Relevant Implications

The 'material implication' of classical logic does a fairly poor job of capturing our intuitive understanding of if-then.  It holds X -> Y [if X then Y] to be logically equivalent to ~X v Y [either not-X, or Y].  So an implication will be true whenever the antecedent is false (regardless of the consequent), or else whenever the consequent is true (regardless of the antecedent).  But it just seems wrong to say "if today is Tuesday then today is Wednesday".  We are more inclined to interpret this as a counterfactual / subjunctive conditional, i.e. "if today were Tuesday then today would be Wednesday" - and this we can see is obviously false.

Further, there are true material implications where the antecedent has nothing to do with the consequent at all.  For example, "if I click my fingers, then 2+2=4".  This is true, even if understood subjunctively (since 2+2=4 is true in all possible worlds).  So the problem isn't so much to do with counterfactual truth-values, as relevance.  Our common-sense understanding of if-then seems to involve an explanatory (perhaps even causal) notion.  To say "if A then B" often means that B depends on A, in some sense.  We might even think that if A is false, then B will be too (which is fallacious reasoning according to classical logic).

I'm not suggesting that our intuitive understanding is better than classical logic, or vice versa (though for particular purposes one might be more suitable than the other).  I just think it would be interesting to formalise the intuitive notion, though I'm not sure how this is best done.  As always, feel free to leave a comment if you know more about all this stuff.

In class today we briefly discussed some oddities about the 'Conditional Proof' natural deduction rule.  The basic idea is that you (temporarily) assume X, somehow or other derive Y, then you can cancel the assumption and conclude that X -> Y.  The oddity arises from the fact that you need not use X when deriving Y.  So if you have already proven Y, then you can use CP to prove X -> Y for any proposition X at all, no matter its truth value or relevance.

One suggestion a classmate offered was that we could modify our CP rule to require that the assumption X must be used in the derivation of Y.  The problem with this (as our lecturer pointed out) is that it is too weak.  One could always just append X using &-intro, then remove it again using &-elim.  It would then be used in the derivation, though in an utterly redundant way.

What we really want here is some way of telling that X needs to be used to derive Y.  My first thought was that we could formalize this requirement as ~(~X -> Y), but in classical logic that is equivalent to X & ~Y, which is clearly not the meaning I was after!  I was really thinking of something like "If X were false then you would not be able to derive Y any longer".  But that begs the question of how to represent counterfactuals.

The closest truth-functional approximation I could think of was ~X -> ~Y, but that's a bit too strong really.  Combined with the original CP, that would prove the biconditional X <--> Y.  Instead, I want to say that if X is false then we cannot (yet) know that Y is true.  Note that this doesn't necessarily mean that Y is false, however.  Hmm... This might work according to some sort of constructivist logic, I'm not sure (I know very little about that stuff).  Suppose "~X" means "I cannot construct a proof of X" (rather than the usual "X is false").  Then I think this formula would capture what I want: If you lack a proof of X, then you cannot construct a proof of Y. [Should this instead be done subjunctively too?  I'm hoping I can get away without it.]

Perhaps it isn't possible to formalize a relevance requirement on CP whilst remaining within classical logic.  Framing relevance in terms of what we can know turns it into an epistemic question (seemingly requiring constructivist logic?), whereas the earlier notion of what we need raises modal issues (and so we need a logic of counterfactuals, to deal with other possible worlds). Truth-functional logic is merely concerned with 'brute' truth-values.  The concept of relevance goes beyond this, so I guess it would make sense for the concept to be unanalysible in truth-functional terms.

Incidentally, a quick google turned up the Stanford encyclopedia entry for Relevance Logic.  I've only skimmed it, but it looks fairly interesting.

Retreat

I'm off to Kaikoura for the weekend, as our university is having another 'philosophy retreat' thingamajig. Should be fun - especially since I've been asked to present my paper on free will.

If you're looking for something else to read in the meantime, have a glance through this huge list of weblogs.

Pseudo-Philosophy

Wouldn't we think it odd if a science magazine had as their 'top story' a link to a creationist website? Or an article espousing the wonders of astrology? Yet the Philosophy News Service had as their previous 'Top Story' this nonsense about gay marriage, full of straw men, logical fallacies, and pseudoscience.

Consider his portrayal of the liberal argument:

Homosexuals argue they are born this way, they say. Therefore, since we can't change who we are, then we should be just as protected as minorities and females.

That's a stupid argument, and no intelligent liberal should use it. For starters, it's politically foolish to rest our case on such a flimsy premise which could well be disproved by new evidence. More importantly, the argument isn't even valid. If some people are born with a predilection for violence, that doesn't mean we should allow them to commit murders. The "it's natural!" argument is just as idiotic as the "it's unnatural!" argument; neither justifies any moral conclusion whatsoever.

Homosexuality is perfectly acceptable regardless of whether it arises from nature, nurture, or is a freely-made choice. Better arguments will reflect this. For example, one could appeal to J.S Mill's harm principle: that is, the value of liberty, and the utter lack of any decent justification for curtailing it in this case.

But the writer of this article ignores such arguments, preferring to attack the straw man. Remarkably, though, he can't even do that properly.

Why am I so sure that a gay gene "isn't there"? It's a simple matter of (surprise) Darwinism... In the case of homosexuals, same-sex couples cannot reproduce. This is the fatal flaw in the homosexual argument that homosexuality is genetic. Therefore, even if you grant that a "gay gene" did exist at one point, it couldn't have been passed along to future generations since same-sex couples can't reproduce.

There is so much wrong with this I'm not sure where to start. But I'll try anyway. (1) Genetic influences are rarely so blunt as a single "gay gene". [Note: follow the link in the update for my explanation of precisely why this is a problem for his argument.] (2) Recessive alleles can be passed on by 'carriers' who do not themselves experience its phenotypic effects. (3) Gay people can still have children the usual way. (4) Non-reproducing family members can help their relatives to pass their genes into the next generation. It's also rather telling that homosexual behaviour has been observed in other animals.

As I mentioned above, the genetic question is rather irrelevant anyway. But it's worth pointing out that this guy doesn't know what he's talking about. Just look at this fallacy:

If the presumption [in the "liberal argument"] is not true, then homosexuals do not have a right to marry.

Um, no. A false premise invalidates an argument, but it does not guarantee the falsity of the conclusion. There could always be some other argument which establishes the conclusion.

Lastly, there's the usual tripe about homosexuality being "unnatural":

[S]ame-sex sexual relations deviates from the biological standard of male-and-female sexual relations. Any other type of sexual relations (e.g., human-and-animal) is similarly deviant. Pretty black-and-white, folks... As everyone knows, male homosexual relations involves the anus. And everyone also knows that the biological function of the anus is not sexual.

It's pretty stupid to assert that just because an object has some function, it's immoral to use it for any other purpose. The biological function of the mouth is the consumption of food and drink. Does that make kissing immoral? I use my hands to play the piano. I'm pretty sure hands didn't evolve for that specific purpose though. As for the reproductive purpose of sex, I take it that the "biological standard" is fertile male-and-female couples. So does that make it "deviant" for the infertile to have sex? Should post-menopausal women not be allowed to marry either?

This article is no more philosophy than creationism is science. The Philosophy News Service should be ashamed of itself for implicitly endorsing such rubbish.

Update: For more on the first article, see the PNS forum.

Mixed Metaphysics

I really enjoyed the metaphysics course I took last semester, though it often struck me as a deeply mistaken enterprise. We would take some concept central to our common-sense understanding of the world (such as 'properties'/universals), and then treat it as though it really existed. This seems to ignore the central lesson of modern science (particularly 20th century physics): that our intuition or "common sense" is a very unreliable indicator of the actual nature of reality. See also this post by John Wilkins on the foolishness of philosophers who understand biological species' as metaphysical 'kinds'.

Our metaphysics course stood squarely in the Aristotelian tradition of examining the nature of reality, or "being qua being". An alternative approach was mentioned, however. As I understand it, the Kantian tradition instead limits itself to examining mental constructs, or the conceptual schema which underlies our understanding of the world.

Based (perhaps loosely) upon those two opposing traditions, I find myself thinking that there are two 'ideal' ways of approaching metaphysics. Both should be scientifically informed, but in different ways: the one by physics, the other, psychology.

My first ideal is modelled on the Aristotelian tradition of pursuing the fundamental nature of reality. This approach absolutely must be based upon physics if it is to have any hope of success. What it should do, I think, is take our best scientific theories (general relativity, quantum mechanics, perhaps soon String Theory), and provide us with the bare minimum of metaphysical constructs required to support the scientific theory. Baroque two-worlds ontologies, or unscientific treatments of space and time, would have no place here. What we need instead, is a presentation of the metaphysics underlying scientific materialism/physicalism. (Please let me know of anyone who has undertaken such a project - I'm sure it must have been done, but I haven't heard of it yet!)

My second ideal is essentially tied to cognitive science. It is the attempt to understand the human mind, and how we categorise/analyse reality. "Folk metaphysics", in other words. This approach would take our common-sense concepts of properties, truth, time, etc, and try to spin them together into a coherent, conceptually simple, system of understanding.

The 'folk metaphysician' would make no claims about the nature of reality. Instead, he would help us to understand ourselves. Furthermore, he would play a crucial role in the development of Artificial Intelligence. For without some coherent conceptual schema for analysing the flood of data provided by its input sensors, an AI would have little hope of understanding the world around him. In this way, metaphysics - seemingly one of the most abstract branches of philosophy - would in fact serve an important practical purpose.

The problem with the metaphysics we studied, is that it mixed these two ideals. It was based on the common-sense, rather than scientific, structures. Yet it still made claims to objectivity, to representing the fundamental nature of reality. I found that mix to be rather implausible. It would be better, I believe, to pick one of the two 'ideals' outlined above, and stick to that. You can have reality, or common sense, but not both.

Logic Translations

An issue about translations came up in our Logic lecture earlier this week.  Consider the sentence S = "Either Fred will eat out and Martha will eat with him, or Fred will go home and eat alone".  How should this be translated into truth-functional logic?
 
The obvious approach is to simply go for (O & M) v (H & A), where O = Fred will eat out, M = Martha will eat with Fred, H = Fred will eat at home, A = Fred will eat alone.  Or something along those lines.
 
However, as another student pointed out, this seems incomplete, failing to capture the meaningful connections between elements of the sentence.  For example, it seems clear to us that A and M are mutually exclusive, as are O and H, yet the syntax of the simple translation offers no hint of this.
 
I think, then, a better translation may be:
[(O & M) v (H & A)] & (O → ~H) & (M → ~A) 
 
It might be objected that this unnecessarily complicates things.  Perhaps our basic translation should be kept as simple as possible, and any "extra information" or inferences between its parts, ought to be introduced as new premises if necessary.  Though if this principle were taken to extremes, we might be tempted to just offer the entire sentence S with no internal analysis whatsoever.  That would obviously not be helpful.
 
I would have thought that a translation should try to capture as much of the analytic/syntactical information provided in the sentence as possible.  That is, our translation should try to spell out the appropriate inferences between the words and phrases used.  However, we should avoid cramming in 'synthetic' or contextual information, for these might include unwarranted assumptions - far better to introduce them as new premises. 
 
In this case, however, I think the information added is best understood as analytic.  If Fred eats alone, then it is not the case that he eats with someone else (e.g. Martha).  This is a simple analytic truth that we can know from the words themselves, without having to reference contingent real-world facts.
 
To re-inforce this point, consider how we would translate the sentence into first-order logic.
We can do this with the single predicate - Wxyz: x will eat at (location) y with (person) z; 
We will also need the constants f = Fred; m = Martha; h = (Fred's) home.
Using 'E' as the existential quantifier ("there exists"), and 'V' for universal quantification (i.e. "for all..."), our translation of S could now be as follows:
 
Ex(Wfxm & x ≠ h ) v (Wfhf & VxVy(Wfyx → x = f))
 
This literally says "Either Fred will eat at somewhere that is not home, with Martha, OR Fred will eat at home (with himself), and he will eat with nobody but himself."
I think that captures the meaning of S reasonably well.  Notice that it includes the analytic implication that if Fred eats alone then there is no-one else he eats with - not even Martha.
 
However, this particular translation doesn't rule out the possibility of eating at several places, i.e. both at home and not and home.  This could be introduced easily enough with some more quantification, but it's starting to get a bit messy.  Also, the 3-place predicate might not be entirely faithful to the original syntax of S, which appeared to treat locations and companions quite separately.  This too is easily solved, this time by replacing it with two 2-place predicates, but that will make things messier yet. 
 
I think a better approach - solving both problems - might be to introduce a function, to map from a person to the place where that person "will eat" (at whatever particular time we're implicitly talking about). So here's a stronger translation...
 
Let px = the place where x will eat; f = Fred; m = Martha; h = (Fred's) home;
Wxy: x will eat with y.
(Wfm & pf ≠ h) v (pf = h & Vx(Wfx → x = f))
 
i.e. "Either Fred will eat with Martha, at a place which is not home, OR Fred will eat at home with nobody else".
 
Alternatively, we could lose the predicates altogether, and just use the function...
(pf = pm & pf ≠ h) v (pf = h & Vx (px = pf → x = f))
i.e. "Either Fred and Martha will eat at the same place, which is not home, OR the place where Fred will eat is home, and there is no-one but him at that place".
 
I quite like that last translation actually, though one might complain that there is a conceptual distinction to be made between eating with someone, and merely eating at the same place as them.
 
Anyway, my enthusiam about various translating strategies aside, the point is that these first-order logical translations seem to capture the meaning of S, better than the basic truth-functional one did.  These analytic implications can be captured truth-functionally, however, we just need to append them manually to the end of the basic translation.
 
The big question, then, is "should we?".  I think so, but I'm not entirely sure.  One could object that I've just assumed the sentence is talking about a single temporal period, and that this may not be so - perhaps the pairs are not mutually exclusive after all.  Fair enough.  But suppose I picked a better example to illustrate this point, one with no ambiguities here (e.g. reword the sentence to be entirely in the present tense).  The general question remains: How much detail should we seek to include in a logical translation?  Just the basic elements?  Or also the analytic implications we can draw between them?
 
As always, comments are most welcome.

Israeli Spies

International spies, here in li'l ole New Zealand! How exciting ;)
Details available from NBR, NZ Herald and The Press. Some excerpts:

Prime Minister Helen Clark launched a blistering verbal attack and slapped diplomatic sanctions on Israel last night after two suspected Mossad agents were imprisoned for six months for illegally trying to obtain a New Zealand passport.

"The Israeli agents attempted to demean the integrity of the New Zealand passport system. The Israeli government was asked for an explanation and an apology three months ago. Neither was received."

"This type of behaviour is unacceptable internationally by any country. It is a sorry indictment of Israel that it has again taken such actions against a country with which it has friendly relations."

Helen Clark said it was not the first time Israel had sought fraudulently to obtain passports from another country.

"Israeli agents caught in an unsuccessful assassination attempt in Jordan in 1997 were found to be carrying fraudulent Canadian passports," she said. "While we regret the need for it, New Zealand has no option but to take the actions that it has in response to a deliberate breach of its sovereignty."

The response of our government has been to:

Suspend high-level visits to and from Israel.

Decline permission for Israel's President to visit later this year.

Implement visa restrictions on all Israeli government officials visiting New Zealand.

Suspend Foreign Ministry consultations with Israel.

Delay indefinitely the appointment of a new Israeli ambassador.

Ha, this even makes me feel patriotic - I imagine it'll do wonders for Clark and the Labour party in the polls...

Questions or Pretensions?

One of the great things about my Semantics class is the high level of student participation (i.e. questions and comments) in the lectures. Though some see it differently. A linguistics student I know complained to a mutual friend about the "pretentious philosophy prats" in the double-coded course. That got me thinking.

I sometimes find other students' questions/comments annoying. But only if they are bad questions. There are, after all, some people who make utterly random or irrelevant comments, and that can be quite frustrating. Fortunately, our lecturer (Kate Kearns) is very skilled at handling them, managing to treat the student respectfully yet quickly steering the lecture back on track.

Other times, however, student participation can really add a lot to a lecture. For example, in our last lecture Kearns suggested that the 'sense' of a word limits the appropriate uses of it - we can't just use words however we please - to which one girl pointed out the dynamic aspects of language, especially in slang and idioms. After all, if a particular 'inappropriate' use of a word becomes widespread, then the meaning of the word is expanded, and such use becomes 'appropriate' after all. This helpful interjection led the lecturer to discuss the issue in a bit more detail, clarifying matters and treating the rest of us to a very interesting discussion. I, for one, greatly appreciated it.

I'm interested in the difference between good and bad questions because I often ask questions in lectures myself. Of course, I always think they're fairly interesting and relevant, or I wouldn't ask them. But I'm not sure how they come across to others. I get particularly worried when I find myself inwardly cringing at someone else's pointless interjection, since I'm faced with the niggling doubt, what if others react similarly to mine?

I think student comments can be categorised as belonging to one of three broad groups. These are, in increasing order of intelligence displayed:

Missing the point entirely - These are always annoying, though the severity of idiocy displayed varies. It's worst when the stubborn speaker is unable to realise that he's barking up the wrong tree.

Requests for clarification - Though mildly tedious for those who understood it all the first time through, these are probably good questions to ask. Chances are others will benefit from having a tricky concept clarified.

Making a point/suggestion/objection - Here I intend to exclude those who aim for this group, but instead fall under group one.

That last one is the group I'm interested in - and, I assume, the target of the "pretentious prats" remark. I guess when someone raises a technical issue, it may seem rather pedantic, appearing to serve no point but to say 'look how smart I am!'. Being a question-asker myself, I'm generally more sympathetic here, but even I sometimes reach this uncharitable conclusion about a speaker.

Sometimes it may be justified. But I think it is far too easy for people to conflate intellectual curiosity with arrogance. If I ask a question, it's because I'm interested in what the answer is. Seemingly 'pedantic' technicalities can decide the viability of a theory or explanation. If it has some potential flaw in it, this is worth exploring. Delving into such details requires a narrower focus than the broad sweep of generalisations we undergrads are used to. But it would be a mistake to think that breadth is always preferable to depth, or to conclude from this that such questions are not worth asking. And it would certainly be a mistake to think that the only motivation for asking them is egotism.

P.S. On a semi-related topic, Leiter has a couple of great posts on scientists being called "arrogant" for having specialized knowledge.

New blog

Check out the a new Philosophy of Biology blog. They already have a couple of interesting posts on Game Theory, and the teleological nature of biology.

Blogging Survey

Via Pharyngula, I noticed a blog survey some graduate students are carrying out. It's quick (took me five minutes), easy, and even fairly interesting (as far as surveys go). If you have a blog of your own, go do it!

Though I actually think finding out the personality traits of bloggers would be far more interesting. But by the meagre response so far, it seems not many others agree with me here. A pity.

Template Changes

I've been playing around with my blog template a bit recently, you might have noticed. My sidebar in particular now looks much better (it was a bit of an eyesore before!).

Changes include:

Cooler bullet points

Cooler blockquotes

A list of links to 'previous posts' on the sidebar, which should make travelling around the site quicker and easier.

New fonts

Aligned post text (now 'justified' on both margins, not only the left)

Slightly wider margins, for easier reading

Google search (of site or web) available at the bottom of the page.

Blogrolls displayed on main page only

Archives condensed down to just links, rather than entire posts.

Leave a comment if you have further suggestions - any feedback is much appreciated.