Re: [ontolog-forum] A Question About Logic

[Thomas Johnston <tmj44p@xxxxxxx>]

Pat,

This addresses just one point you made in your lengthy response. (BTW, I am studying the Ontological Commitment article in SEP. It is, unsurprisingly, excellent.)

10/16/2015.

Pat asked me what a predicate logic that I liked would look like. Well, I can't spell it out (and if I did, I might find myself wandering down dead-end alleys), but here are two things I'd like to see included (or to understand why including them wouldn't work):

First, as I suggested yesterday, why not distinguish predicates in predicate logic as (i) predicates identifying the type that the particular(s) in the subject position of the statements belong to (if any such should exist), and (ii) predicates which ascribe properties or relationships to those particulars.

Second, why not separate the "All" and "Some" distinction from the "exists" attribution. The first distinction would qualify statements with "All" ("A") or "Some" (S"); the second would qualify statements with "Exists" (E), "Does not exist" (~E)" or "May or may not exist" (?E).

Here are eight examples. Each one has, in order, (i) the natural language statement to be expressed formally; (ii) the formal _expression_; and (iii) the literal translation from the formal _expression_ back into natural language.

Using my eight statements as examples (extension to Pat's examples should be straightforward):

(i) "There are dogs." ExSx(D_tx). "There exists an x whose type is Dog". (I think that ExAx(D_tx) is equivalent, and that this supports my contention that if there are some dogs, there are also all dogs.)

(ii) "There aren't any dogs." ~ExAx(D_tx). "There exists no x whose type is Dog". (I think that ~ExSx(D_tx) is equivalent, and that this supports my contention that if there are all dogs, there are also some dogs.)

(iii) "If there are any dogs, some of them will be friendly." ?ExSx(D_tx & F_px). "There may or may not exist xs whose type is Dog, and if any such exist, some will have the property of being friendly."

(iv) "If there are any dogs, all of them will be renates". ?ExAx(D_tx & R_px). "There may or may not exist xs whose type is Dog, and if any such exist, all of them will have the property of being a renate."

(v) "There are dogs, and some of them are friendly." ExSx(D_tx & F_px). "There are xs whose type is Dog, and some of them have the property of being friendly."

(vi) "There are dogs, and all of them are renates." ExAx(D_tx & R_px) There are xs whose type is Dog, and all of them have the property of being renates."

(vii) "If there are any dogs, some of them will not be black". ?ExSx(D_tx & ~B_px). "There may or may not exist xs whose type is Dog, and if any such exist, some of them will have the property of not being black."

(viii) "If there are any dogs, none of them will be made of gold". ?ExAx(Dtx & ~G_px). "There may or may not exist xs whose type is Dog, and if any such exist, all of them will have the property of not being made of gold."

Of course, this is just the beginning of sketching out a predicate logic with these features. One thing to note is that it will require a three-valued logic. Another thing to note is that rules of inference have not been stated, nor deMorgan-like equivalences. But I still believe that this approach allows us to express natural language statements in a formalism, without forcing us to say things we don't want to say.

On Tuesday, October 13, 2015 9:04 PM, Pat Hayes <phayes@xxxxxxx> wrote:

On Oct 13, 2015, at 10:53 AM, Thomas Johnston <tmj44p@xxxxxxx> wrote:

> Oct 13, 2017.
>
> My intuitions tell me that anyone who asserts "All dogs are renates" believes that there are dogs (i.e. is ontologically committed to the existence of dogs) just as much as someone who asserts "Some dogs are friendly".

I think your intuition needs some pumping with more examples. Try these:

All unicorns are imaginary.
All large unicorns are male.
All composite prime numbers are larger than 13.

Your intuition has some very strange consequences. You claim that  (Ux Px -> Qx)  entails (Ex Px). But the first is equivalent to (Ux. ~Qx -> ~Px), so it must also entail (Ex ~Qx) So "All dogs are renates" entails the existence of some non-renates. But if so, then "All dogs are real entities" (or some other vacuously true property in place of Q), must entail that some non-real entities exist, which is false. So we can derive a falsehood (perhaps necessarily false) from a truth (perhaps a necessary truth), using your intuitions as a guide. 

> Suppose someone else asserts, instead, that "No dogs are renates". Certainly, to do that, that person must believe that there are such things as dogs

?? Of course not.  For example:  "No blue rainbows have gold highlights."  Which, by the way, I believe and am quite sure is true, precisely *because* there are no blue rainbows. Or, a more realistic example, one that actually does arise in some conversations here in the deep south, "No spirit guide will cause you harm."

> and, in addition, believe that some of them are not renates (a false belief, of course).
>
> Now for "Some dogs are friendly", and also "Some dogs are not friendly". In both cases, we all seem to agree, someone making those assertions believes that there are dogs.
>
> Now I'm quite happy about all this. If I make a Gricean-rule serious assertion by using either the "All" quantification or the "Some" quantification, I'm talking about whatever is the subject term in those quantifications – dogs in this case. I'm particularly happy that negation, as it appears in the deMorgan's translations between "All" statements and "Some" statements, doesn't claim that a pair of statements are semantically equivalent, in which one of the pair expresses a belief that dogs exist but the other does not.
>
> But in the standard interpretation of predicate logic, that is the interpretation. In the standard interpretation, negating a statement creates or removes the _expression_ of a belief that something exists.

Well, negating the statement expressing that belief yields another statement denying that belief. Why would anyone expect otherwise? Surely that is the whole point of negation, that not-P expresses the exact opposite of what is expressed by P, so they cannot both be true.

> My beliefs in what exist can't be changed by the use of the negation operator.

So if you say "Foos exist", and I respond, disagreeing with you, "There are no foos", then we are in fact agreeing with one another! Do you really find this a reasonable interpretation?  If you do, then what could I possibly say, in order to disagree with you about such a claim of existence?

>  Apparently, John's beliefs can, and so too for everyone else who feels comfortable with predicate logic as a formalization of commonsense reasoning, and with the interpretation of one of its operators as "There exists ....".
>
> I usually don't like getting into tit for tats. Those kinds of discussions always are about trees, and take attention away from the forest. But I'll make exceptions when I think it's worth taking that risk (as I did in my response to Ed last night).
>
> So:
>
> From John Sowa's Oct 12th response:
> <<<
> TJ
> > why, in the formalization of predicate logic, was it decided
> > that "Some X" would carry ontological commitment
>
> Nobody made that decision.  It's a fact of perception.

I wish John had not mentioned perception here, as it muddies the discussion with irrelevant ideas. It is not a fact of perception. but a fact of the truth-conditions of the sentences involved. You may cite medieval scholars as much as you like, but I would be more impressed by a brief account of what you take the truth-conditions of a sentence of the form (Ux Px -> Qx) to actually be, and show us how this will entail Ex Px.

>  Every
> observation can always be described with just two operators:
> existential quantifier and conjunction. No other operators can
> be observed. They can only be inferred.
> >>>
> (1) If all ontological commitments have to be based on direct observation, then we're right back to the Vienna Circle and A. J. Ayer.
>
> (2) And what is it that we directly observe? A dog in front of me? Dogs, as Quine once pointed out, are ontological posits on a par with the Greek gods, or with disease-causing demons. (I am aware that this point, in particular, will likely serve to reinforce the belief, on the part of many engineering types in this forum, that philosophy has nothing to do with ontology engineering. That's something I want to discuss in a "contextualizing discussion" I want to have before I pester the members of this forum with questions and hypotheses about cognitive/diachronic semantics. What does talk like that have to do with building real-world ontologies in ontology tools, in OWL/RDF – ontologies that actually do something useful in the world?
>
> (3) I wouldn't talk about some dogs unless I believed that some dogs exist.

Suppose you believe that dogs do not exist, ie that there are no dogs. And you wish to say this to someone, perhaps to enlighten them about the true nature of their animal pet. What will you say? You have to say something like "There are no dogs", or "Nothing is a dog". Rendered into logic, you have to say Ux.~Dx  or ~Ex.Dx. Either way, you have to talk about dogs, in order to deny their existence. So someone might well want to talk about dogs - perhaps it would be better to say, to use dog language - when they do not believe that dogs exist.

> And if some dogs exist, then all dogs do, too.

Clearly this is false. I had a dog once, called Sally. Sally existed. On the other hand, the Hound of the Baskervilles was an imaginary dog. That dog did not exist. Similarly Rin-Tin-Tin was a fictional dog that did not exist.

> Either there are dogs, or there aren't. If there are, then I can talk about some of them, or about all of them. If there aren't, then unless I am explicitly talking about non-existent things, I can't talk about some of them nor can I talk about all of them, for the simple reason that none of them exist. To repeat myself: if any of them exist, then all of them do.
>
> (4) And I am, of course, completely aware that trained logicians since Frege have been using predicate logic, and that, at least since deMorgan, have been importing to negation the power to create and remove ontological commitment.
>
> (5) Here's a quote from Paul Vincent Spade (very important guy in medieval logic and semantics):
>
> "This doctrine of “existential import” has taken a lot of silly abuse in the twentieth century. As you may know, the modern reading of universal affirmatives construes them as quantified material conditionals. Thus ‘Every S is P’ becomes (x)(Sx ⊃ Px), and is true, not false, if there are no S’s. Hence (x)(Sx ⊃ Px) does not imply (∃x)(Sx). And that is somehow supposed to show the failure of existential import. But it doesn’t show anything of the sort .... "
> http://pvspade.com/Logic/docs/Thoughts,%20Words%20and%20Things1_2.pdf

It is worth reading the rest of that footnote. He agrees that  (x)(Sx ⊃ Px) does not imply (∃x)(Sx), but points out that (Ux)(Px) implies (Ex)(Px), and that *this* is the real existential import of modern logic.

"The modern equivalent of existential import, therefore, is not: (x)(Sx ⊃ Px) ∴ (∃x)(Sx), but rather (x)(Px) ∴ (∃x)(Px). And that holds in standard modern logic, which is therefore just as much committed to existential import as traditional logic is."

Perfectly correct, but has no bearing on the issue you are raising here.

>
> So Spade approaches this as the issue of the existential import of universally quantified statements. He points out that, from Ux(Dx --> Rx), we cannot infer Ex(Dx & Rx). The rest of the passage attempts to explain why. I still either don't understand his argument, or I'm not convinced by it. Why should "All dogs are renates" not be expressed as Ux(Dx & Rx)?

Because that would then also express "All renates are dogs", by the symmetry of conjunction, and would entail "Everything is a dog", by &-elimination.

Best wishes

Pat Hayes

PS there is a wonderful extended essay here on this general topic:  http://plato.stanford.edu/entries/ontological-commitment/

> From John's reply, I think he would say that it's because we can only observe particular things; we can't observe all things. But in the preceding points, I've tried to say why I don't find that convincing.
>
> (6) Simply the fact that decades of logicians have not raised the concerns I have raised strongly suggests that I am mistaken, and need to think more clearly about logic and ontological commitment. But there is something that might make one hesitate to jump right to that conclusion. It's Kripke's position on analytic a posteriori statements (which I have difficulty distinguishing from Kant's synthetic a priori statements, actually -- providing we assume that the metaphors of "analytic" as finding that one thing is "contained in" another thing, and of "synthetic" as bringing together two things first experienced as distinct, are just metaphors, and don't work as solid explanations).
>
> All analytic statements are "All" statements, not "Some" statements. Kripke suggests that the statement "Water is H2O" is analytic but a posteriori. In general, that "natural kind" statements are all of this sort. Well, a posteriori statements are ones verified by experience, and so that would take care of John's Peircean point that only "Some" statements are grounded in what we experience.
>
> I don't know how solid this line of thought is. But if there is something to it, that might suggest that if we accept Kripke's whole referential semantics / rigid designator / natural kinds ideas (cf. Putnam's twin earth thought experiment also), then perhaps we should rethink the traditional metalogical interpretation of "All dogs are renates" as Ux(Dx --> Rx), and consider, instead, Ux(Dx & Rx).
>
> Well, two summing-up points. The first is that Paul Vincent Spade thinks that my position is "silly", and John Sowa thinks that it's at least wrong. The second is that such discussions do indeed take us beyond the concerns of ontology engineers, who just want to get on with building working ontologies.
>
> As I said above, I will address those concerns of ontology engineers before I begin discussing cognitive semantics in this Ontolog (Ontology + Logic) forum.
>
> Regards to all,
>
> Tom
>
>
>
>
>
> On Monday, October 12, 2015 10:49 PM, John F Sowa <sowa@xxxxxxxxxxx> wrote:
>
>
> Tom, Ed, Leo, Paul, Henson,
>
> TJ
> > why, in the formalization of predicate logic, was it decided
> > that "Some X" would carry ontological commitment
>
> Nobody made that decision.  It's a fact of perception.  Every
> observation can always be described with just two operators:
> existential quantifier and conjunction. No other operators can
> be observed. They can only be inferred.
>
> EJB
> > I was taught formal logic as a mathematical discipline, not
> > a philosophical discipline. I do not believe that mathematics
> > has any interest in ontological commitment.
>
> That's true.  And most of the people who developed formal logic
> in the 20th c were mathematicians.  They didn't worry about
> the source or reliability of the starting axioms.
>
> Leo
> > most ontologists of the realist persuasion will argue that there
> > are no negated/negative ontological things.
>
> Whatever their persuasion, nobody can observe a negation.  It's
> always an inference or an assumption.
>
> PT
> > on the inadequacy of mathematical logic for reasoning about
> > the real world, see Veatch, "Intentional Logic: a logic based on
> > philosophical realism".
>
> Many different logics can be and have been formalized for various
> purposes.  They may have different ontological commitments built in,
> but the distinction of what is observed or inferred is critical.
>
> HG
> > I keep wondering if this forum has anything useful to offer the
> > science and engineering community.
>
> C. S. Peirce was deeply involved in experimental physics and
> engineering.  He was also employed as an associate editor of the
> _Century Dictionary_, for which he wrote, revised, or edited over
> 16,000 definitions.  My comments below are based on CSP's writings:
>
>  1. Any sensory perception is evidence that something exists;
>    a simultaneous perception of something A and something B
>    is evidence for (Ex)(Ey)(A(x) & B(y)).
>
>  2. Evidence for other operators must *always* be an inference:
>
>    (a) Failure to observe P(x) does not mean there is no P.
>
>        Example:  "There is no hippopotamus in this room"
>        can only be inferred iff you have failed to observe
>        a hippo and know that it is big enough that you would
>        certainly have noticed one if it were present.
>
>    (b) (p or q) cannot be directly observed.  But you might infer
>        that a particular observation (e.g. "the room is lighted")
>        could be the result of two or more sources.
>
>    (c) (p implies q) cannot be observed, as Hume discussed at length.
>
>    (d) a universal quantifier can never be observed.  No matter
>        how many examples of P(x) you see, you can never know that
>        you've seen them all (unless you have other information
>        that guarantees you have seen them all).
>
> TJ
> > But now notice something: negation creates and removes ontological
> > commitment. And this seems really strange. Why should negation do this?
>
> The commitment is derived from the same background knowledge that
> enabled you to assert (or prevented you from asserting) the negation.
>
> > I'd also like to know if there are formal logics which do not
> > impute this extravagant power of ontological commitment /
> > de-commitment to the negation operator in predicate logics.
>
> Most formal logicians don't think about these issues -- for the
> simple reason that most of them are mathematicians.  They don't
> think about observation and evidence.
>
> CSP realized the problematical issues with negation, but he also
> knew that he needed to assume at least one additional operator.
> And negation was the simplest of the lot.  Those are the three
> he assumed for his existential graphs.  (But he later added
> metalanguage, modality, and three values -- T, F, and Unknown.)
>
> John
>
> PS:  The example "There is no hippopotamus in this room" came from
> a remark by Bertrand Russell that he couldn't convince Wittgenstein
> that there was no hippopotamus in the room.  Russell didn't go
> into any detail, but I suspect that Ludwig W. was trying to
> explain the point that a negation cannot be observed.
>
>
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