Category Archives: Plato

Logical Pairings

In previous posts I’ve tried to interpret the canonical Tagalog sentence (e.g., maganda si Taylor Lautner) in terms of an equality relation, GORGEOUS_EQUALS_GORGEOUS.  Conceptually, the relation is formed by logically pairing each member of the set GORGEOUS (MAGANDA) to each of the members, then taking a subset of the set that results from this logical paring.  That subset comprises those logical pairings in which each member of the pair is identical with the other.

What do I mean by ‘logical pairing’?  In the real world, to pair one thing with another is to bring the two things together in some way.  One may pair, for example, some particular matte board, with its particular color, with the painting one is getting framed.  Here, the matte board and painting are getting physically paired.  Or one may pair John with Bill by picturing them in the mind’s eye as together as a couple.  Or one may pair John with John by first seeing him double (i.e., seeing him twice but simultaneously), then by realizing the two Johns are in fact one.

To get a logical pairing, abstract from any concrete form of pairing, that is, ignore any particular way in which the bringing together is done.  Ignore in fact everything about them except that they go under the heading ‘bringing together’ (since maybe that is the only single thing they all have in common.)   Then be content with the fact that, while each member of the set MAGANDA can potentially be brought together with every member of that set,  any actual pairings will be performed just every now and then, and only for a few members.  (For example, in a particular article, Dan Savage pictures Ashton Kutcher and Matt Damon together.)  A logical pairing is a bringing together in which all concrete details of the bringing together (how it is done, in what sense the things are brought together?  Physically?  In the imagination only?  By already knowing that the “objects” of one’s double vision are in fact one and the same?) are ignored.  One salient detail in particular is ignored:  is the pairing actually being done in any given instance, or is it just something that could be done?

If one does not want to rest content with each member of the set being brought together just potentially with every other member of the set, they (plural third person intentionally being used here as a neutral singular third person) are free to imagine a Demiurge ala Plato or a God ala the medievals whose cognitive capacities are sufficiently large as to simultaneously bring together in its mind’s eye every member of the set MAGANDA with every member of that set, so large, in fact, as to be able to see Matt Damon twice with the mind’s eye but already know that Matt Damon is, well, Matt Damon.

I will end by confessing that I like to think of projection as the Demiurge’s ignoring one or more attributes of a relation, and of restriction as the Demiurge’s ignoring one or more tuples in the relation.

Today, my homage to Plato’s SYMPOSIUM (first, gorgeous guys, then the Relational Algebra, then the form Beauty itself) will take the form of a concrete (not just a logical) pairing of Matt Damon and Ashton Kutcher:

matt_damon_splashnews--300x300

ashton_kutcher-4036

Sigh.  There is too much beauty in the world.

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A Doubtlessly Lame Attempt At Explaining The Awkwardness

Caution:  The following belongs to the category of ‘let’s see how long I can get away with this before it convincingly gets shot down’.  Either that, or to the category ‘This is so obvious and has been stated so many times in the past that it is a puzzle why you bother mentioning it.’

The motivation for the following blather:  In a previous post I was bothered by the (I think true) assertion that one can reduce propositions to states of affairs…my botheration arising from the fact that while propositions are always either true or false, it seems awkward to say things like “Don’s standing to the right of Genghis Khan is a true state of affairs.”

The blather itself:  Let’s suppose that we could describe a state of affairs as either true state of affairs, or a false state of affairs, using the ‘state of affairs’ vocabulary, only if any state of affairs could be so described.  Not every state of affairs can be described as either true or false:  for example, “Guile riding his bicycle.”  This is a state of affairs that occurs at any given moment, when Guile is riding his bicycle at that moment, or that fails to occur at that moment; but it is neither true nor false.

So if we try to describe “Guile rode his bicycle a moment ago in Angeles” as a true (or false) state of affairs, we will be beating our head against the wall, for we will be using a vocabulary that cannot be applied to any state of affairs, but only to some of them.  So instead of beating our head against a brick wall, we will invent a new kind of thing:  propositions, which are either true or false.  And we will reduce propositions to states of affairs by saying propositions are nothing but a subset of the set of states of affairs, namely, those that always occur (or fail to occur).  (“Guile rode his bicycle a moment ago, i.e., at time t_1, in Angeles” is a state of affairs that will always occur if Guile did ride his bicycle during that time and at that place, or it is a state of affairs that will never occur.)

Conclusion:  This way we can reduce propositions to a subset of states of affairs without having to talk about true or false states of affairs.

My homage to Plato’s SYMPOSIUM for this post is Brad Pitt again:

Brad-Pitt-a-River-Runs-Through-It

How can anyone get anything done with beauty like that walking the earth?

 


Oh My God, I Get Tons Of Likes When…

Oh my god, I get tons of likes when I put Ashton Kutcher in relation to a Cavafy poem, but none at all when I talk about the Relational Algebra.  🙂

Male Pilipino total gorgeousness:

dennis_trillo 4

 

This is a high-minded homage to Plato’s SYMPOSIUM, of course.


Some More Clean-Up Work: Propositions And States Of Affairs

Following Chisholm, I have been identifying propositions with states of affairs.  A proposition is a subset of the set of states of affairs.  The state of affairs of John grasping a doorknob at time t_0 in Chicago is a state of affairs that always occurs (or always fails to occur).   States of affairs like this one are propositions.  The truth (falsity) of a proposition is nothing but a certain state of affairs occurring (failing to occur).  I am ignoring the question, which is pestering me right now, of why then it seems so awkward to talk about a ‘true’ (‘false’) state of affairs.  From The Stanford Encyclopedia of Philosophy article on Roderick Chisholm:

Consider the state of affairs that is expressed by the sentence ‘Someone is walking’. Chisholm wanted to say that this state of affairs occurs whenever someone walks, and fails to occur at times when no one is walking. Other states of affairs are not like this. For them, it is impossible to sometimes occur and sometimes fail to occur. Chisholm claims that this provides the opportunity for an ontological reduction. We can define a proposition as a state of affairs of this latter sort — it is impossible for there to be times when it occurs and other times when it does not occur. A true proposition is thus one that occurs; and afalse proposition is one that does not occur. Chisholm thinks that we may understand the principles of logic to be about these propositions. By saying that a fact is a true proposition, Chisholm gains yet another ontological reduction ([P&O], 123).

Chisholm thought that in some cases it makes sense to speak of the location at which a state of affairs occurs. Suppose John walks in Chicago at a certain time. Then Chisholm would be willing to say that the state of affairs of John’s walking occurs in Chicago and at that time.

Those states of affairs that are not propositions are events.   I am going through this stuff a bit impressionistically; the chances of my making an error someplace are high.

The tuples in the body of a database relation are propositions.  That is to say, they are states of affairs.  In a conventional database, these are always states of affairs occurring now, and now, and now…. John is an employee of WIDGETS_R_US now, the ‘now’ being implicit in the presence of that tuple in the relation.   In a temporal database as described by Date and Darwen (TEMPORAL DATA AND THE RELATIONAL MODEL),  these are states of affairs that occurred during a time period, or are occurring now (“Since t_0….”), the relevant time periods being explicitly stated in the tuple.

Since propositions are nothing but states of affairs of a certain kind, the operations of the Relational Algebra are operations on states of affairs of that kind.  On the relation ‘Standing_To_The_RIGHT_Of’, for example, we can perform a RESTRICT operation that delivers to us the state of affairs of Don standing to the right of Genghis Khan, then perform a PROJECT operation on that derived relation to obtain just Don.

We will figure out later what to do with Don now that we have him.

My homage to Plato’s SYMPOSIUM for this post will be Matt Damon.  This time we are a bit further along on the way towards eros for mathematical beauty:

mattdamon2a

But let’s not forget it all originally stems from eros for gorgeous young men.


Some Clean-Up Work: Why A Name Needs A Selector If One Is To Be Fully Explicit

Let me unpack a bit the NAME() selectors I have been using.  A selector such as NAME(‘Tom’) takes as an argument the string ‘Tom’ and returns the name Tom.(Tom is being mentioned here, not used.  The arguments surely have to be syntactic arguments.)  A string comprises 0 or more written characters (henceforth  just ‘characters’).  A character is an abstract object:  the character ‘e’, for example, can be instantiated by a blob of ink, a pencil mark, a set of pixels….  So a string is an abstract object comprising other abstract objects, and exists at one level-of-abstraction higher than they.

A string of characters is not itself a name, since a name can also be instantiated by a zero or more sounds.  I say “0 or more” because I can imagine a language that uses the glottal stop as a name, whatever the merely practical difficulties might be in doing so.  (A name that could never be pronounced by itself, but only within a stream of other sounds?)

(Perhaps — to jump back to characters for a moment — this language could write the name as ”.  So a name could be instantiated by strings comprising 0 or more characters. )

(Perhaps — to jump back to sounds for a moment — if I tried hard enough I could turn a sound into an abstract object (perhaps one sound can be instantiated by any number of configurations of sound waves?), but I will not try this at the moment. )

Instantiated, as I was saying, by either strings or sounds, a name is an abstract object, one existing at one level of abstraction higher than the abstract object STRING, which itself is one level of abstraction higher than the abstract object CHARACTER.   Not identical with either a sound or a string, a name is best represented not by, for example, ‘Tom’ or <<some sound>>, but by NAME(‘Tom’) or NAME(<<some representation of a string of sounds>>).

This, then, is why, when I am trying to be fully explicit, I refer to a name not as, e.g., ‘Tom’, but as NAME(‘Tom’).

Today’s homage to Plato’s SYMPOSIUM is Ashton Kutcher:

Ashton_Kutcher

There is too much beauty in the world.  How can one concentrate on anything at all with gods like this walking the earth?


Just For The Fun Of It: A Quibble With C.J. Date

In the previous post, I said “I get the funny feeling that for him [C.J. Date] a sentence or expression functions normally at first, but when he stares at it too long it suddenly loses its transparency….”  Here I want to unpack that doubtlessly cryptic statement a bit.  I am doing this as a kind of finger-exercise, mainly for the fun of it, and not necessarily because it will help resolve some important issue.

In “SOME OPERATORS ARE MORE EQUAL THAN OTHERS”, the second chapter of his LOGIC AND DATABASES:  THE ROOTS OF RELATIONAL THEORY, C.J. Date introduces the names NAME(‘TRUE’) and NAME(‘FALSE’) to use as symbols for BOOLEAN(NAME(‘TRUE’) ) and BOOLEAN(NAME(‘FALSE’)).  He doesn’t use, of course, the selectors I am using (to avoid ambiguity); instead, he mentions the names.  That is to say, he is talking about the symbols TRUE and FALSE.  All  page references will be to LOGIC AND DATABASES:

If p is a proposition, it has a truth value.  For the purposes of  this chapter, if p is true, I’ll denote its truth value by TRUE; it it’s false, I’ll denote its truth value by FALSE.  In computing terms, we might say we have a data type called BOOLEAN that contains just these two truth values, and the literal representations of those values are TRUE and FALSE, respectively.

(p. 42)

Then for a moment, he uses the symbols to talk about the truth values themselves.  For a moment, the symbols stop being the things seen, and function transparently as windows through which we see the things denoted.  But, feeling conflicted,  he immediately labels his use of the symbols as a kind of perhaps-understandable confusion:

We might also say, more simply but less accurately, that the legal values of type BOOLEAN just are TRUE and FALSE; strictly speaking, however, TRUE and FALSE aren’t values as such — rather, they’re literals, or symbols, that denote certain values, just as, e.g., the numeral 3 isn’t a number as such but rather a literal, or symbol (more colloquially, a digit or numeral) that denotes a certain number.

(p. 42)

But no, when ‘TRUE’ and ‘FALSE’ are being used to denote truth values, it is perfectly accurate to say that the legal values of type BOOLEAN just are TRUE and FALSE.  Likewise, when ‘3’ is being used to denote a number, 3 is a number as such.  It is as if Date were peering through the symbol-as-window, so to speak, letting it perform its function of letting us see the thing the symbol denotes….when, all of a sudden, the window going all milkily opaque on him, all he sees is the now suddenly non-transparent glass.  It is as if he said, first, ‘Cleopatra was a Ptolemaic ruler of Egypt’, but then immediately felt the need to correct himself and say, ‘Well, this is a bit inaccurate, since the name ‘Cleopatra’ wasn’t a Ptolemaic ruler of Egypt’.

Similarly, Date is brought up short when he sees the expressions x + 4 and 2x – 1 in the equation x + 4 = 2x – 1:

But the trouble is, we use the symbol “=” to mean other things as well — other things, that is, in addition to identity as such….  By way of illustration, consider the following equation:

x + 4 = 2x -1

The symbol x here is meant to denote some number, and it’s easy to see by solving the equation that the number in question is five.  But the expressions on the two sides of the “=” symbol are self-evidently not identical:  that is, the “=” symbol here does not denote identity.  Rather, it is the values denoted by the expressions on the two sides of the “=” symbol that are identical.

(p. 45)

But the “=”, rather, MATH_CONSTANT(‘=’), does denote identity, because the expressions are being used to denote the number that each expression resolves to, given the expression on the other side.  The expressions are not being mentioned — for example, the author of the equation is not talking about the fact that each expression contains the symbol ‘x‘ and references at least one number.   ‘x + 4′ names a quantity when made resolvable by equating it with ‘2x – 1′, and vice versa.   MATH_VARIABLE(‘x’) MATH_OPERATOR(‘+’) NUMBER(NUMERAL(‘4’)) MATH_CONSTANT(‘=’) NUMBER(NUMERAL(‘2’)) MATH_OPERATOR(‘*’) MATH_VARIABLE(‘x) MATH_OPERATOR(‘-‘) NUMBER(NUMERAL(‘1’)).  The quantity named by the one expression is identical with the quantity named by the other expression.  The hermeneutical principle of charity compels us to interpret the expressions this way:  otherwise, we don’t get a symbol “=” that means something other than identity; instead, we simply get a blatantly false statement when there is an absurdly easy path to getting a true statement.

Not so, then, that “…we use the symbol ‘=’ to mean other things as well — other things, that is, in addition to identity as such…”  It is used to mean just identity. It would continue to denote identity even if, perversely, we chose to mention the expressions instead of using them, for we would end up with, not a true statement that relied on a different meaning of ‘=’, but with a false statement using ‘=’ to denote identity.  Notice that I am not exercising enough control here to avoid vacillating between using single quotation marks and double quotation marks to indicate I am talking about the symbol for the equals constant.

In spite of saying a number of false things on the way towards his conclusion, it’s hard to argue with that conclusion:   But saying this does not commit me to asserting that nothing is subtly wrong with it. I am too lazy now to try to dig and see if there is a subtle, non-obvious but not necessarily unimportant way in which Date is going wrong here; and anything I came up with would probably seem rather scholastic anyhow.

To sum up:  Equality means identity.  An expression of the form x = y is almost always shorthand for one of the form value_of(x) = value_of(y).

(p. 47)

Except I would add that even in the case of expression(x) = expression(y) — for example, ‘2x -1′ = ‘2x – 1′, equality also means identity.

We can perhaps forgive Date for using (with whatever qualms) one moment the symbol as a window through which he sees the thing denoted, but seeing the next moment nothing but a window become thoroughly milky and completely opaque.  For ordinary language can start out using a name, then shift mid-stream to mentioning the same occurrence of the name by the time one gets to the middle of the sentence.  I think the example is Quine’s:

Giorgione was so called because of his size.

And on that note, I will show one of Giorgione’s paintings:

300px-Giorgione_019

and since we are now on the topic of beauty, I will go all Plato’s SYMPOSIUM on my few (0, 1, 2…) readers:

936full-a-river-runs-through-it-screenshot

Maganda si Brad Pit.  Being that gorgeous should be illegal.  I will attempt to say that in Tagalog this way:  Dapat ilegal ang magiging ganoong maganda, but that is almost certainly either ungrammatical or unidiomatic (hindi pangwikian).  Perhaps someone with better Tagalog may correct me.

Update:  12/18/2012:  Made various corrections.


Semantic vs. Syntactic Arguments

In a previous post, playing the role of a would-be ordinary-language philosopher working in Tagalog (which language, to the the total consternation and perplexity of the Spanish grammarians during the 1600s, lacks the verb ‘to be’), I tried to provoke the suspicion that there is no single relation IS that could be pictured as follows:

IS (0)
THING PROPERTY
NUMBER( NAME(‘3’) ) PROPERTY( NAME(‘Prime’) )
CAR( NAME(‘Car With Serial Number 1235813’) ) PROPERTY( NAME(‘Red’) )
FLOWER( NAME (‘Rose With Barcode 3185321’) ) PROPERTY( NAME(‘Beautiful’) )
MINERAL_ITEM( NAME(‘Grain Of Salt Mentioned By Hegel’) ) PROPERTY( NAME(‘Cubical’) )
MINERAL_ITEM( NAME(‘Grain Of Salt Mentioned By Hegel’) ) PROPERTY( NAME(‘White’) )

Contra John Duns Scotus, for example, there is no single relation ‘contracts’ holding between a universal existing as always-already contracted into a particular. (To back up for a moment, the property denoted by NAME(‘three-sided’)is the universal denoted by NAME(‘three-sidedness’)existing as already contracted into a particular triangle.) Nor is there any other single relation which we can identify with the verb ‘is’. Or … letting my Tagalog ordinary-language suspicions run wild for the moment … so I will suppose.

There is no semantic relation (we are supposing) between the particular thing and the particular property.  But there is a syntactic relation between two names, pictured as follows;

IS (0)
THING PROPERTY
NAME(‘3’) NAME(‘Prime’)
NAME(‘Car With Serial Number 1235813’) NAME(‘Red’)
NAME(‘Rose With Barcode 3185321’) NAME(‘Beautiful’)
NAME(‘Grain Of Salt Mentioned By Hegel’) NAME(‘Cubical’)
NAME(‘Grain Of Salt Mentioned By Hegel’) NAME(‘White’)

Voila:  here is the distinction between semantic vs. syntactical arguments to a verb aka predicate that puzzled me in an earlier post.  NAME(‘3’) and NAME(‘Prime’) are syntactic predicates to the verb/predicate ‘is’.  NUMBER( NAME(‘3’) ) and PROPERTY( NAME(‘Prime’) ) are the semantic predicates to the verb ‘is’ — or would be if there were such a verb ‘is’ that took semantic arguments.

In the spirit of ‘let’s see how long I can get away with this’, let me propose the following chain of events for verbs such as eats that do take semantic arguments.  Consider a relation like the one pictured here:

EATS (0)
NAME_OF_PERSON_EATING NAME_OF_FOOD_ITEM_BEING_EATEN
NAME(‘Joe’) NAME(‘This egg’)
NAME(‘Khadija’) NAME(‘This souffle’)
NAME(‘Juan’) NAME(‘This fajita’)
NAME(‘Kha’) NAME(‘This bowl of Pho’)
NAME(‘Cliff’) NAME(‘This plate of Thai food with a 5-star Thai-spicy rating’)

When used in ordinary discourse, rather than mentioned as sentences with whatever syntactic properties, these tuples with their syntactic arguments get transformed into the following tuples with their semantic arguments:

EATS (0)
PERSON_EATING FOOD_ITEM_BEING_EATEN
PERSON( NAME(‘Joe’) ) FOOD_ITEM( NAME(‘This egg’) )
PERSON( NAME(‘Khadija’) ) FOOD_ITEM( NAME(‘This souffle’) )
PERSON( NAME(‘Juan’) ) FOOD_ITEM( NAME(‘This fajita’) )
PERSON( NAME(‘Kha’) ) FOOD_ITEM( NAME(‘This bowl of Pho’) )
PERSON( NAME(‘Cliff’) ) FOOD_ITEM( NAME(‘This plate of Thai food with a 5-star Thai-spicy rating’) )

If we think of the intransitive and transitive verbs ‘eats’ as really being the same verb on the semantic level (though it is not clear to me that they are the same), ‘Joe eats’ would be ‘Joe eats something’.  We can derive the corresponding tuple from  the EATS relation first by projecting on the attribute PERSON_EATING:

EATS(1)
PERSON_EATING FOOD_ITEM_EATEN
PERSON( NAME(‘Joe’) ) FOOD_ITEM( NAME(‘This egg’) )
PERSON( NAME(‘Khadija’) ) FOOD_ITEM( NAME(‘This souffle’) )
PERSON( NAME(‘Juan’) ) FOOD_ITEM( NAME(‘This fajita’) )
PERSON( NAME(‘Kha’) ) FOOD_ITEM( NAME(‘This bowl of Pho’) )
PERSON( NAME(‘Cliff’) ) FOOD_ITEM( NAME(‘This plate of Thai food with a 5-star Thai-spicy rating’) )

Then by doing a RESTRICT on Joe:

EATS(2)
PERSON_EATING FOOD_ITEM_EATEN
PERSON( NAME(‘Joe’) ) FOOD_ITEM( NAME(‘This egg’) )
PERSON( NAME(‘Khadija’) ) FOOD_ITEM( NAME(‘This souffle’) )
PERSON( NAME(‘Juan’) ) FOOD_ITEM( NAME(‘This fajita’) )
PERSON( NAME(‘Kha’) ) FOOD_ITEM( NAME(‘This bowl of Pho’) )
PERSON( NAME(‘Cliff’) ) FOOD_ITEM( NAME(‘This plate of Thai food with a 5-star Thai-spicy rating’) )

The existence of verbs that are sometimes transitive, sometimes intransitive  is what motivated Santorini’s distinction between semantic and syntactic arguments to a verb.

Although I have labored over making this distinction for an embarrassing amount of time, it becomes quite easy to make once one has the notion of a SELECTOR available as a conceptual tool.

I’d like to mention as a final note that possibly we should think of the arguments of eats as always already nested inside the selectors PERSON and FOOD_ITEM; in other words, the names are always transparent, so to speak, letting us see through them the persons and food items, the semantic arguments, named.  It is only under special circumstances — say when the transparent denoting function of the sentence breaks down … maybe one has been staring at the sentence for too long — that the selectors PERSON and FOOD_ITEM get stripped away and we see the names, the syntactic arguments, doing the denoting.  (The sentences always seem to be breaking down this way for C.J. Date in his article “SOME OPERATORS ARE MORE EQUAL THAN OTHERS” in his LOGIC AND DATABASES:  THE ROOTS OF RELATIONAL THEORY.  I get the funny feeling that for him a sentence or expression functions normally at first, but when he stares at it too long it suddenly loses its transparency and becomes an opaque relation between names.  See pages 42 and 45, and see if you get the same impression.)  This final note has been brought to you by the balefully compromised spirit of Martin Heidegger, which was nagging me as I wrote the above.

And now, in the spirit of Plato’s SYMPOSIUM, I would like to picture something a little less dry than the pictures of Relations shown above.  Today it is Kellan Lutz who is serving as my stepping stone, first, to the Relational Algebra, then, finally, to the form of Beauty itself:

Kellan_Lutz

(Rapturous sigh.  How can one live in this world when there is so much beauty in it?)

Update 12/16/2012:  Corrected some errors in the names of some of the Relations; tried to improve the flow of the writing.