either premise may be removed: $$p$$ $$q$$ $$\therefore p \vee q$$ Although you may remove either the premise $p$ or the premise $q$ from that argument, you may not remove both. The argument is no longer redundant once one of them is removed.
2:24 PM
Note that, while premises that can be inferred from other premises may be removed, and there are various other situations where premises may be removed, premises from which other premises can be inferred cannot be removed... or in the particular cases when it turns out they can, that's not why. $$p$$ $$p \vee q$$ $$\therefore p$$ From $p$ we can infer $p \vee q$, but $p \vee q$ is strictly weaker than $p$: we cannot infer $p$ from $p \vee q$ (since it might be that $q$ is true and $p$ is not).
When no premise follows from (i.e., can be inferred from) the others, the premises are said to be independent. This terminology applies to sentences in general, i.e., a collection of sentences is said to be independent when none of the sentences can be inferred from the others. In particular, when developing an axiomatic system, it is often considered desirable, and related to the goal of elegance, to choose independent axioms.
If you have an axiomatic system $\mathcal{S}$ whose axioms are not independent, then one of them may be inferred from the others; if you remove that axiom to produce the other system $\mathcal{S'}$, then $\mathcal{S}$ and $\mathcal{S'}$ have all the same theorems; i.e., every sentence that can be proved in one can be proved in the other.
(Whether or not $\mathcal{S'}$ is really a different system from $\mathcal{S}$ depends on how you formalize the notion of a system. In general, I've use "theory" formally and "system" informally or, in this case, semi-formally informally with symbolic notation. In case you're wondering why I sometimes say "system" and sometimes say "theory," this is why. Although this pattern of use isn't original to me, it is in no way established that one must to use those words with that distinction.)
When it is deductively valid to infer a particular conclusion from particular premises -- that is, when there is no way for the premises to be true and the conclusion false, and this relationship between the premises and the conclusion is identifiable from the form that the premises and conclusion take, so that the argument with those premises and that conclusion is a valid argument -- we say the premises imply the conclusion.
When you have more than once premise, you can express them as a conjunction, as shown above, so for every valid argument there are corresponding sentences $S$ and $T$ such that $S$ implies $T$. That is, if an argument is valid and concludes $T$ from $S_1, S_2, \ldots, S_n$, then for $S$ we can put $S_1 \wedge S_2 \wedge \ldots \wedge S_n$. The sentence $S_1 \wedge S_2 \wedge \ldots \wedge S_n$ implies the sentence $T$.
1 hour later…
4:02 PM
Implication is often simply defined as validity of the conditional. By this, I mean that implication in the sense of a sentence implying a sentence, or a schema implying a schema, is often defined that way.
Note that implication is not the conditional. Given sentences $S$ and $T$, the sentence $S \rightarrow T$ is true when $S$ is false or $T$ is true. That does not mean $S$ implies $T$. This is related to how an argument with a false premise (and conclusion of any true value) or a true conclusion (and premises of any true values) need not be valid, though it could be.
It might be bad that I am using $T$ to represent a sentence, considering that the symbol $T$ is also used to represent truth! However, I have TeX, so to represent truth and falsity, I'll use $\top$ and $\bot$.
@EliahKagan An argument's corresponding conditional may be true without the argument being valid. It is specifically when an argument's corresponding conditional is valid (in the sense of validity that applies to sentences) that the argument is valid (in the sense of validity that applies to arguments).
However, as is the case with many different words and phrases, the words "implication" and "implies" are used in different ways by different people, and not just in these very closely and thus non-confusingly related ways (applying to arguments, sentences, or schemata) detailed above. Some people refer to the conditional as implication, and even say "implies" in place of "only if."
I recommend against such usage. ("I showered late this morning, which implies Pluto is no longer considered a planet.") It is fairly popular among mathematicians, but it seems to have originated in older (i.e., very early 20th century and earlier) works in modern logic. Some contemporary logicians use it this way when they're talking to mathematicians.
Because "implies" is used in at least two radically different ways (to mean validity of the conditional and closely related ideas, as expounded on above, and to mean "only if"), it is useful to be able to distinguish between them. Plus, mathematicians need some economical and unambiguous way to indicate validity of the conditional. So implication, in the sense of validity of the conditional, is also called logical implication, and the conditional is also called material implication.
(Given any two statements, at least one of them materially implies the other. But it is quite possible and in fact typical that neither logically implies the other.)
@EliahKagan I of course meant "any truth value" there, both times I said "any true value." That is, what I meant to say is better expressed this way:
> Note that implication is not the conditional. Given sentences $S$ and $T$, the sentence $S \rightarrow T$ is true when $S$ is false or $T$ is true. That does not mean $S$ implies $T$. This is related to how an argument with a false premise (and conclusion that is either true or false) or a true conclusion (and premises consisting of any combination of truths and falsities) need not be valid, though it could be.
Recall that the distinction between a sentence and a schema is that non-logical symbols in a schema are uninterpreted, so that the schema represents many different possible sentences. Infinitely many, actually. A schema may be regarded as a pattern that matches sentences that take its form, or as a rule for generating sentences that take its form, or even as a form sentences may take.
The sentences that are of that form--so that one might say the schema matches or generates them--are called the schema's instances. The plural of "schema" is "schemata." Some people call schemata schemes (and thus call a schema a scheme).
There is more than one kind of schema in logic, but the truth-functional kind is the kind I've covered so far. Truth-functional schemata consist of uninterpreted sentence letters connected with zero or more truth-functional sentential connectives (and may have parentheses for grouping). I say "zero or more" because a schema may consist of a single sentence letter, e.g. $p$ is a schema when $p$ is an uninterpreted sentence letter, and such a schema has no connectives.
An interpretation of a truth-functional schema is an assignment of actual sentences to each of its sentence letters, where the same sentence letter is always given the same meaning. For example, in an interpretation of $p \wedge (p \rightarrow q)$, each occurrence of $p$ must be taken as the same sentence. The sentence $q$ is taken as may be the same or different from the sentence $p$ is taken as.
Of course, this requirement applies separately for each interpretation. In one interpretation of that schema, $p$ may be taken to mean "Smith is away." In another, it may be taken to mean "Jones is ill." That's fine. But we cannot simultaneously taken one occurrence of $p$ to mean "Smith is away," and the other occurrence of $p$ to mean "Jones is ill."
Strictly speaking, that's what an interpretation of a truth-functional schema is. But when we speak of an interpretation of a truth-functional schema, we sometimes mean something different: an ascription of truth values to each sentence letter in the schema, obeying the rule that different truth values may not be given to occurrences of the same sentence letter at the same time.
Note that claims that use phrases like "under all interpretations" should only be asserted when they work for both meanings of "interpretation," and this is typically easy to achieve even without trying, which may shed some light on why this ambiguity of meaning is widely regarded to be acceptable.
In particular, note that for a schema to be valid is for it to be true under all interpretations (as stated above) and that, whichever meaning of "interpretation" you use, the notion of the validity of a truth-functional schema has the same meaning (in the sense that all the same schemata are valid and invalid, regardless of which meaning of "interpretation" you use).
You will notice the close relationship between truth tables and and the all combinations of truth values for sentence letters sense of "all interpretations." This is why you can tell if a truth-functional schema is valid by constructing a truth table. If the column for the schema has all $\top$s, the schema is valid. Otherwise (i.e., if it has one or more $\bot$s), the schema is not valid.
A sentence is said to be valid when it is true, and true by virtue of its logical structure. Validity of a sentence is thus analogous to validity of an argument, which makes sense, since with an actual argument with actual sentences as premises and an actual sentence as a conclusion, the argument's corresponding conditional is an actual sentence (not a schema), and the argument is valid if and only if that corresponding conditional is valid.
Another way to put it is that the valid sentences are those that take the form of some valid schema. That is, a sentence is valid if and only if there is some valid schema whose form it takes (i.e., that it matches, that generates it).
Note, however, that any sentence (I am talking about sentences that can be construed to have truth value, i.e., statements), regardless of its complexity, can be denoted merely with a single sentence letter. So every sentence takes the form of the truth-functional schema $p$, whatever other forms (if any) it takes. Yet the schema $p$ is not valid. Nonetheless some sentences are valid.
What's going on is that valid sentences are valid, but not by virtue of being able to be expressed as a single sentence letter. Rather, they're valid by virtue of being able to be expressed in some other form, i.e., by virtue of matching some other schema, one that is valid. Every valid sentence matches a valid schema, but it may (and in fact will always) match other schemata as well, some of which are not valid.
« first day (876 days earlier) ← previous day next day → last day (1523 days later) »
