BrianO

In its role as the square root of $-1$, $i$ is not best thought of as a direction.

The axioms are tautologies, and the rules of deduction when applied to tautologies yield only tautologies.

Bourbaki's set theory is roughly ZC without Foundation. That is, it's ZFC, minus Foundation, minus Replacement (the 'F' in ZFC, for Frankel), and with a global choice principle (their $\tau$ operator, which was introduced in Hilbert & Ackermann as the $\sigma$ ('selector') operator). Their official 2-dimensional syntax is unique to their presentation - nobody else can take credit or blame. But don't let any of this stop you from checking out their volumes if your goal is to see mathematics presented from the ground-up starting from modern set theoretic foundations.

Principia Mathematica is way too old and idiosyncratic to be of use. It's not ZF or ZFC. Reading its notation in 2000+ is like trying to read Chaucer or Old French. It's encumbered by concerns over predicativity, leading to heirarchies within the type heirarchy, which then are flattened with an arbitrary axiom required to get any real math done. Russell & Whitehead are not as fastidious as the logicians of the next generation or two -- mathematicians, not philosophers -- who cleaned things up considerably.

The most ambiguous term that has come up in the discussion is "predicate". Not "predicate symbol", which clearly means an element of a 1st order language, but "predicate". Over the decades (more than 10 of them), various authors have used it in various ways. Some have used it to refer syntactic things; others have used it to refer to the possibly intensional denotations of syntactic things. IIRC Montague's writings on semantics of natural language are examples of the latter.

Hi. I finally finished other things. Not much to add, really: My working definitions:: the interpretation of an n-ary predicate symbol in a model with a universe U is an n-ary relation on U, meaning, a subset of $U^n$. Whether one construes $U^n$ to be iterated ordered pairs, or functions $n \to U$, is an implementation detail that shouldn't be of great concern.

Again, I never used the word "collection". But no matter. Thanks for the un-downvote.

Overly casual usage. Please, I've had more than enough of this pedantry and going on about an answer that doesn't exist anymore. You can have the last word if you want it, I will not reply.

Oh, that's what's still rubbing -- not enough that it's preceded by "(the interpretations of)", huh. OK I'll remove the parens.... There, I removed them. Happy now?

You mean the "error" resulting from answering the original question? Do you see the word "collection" anywhere in my answer? It seems you're using "relation" to mean part of the formal language, whereas most mathematicians use it to mean an extensional thing, a set of ordered pairs in the typical case, or a set of n-tuples as in "n-ary relation". Such a thing can be an interpretation of a predicate symbol, or the set of all n-tuples of a model that satisfy a formula. ¶ Finally, I don't need a lecture about the purpose and value of formalization, thank you.

There is no such identification, or casual abuse of terminology, in the answer as it now stands. It does still answer the original question, which I'll now fix.

Well, thanks for the pedantry. Obviously I meant "the interpretation $P^M$ of $P$ in such a model $M$" and so on. But I emended the language to be more precise.

Yes of course "slides" not "paper". Ok, I'll write it up. Thanks!

If not, I can talk around it

Can I cite the paper you linked to?

So you don't have to add anything to guarantee that the 3 distinctly named pets are in fact distinct — you get that "for free".

That is, distinct names are assumed to denote distinct individuals, in every interpretation.

Lol, OK let me examine that...

Very well

I'm almost embarrassed to do so: it seems I have questions but no answers! And I don't know the notations involved (ABox, TBox, things that look like first order logic, from a distance, but aren't, etc.) When I search ALCQ, I don't find much, and what little I do find is behind paywalls. You only have hardcopy (paper), not a link?

I have no answer beyond that. You can't determine whether identity is allowed in the KB system you're using???

I'm just guessing. You, however, know something about ALCQ KBs... The question must have an easy answer.

Can you use identity? And thus add e.g. "TSIPRAS $\ne$ VAROUFAKIS"?

You're welcome & thank you too. Email me if you don't hear back in a reasonable interval. & you have a good night too.

I'm going to call it a night now, it's been a very long day. You're Julian, I take it - good to meet you. (I'm Brian but you probably guessed that ;/ )

Thank you :)

Hmmm let me ponder where you might take it from here. I haven't given out my email via stackexchange yet but ... what the hey: [email protected]

"I to" ? (Assuming I know what you mean, yes)

Doesn't that do the trick?

^^^ is a restatement of Higman's lemma.

Well now, there's that universal characterization of $(A^*, \ll)$. Does this make that structure "the free pomonoid on A"? Looks like that is so. Happily. Thus: if A is wqo then so is the free pomonoid on A.

That the induced monoid morphism A* -> B is also order-preserving :) I meant ≤ to be the order on B.

Thus f(u_1 ... u_m) = f(u'_1 ... u'_n) = f(u'_1) ... f(u'_n) ≤ f(v_1) ... f(v_n) = f(v_1 ... v_n)

So if f: A \to B is order-preserving, then f(e) = e_B, and for all j = 1, ... n, we have f(u'_j) ≤ f(v_j)

Then m <= n, and we can pad the finite sequence of u_i with e to get a finite seq u'_j of length n, such that u_1 ... u_m = u'_1 ... u'_n, and u'_j \preceq v_j for all j =1, ... n

Suppose u_1 ... u_m \ll v_1 ... v_n.

Right (I'll drop LaTeX/Mathjax niceties here) Thinking aloud again but I think it's true...

Does that unique monoid morphism also preserve the ordering $\ll$ ? Yes:

Any function $A\to B$ lifts to a unique monoid morphism $A^* \to B$, as A* is the free monoid on A.

(as part of the definition of 'pomonoid', I'm guessing)

The monoid op on B respects the order on B, right?

Uh huh that starts to go somewhere I think.

So that may be relevant, useful

Note that the monoid operation is compatible with the ordering $\ll$: for q, r, s, t in A*, if $q \ll r$ and $s \ll t$ then $q s \ll r t$.

Not sure yet

Welcome back

In any case this doesn't use anything about the order $\ll$ on A*. The same can be said about many order-preserving maps (functors) $A \to B$, $B$ an arbitrary preorder... and there are plenty of non-wqo $B$ that a wqo $A$ can be mapped into order-preservingly..... in particular, if $A$ is a singleton :)

No?

For u, v in A, F(u) and F(v) are both of length 1 in A*, so $F(u) \ll F(v)$ iff $u \preceq v$. But as categories, both A and A* are induced by the preorders, so the Hom sets have one morphism or are empty. Thus if $u\preceq v$ then both Hom_A(u,v) and Hom_{A*}(F(u), F(v)) have one morphism.