Category Archives: Sets

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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Selectors And Semantic vs. Syntactic Arguments

In case anyone wonders (“feel free to come to the point when you finally decide what it is”), the point of the following ramblings is to arrive at a place where I can make a distinction between semantic arguments and syntactic arguments.  The point of making this distinction will become clear (or not) in a later post.  Making the distinction is part of my attempting to put in my own words the argument that Tagalog lacks a subject.

In the previous post, I argued (or claimed, or made the completely unsupported, nay, spurious assertion, as the case may be) that the semantics of Maganda si Robert Pattinson can also be given by the following statement in the database language Tutorial D:

GORGEOUS_EQUALS_GORGEOUS{THIS_ONE, THAT_ONE} where THIS_ONE = PERSON(NAME(‘Robert Pattinson’))

This statement includes the Selector PERSON(NAME(‘Robert Pattinson’)).  Let me unpack a bit what this is. Before I start, I’d like to point out that I THINK that it is  legal in Tutorial D to nest one selector inside another…

NAME(‘Robert Pattinson’) is a operator or function that takes the string ‘Robert Pattinson’ and selects one and only one name.  I will take the concept ‘selects’ as primitive here.  Any implementation of this selector in a physical computer would involve shuffling around ones and zeros until the computer spits out, i.e., returns, one member of the set NAME.  NAME would include strings, but subject to certain limitations.  For example, I assume a  name would have to be, at least, less than 1 billion characters long.  NAME would also include more than strings (that is, representations of text):  a name can be selected by a sound.  So NAME(<<some representation of a sound>>) could also select the name Robert Pattinson. (The reader will notice that I have not yet decided on how to represent, in the absence of a formal selector, a name as opposed to a string as opposed to the person himself…)

PERSON(NAME(Robert Pattinson)) would take the name selected by NAME(‘Robert Pattinson’) and return a member of the set PERSONS, i.e., Robert Pattinson himself.  I don’t know how a computer would implement this operator, but a human being would be implementing that operator in the following type of circumstance:  say, I am sitting in a restaurant.  Someone in the table next to me says:

 I hereby officially declare myself to belong to Team Edward because Robert Pattinson is just too gorgeous.

One part of that utterance, the part that I hear as the word ‘Robert Pattinson’, is the end point of a long causal chain that begins, say, when the parents of Robert Pattinson, after endless wrangling and indecision, finally agree to call their baby ‘Robert’; the doctor in the Maternity Ward crosses out the ‘baby boy’ in ‘baby boy Pattinson’ and writes in  ‘Robert’ on the birth certificate (call this the ‘baptismal event’) … endless events … a director or producer chooses the person named by ‘Robert Pattinson’ to play Edward Cullen in TWILIGHT … endless events…the person sitting at the table next to me sees TWILIGHT…he reads in a magazine he buys at the supermarket that Robert Pattinson played the part of Edward Cullen…he emits a set of soundwaves at the table next to me, which in turn trigger God-only-knows what processes in my brain, until I hear ‘…Robert Pattinson….’  That entire causal chain, ending up in the wetware of my brain, selects the person Robert Pattinson.  THAT’s the implementation of the selector PERSON(NAME(<<some representation of certain sound waves>>)).  Speaking metaphorically and a bit picturesquely, the selector spits out, or returns, Robert Pattinson himself, the flesh-and-blood Robert Pattinson who lives in (I would say ‘Valencia, California’, but that is where Taylor Lautner lives)…. Speaking literally, the selector selects Robert Pattinson himself.

(See Saul Kripke, who apparently never explicitly endorsed this causal theory of reference aka selection.  Gareth Evans would apparently deem this theory, as stated by me, to be naive, but it seems perfectly intuitive to me.)

Invocations of selectors produce literals (more accurately, I guess, are literals).  So whatever else Robert Pattinson himself may be, he is a literal value.

Let me take the liberty of allowing selector invocations as arguments supplied to the parameters of functions, so that we can replace x with the argument PERSON(NAME(‘Robert Pattinson’)) in the function x EQUALS x to produce a true proposition.  Below, I have identified, ala Chisholm, propositions with states of affairs in the world:  here, with Robert Pattinson being identical with Robert Pattinson.  This proposition gives us the semantics of the utterance “Robert Pattinson equals Robert Pattinson.”

I will therefore call the invocation of PERSON(NAME(‘Robert Pattinson’)) a semantic argument.  By contrast, the invocation of NAME(‘Robert Pattinson’), occuring inside an utterance, spoken or written, is a syntactic argument.  In this way, I make sense of the semantic arguments vs. syntactic arguments distinction I puzzled over in a previous post.

I do not know, of course, whether this is the distinction that Beatrice Santorini wanted to make.

I will end by making another homage to Plato’s SYMPOSIUM, according to which interest in Robert Pattinson, Taylor Lautner, Kellan Lutz et al ultimately leads to interest in the Relational Algebra, and from there, to the Form of Beauty itself:

Robert_Pattinson_2

Wow, I love that slightly-unshaven look…(the reader may  hear a rapturous sigh…)

Now, having briefly lapsed into a lower form of eros, I will go back to eros for the Relational Algebra in connection with Semantics….

Update:  After hitting the publish button, I saw this quote from the first Jewish Prime Minister of Great Britain:

The best way to become acquainted with a subject is to write a book about it.

Benjamin Disraeli

Or blog about it at length.


Some Gorgeous One Equals Robert Pattinson

Below, I have argued that (or, more accurately, attempted to provoke the Aha Erlebniss that)  the following three Tagalog sentences:

Titser ang babae.

Maganda ang lalaki.

Umalis ang babae.

…have as their most literal translation something like the following:

Some teacher one  equals the woman.

Some gorgeous one equals the man.

Some having left one equals the woman.

How would these sentences be expressed in the Relational Algebra?  Let me try to express “Some beautiful one equals Robert Pattinson” (I am switching from Team Jacob to Team Edward for the moment) in the Relational Algebra.  (Notice I am switching from ‘the man’ to ‘Robert Pattinson’.  Can I get away with this?)

A relation is a set of ordered pairs formed by taking the Cartesian Product of two sets, not necessarily distinct, and obtaining a subset (possibly identical with the entire set) of the set of ordered pairs.  Let’s form a particular EQUALS relation, GORGEOUS_EQUALS_GORGEOUS, by taking the Cartesian Product of the set GORGEOUS with the set GORGEOUS, then take from that Product the set of all those ordered pairs in which each member of the pair is identical with the other.  So that the relation can be more easily manipulated (conceptually), add in all the stuff necessary to turn this relation into a database relation, complete with tuples and attributes and all that good stuff.

GORGEOUS_EQUALS_GORGEOUS(0)
THIS_ONE THAT_ONE
Robert Pattinson Robert Pattinson
Taylor Lautner Taylor Lautner
Kellan Lutz Kellan Lutz
Brad Pitt Brad Pitt
Ashton Kutchner Ashton Kutchner

Restrict GORGEOUS_EQUALS_GORGEOUS to just the Robert Pattinson tuple:

GORGEOUS_EQUALS_GORGEOUS{THIS_ONE, THAT_ONE} where THIS_ONE = PERSON(NAME(‘Robert Pattinson’))
More attention needs to be paid to the literal selector PERSON(NAME(‘Robert Pattinson’)).  Will my worries about this, unarticulated here, eventually blow up in my face?

To get the relation pictured by:

GORGEOUS_EQUALS_GORGEOUS(1a)
THIS_ONE THAT_ONE
Robert Pattinson Robert Pattinson

Now project on the attribute THAT_ONEi in addition to performing the RESTRICT:

GORGEOUS_EQUALS_GORGEOUS{THAT_ONE} where THIS_ONE = NAME(‘Robert Pattinson’)

To get the relation pictured by:

GORGEOUS_EQUALS_GORGEOUS(1)
THAT_ONE
Robert Pattinson

(Imagine the surrounding white space as regnant with the matrix from which this relation sprints, namely, the base relation GORGEOUS_EQUALS_GORGEOUS.)

The above relation expresses the proposition that is also expressed in English as:

Some gorgeous one equals Robert Pattinson.

and that is also expressed in Tagalog, I claim, as:

Maganda si Robert Pattinson.

So:

Maganda si Robert Pattinson.

Some gorgous one equals Robert Pattinson

have the same semantics.  (Well, would have the exact same semantics if ‘gorgeous’ were exactly equivalent to ‘maganda’, which of course may be doubtful.)

Now, in the spirit of Plato’s Symposium (eros for gorgeous  young men inspires eros for the Relational Algebra and the Predicate Logic, and from there to the Form of Beauty itself), let me picture some of the members of that set which inspires my forays into the Relational Algebra.  These pictures are a bit more colorful than the pictures of relations shown above.

Do I really have to choose between Team Edward and Team Jacob?

12/04/2012:  Updated to remove problematic assertions about the semantics of ‘is’.


The Mystery Of The Missing IS: Or, Had John Duns Scotus Been An Ordinary-Language Philosopher Working In Tagalog


Below, I have tried to start incubating the suspicion that there is something fishy about treating ‘is’ as a predicate with two parameters accepting one argument each, i.e., a two-place relation.


Tagalog doesn’t have a verb ‘is’, no verb ‘to be’.  Given that more literal translations of Tagalog sentences often display the phrase ang noun phrase structure as:

 

phrase [is] ang noun phrase


For example:

Titser ang babae.

Maganda ang lalaki.

Umalis ang babae.

gets rendered as:

Teacher [is] the woman.

Beautiful [is] the man.

Having left [is] the woman.

or as I prefer (see my attempt below at eliciting the ‘aha erlebniss’):

Some teacher one  [is] the woman.

Some beautiful one [is] the man.

Some having left one [is] the woman.

…given that, one might think that, always, the suspect verb aka predicate aka relation is implicitly in effect in sentences with that structure.  The lack of a verb ‘to be’, of an ‘is’ in Tagalog that so perplexed the first Spanish grammarians of the language (so that, in their total confusion and lack of understanding, they tried to interpret the Tagalog inversion marker ‘ay‘ as the verb ‘is’, a confusion and misinterpretation that has had hilarious consequences lasting to this day), is always there, just unpronounced (or unwritten).  The space between ‘maganda‘ and ‘ang lalaki‘ in the written sentence, or the lack of interruption in the string of sounds (if that is how maganda ang lalaki gets pronounced — I am not strong enough presently in Tagalog to know) or the glottal interruption (if one exists between the ‘maganda‘ and ‘ang lalaki‘)  … the space, or lack of interruption in the continuous stream of sound, or the glottal, these are, as the case may be, an implicit sign of the two-place relation ‘is’.

Following Naylor, Schachter, and my own intuition, I have been treating the space, the lack of interruption in the continuous stream of sound, the glottal as an implicit equals.  For example, I prefer to translate the above three Tagalog sentences as:

Some teacher one  = the woman.

Some beautiful one = the man.

Some having left one = the woman.

Unlike ‘is’, however, which is (if there is such a critter) a two-place relation, ‘equals’ (alternatively, ‘=’ ) is, as I am about to show, a one place relation.  It is not just that the sign corresponding to ‘is’ is lacking in Tagalog:  the (real or putative) semantics of ‘is’ is lacking in Tagalog as well.  Tagalog is working with something completely different.

Clearly the ‘equals’ that is in play here is not given by the ‘equals’ in the following two-place relation:

 

THISTHAT

EQUALS (0)
Morning Star Evening Star
3 3
Rose With Barcode 3185321 Rose With Barcode 3185321
Clifford Wirt Clifford Wirt
The murderer of Jones The butler

…because in sentences such as Maganda si Taylor Lautner, the word ‘Maganda’  does not, at the moment of its utterance, specify, identify, locate, expose, or pick out any one particular thing.   ‘Maganda’ is equivalent to ‘Some beautiful one’, or the part of the formal sentence below that occurs before the ‘=’:

∃x ∈ MAGANDA: x = si Taylor Lautner.

The x that belongs to the set MAGANDA is left unspecified, unidentified, unlocated, unexposed, un-picked-out at the start:  Maganda … though it does get specified at the end:  …si Taylor Lautner.  But a two-place relation requires two identified, specified arguments for its two attributes.

Let me try to capture in D the sentence ‘∃x ∈ MAGANDA: x = si Taylor Lautner’.  Let me posit the following 1-place relation:


MAGANDA (0)
MAGANDANG_BAGAY
Taylor Lautner
Sunset at time t and place p
Rose With Barcode 3185321
Wine Red
The Taj Mahal
Haendel’s Umbra Mai Fu

Taking this relation as my springboard, I capture ∃x ∈ MAGANDA as MAGANDA{} (which gives us TABLE_DEE, or TRUE, or YES), then do a CARTESIAN PRODUCT of that with a restriction of MAGANDA:

with
MAGANDA{} as t_sub_0,
MAGANDA{MAGANDANG_BAGAY} where   MAGANDANG_BAGAY= ‘Taylor    Lautner’ as t_sub_1:
t_sub_0 X t_sub_1

CARTESIAN PRODUCT is a special case of JOIN.  TABLE_DEE JOIN r, where r is any relation, yields r.  So the D statement above yields:

MAGANDA (1)
MAGANDANG_BAGAY
Taylor Lautner

which expresses the semantics of the sentence ‘Maganda si Taylor Lautner’.  In this way, we get rid of the doubtful (I think) verb aka two-place relation ‘is’.

To sum up, a bit impishly:  the semantics of ‘is’ is different in Tagalog than in English because Tagalog really doesn’t have an ‘is’.  Later, I will try to develop this into part of an argument that Tagalog lacks a subject.  Tagalog’s lacking a verb ‘to be’ is related to its lacking a subject.

To stray back for a moment to philosophy:  were Duns Scotus an ordinary-language philosopher working in Tagalog, it may never have occurred to him to try to find a single relation (e.g. ‘contracts’ ) between the entity Beauty, as the argument on one side of the predicate ‘is’, and Taylor Lautner as the argument on the other side of the predicate, and so on for every other proposition formed by supplying arguments to the parameters x and y in the predicate x is y.

11/10/2012:  Updated to make a point a bit more clearly.

11/10/2012:  Updated to parenthetically add some snark about the first Spanish grammarians of the Tagalog language in the 1600’s.

 

Update:  11/25/2012:  Post grayed-out because I am dissatisfied with it.


Predicates And Semantic Roles

The type named in the heading of a relation would typically have to be defined partly in terms of a Semantic Role.  In the Relation INVITES, for example, the person inviting belongs to the set INVITERS, which is a subset of AGENTS, i.e., the set of entities capable of acting volitionally.  AGENTS is in turn is a subset of ACTORS.  An entity belongs to ACTORS when, although not necessarily capable of volition, it “… in some intuitive way performs, effects, or controls the situation.”  (I’ve lost track of the source.)  In turn, the person invited belongs to the set INVITEES, which is a subset of PATIENTS, entities acted upon.

INVITES (0)
INVITER INVITEE
Andrew Chris
Andrew David
Brian Eddie

Agents, Actors, and Patients are all, of course, Semantic Roles.


The Predicate Returns A Relation

We have seen that the predicate:

x is to the left of y

is mapped to the truth value TRUE when Charles is substituted for x and Genghis Khan is substituted for y.  The Relation TO_THE_LEFT_OF comprises all true propositions and only true propositions that get generated when values are substituted for x and y.  So the predicate is a function whose range is the truth value TRUE for every proposition that is included in the relation, and FALSE for every proposition that is not included in the relation.

I think, however, that we would get a slightly simpler account if we see the predicate as a function returning Relations comprising the single proposition TRUE, or the single proposition FALSE.  In the Relational Algebra, we would get a relation comprising the single tuple (and therefore proposition) TRUE if, after doing the Restriction that gives us:

Charles is to the left of Genghis Khan.

we then projected on the null set of attributes (“columns”).  We would then end up with Chris Date’s TABLE_DEE, that is, the Relation with cardinality 0 (o attributes, that is, 0 “columns”) and a single tuple.  TABLE_DEE is the Relation that corresponds to (I guess I should say ‘is identical with’) the weird classical logic proposition TRUE.  The predicate returns the proposition TRUE wrapped in the Relation TABLE_DEE when the Charles and Genghis Khan substitution is made.

Correspondingly, when John is substituted for x and Genghis Khan is subsituted for y, so that we get:

John is to the left of Genghis Khan.

the Restriction selects no tuple in the Relation TO_THE_LEFT_OF.  We then have a Derived Relation with a cardinality of 2 (i.e., the Relation has 2 “columns”) holding the null set of tuples.   If we then project on the null set of attributes, we end up with a Relation of cardinality 0 comprising 0 tuples.  Chris Date calls this Relation TABLE_DUM, and it holds the tuple, that is to say, the proposition FALSE.  The predicate returns the proposition FALSE wrapped in the Relation TABLE_DUM when the John and Genghis Khan substitution is made.

Thinking of the predicate as returning either TABLE_DEE or TABLE_DUM simplifies things a bit, because it means we never have to leave the Relational Algebra when modeling the predicate.  Everything gets explained in terms of just one set of operations, the operations of the Relational Algebra.

 

 

 


The Relational Algebra Gives Us Something (Or Somebody, Or At Least Someone)

Now onto trying to show how the Relational Algebra gives us ‘something’, ‘somebody’, ‘someone’, and so on.

When I talk about database relations in the following, I am, unless I state otherwise, talking about the abstract object, not those relations concretely realized in an RDBMS.

A brief explanation of the Relational Algebra:  Posit a world all of whose people are members of the set {John, Cliff, Charles, Genghis Khan, Leon Trotsky}.  Moreover, suppose that currently, the predicate:

 x is standing to the left of y

generates the Database Relation pictured below when all the members of this set are substituted for the parameters x and y:

TO_THE_LEFT_OF (0)
PERSON_ON_THE_LEFT PERSON_ON_THE_RIGHT
Charles Genghis Khan
Dan Leon Trotsky
Cliff Genghis Khan

(The above picture, by the way, is just that — a picture of the Relation.  It is not the Relation itself.)  As indicated by the number 0 in the name, this Relation is a base Relation, i.e., what we have before any operations are applied to it.

The Relational Algebraic operation RESTRICT is a function that takes the Relation pictured above as input and produces another Relation as output.  For example, the following RESTRICTion, expressed in Tutorial D:

TO_THE_LEFT_OF where PERSON_ON_THE_LEFT = ‘Charles’;  (Yes, I’ve suddenly gone from the flesh and blood Charles as member of a set to the name ‘Charles’; God only knows what confusions this sudden shift will introduce.)

generates the Relation pictured below:

TO_THE_LEFT_OF (1)
PERSON_ON_THE_LEFT PERSON_ON_THE_RIGHT
Charles Genghis Khan
Dan Leon Trotsky
Cliff Genghis Khan

The operation RESTRICT has given us a Relation comprising a single proposition expressed by the sentence ‘Charles is standing to the left of Genghis Khan.’  As indicated by the number 1, this is a Derived Relation, produced as output from a function that took as input the Base Relation.  The charcoal-grayed out portions of the picture are meant to convey that the derived relation is tied to the base relation in a way in which I will discuss later.

As with RESTRICT, the Relational Algebraic operation PROJECT takes the Base Relation as input and generates a Derived Relation as output.  The following RESTRICT and PROJECT operations, expressed in Tutorial D:

(TO_THE_LEFT_OF where PERSON_ON_THE_LEFT = ‘Charles’ ){PERSON_ON_THE_LEFT}

generates the Relation pictured below:

TO_THE_LEFT_OF (2)
PERSON_ON_THE_LEFT PERSON_ON_THE_RIGHT
Charles Genghis Khan
Dan Leon Trotsky
Cliff Genghis Khan

whose body is the set containing the tuple or proposition expressed by the sentence “Charles is to the left of somebody.”

But wait — all we see in this picture is the value Charles.  (Or, more precisely, the name ‘Charles’ appearing as a set of black pixels on a screen.)  Isn’t this a tuple in a one-place relation?  And if it is, wouldn’t it be a proposition belonging to one-place relation, a proposition such as “Charles laughs”, or “Charles runs”, or “Charles eats”?

Well, if it were such, it could be any proposition belonging to a one-place relation.  The only way to constrain which proposition this tuple is to just one proposition is to place it in its context, the source from which it is derived, i.e., the base relation TO_THE_LEFT_OF.  By performing the Projection, we are for the moment blacking-out the identity of Genghis Khan, the person to whom Charles is to the left, so that we can focus on the identity of Charles.  But we haven’t forgotten that we are working with the relation TO_THE_LEFT_OF, so we know that Charles is to the left of somebody.  We haven’t suddenly switched to the relations LAUGHS, or RUNS, or EATS.

To turn for the moment for relations concretely implemented in an RDBMS running in some stuff made out of the same substance as the red paint on the Golden Gate Bridge, complete chaos would ensue, the world would become a topsy-turvey place, objects would start falling up, if, say, a Projection on EMPLOYEE_NAME in the EMPLOYEE (select EMPLOYEE_NAME from EMPLOYEE) would result, not in the set of people employed by the company (more precisely, the set of propositions ‘John, employee of Widgets_R_US’, ‘Jesse, employee of Widgets_R_US’, and so on), but the set of people designated to live on Mars one moment, the set of ambassadors to Vietnam the next moment, and the set of of Pulitzer Prize winners the third moment.

So the meaning of a Projection on an attribute (“column”) of a relation is constrained by the relation from which it is standing out (“projecting”), so to speak.  The derived relation never ceases to, well, derive its meaning from the base relation.  It never ceases to be a derived relation.  Charles never ceases to be one member of a pair whose member on his right is being ignored or blacked-out for the moment.

(Compare this argument with C.J. Date’s argument in LOGIC AND DATABASES, pp. 387-391.)

Let’s trace then what happens, in this relational model, when we plug in Charles to replace x in the predicate:

Person x, to the left of somebody

The ‘somebody’ is not a parameter — no argument gets plugged into it — but it along with the x indicate that the base relation we are dealing with is TO_THE_LEFT_OF.  It tells us that one of the ‘central participants in the situation’ is some person to the right.  The relevant Relational Algebra Operations — the relevant RESTRICT and the relevant PROJECT — are then performed to generate the proposition:

Charles, to the left of somebody.

According to the Closed World Assumption, a Relation contains all and only those tuples — those propositions — those states of affairs — that obtain, and for which plugging in arguments to the parameters of the predicate defining the Relation results in a true sentence.  Therefore, each tuple in the Relation is paired with the truth value TRUE, and of course, within the Range comprising the two truth values, only the truth value TRUE.

So the set of tuples in a Relation and the set of Truth Values is a function.  So, finally — if I may end this string of ‘therefores’ and ‘so’s’ (“Feel free to come to the point when you finally decide what it is, I hear someone say”), when a single tuple is selected, as was done when the RESTRICT and PROJECT were performed on the Relation TO_THE_LEFT_OF, we can see this as the application of the function on that tuple, an application which returns TRUE.  So (this really is the final ‘so’ — I promise) plugging in the argument ‘Charles’ into the parameter x in the predicate:

x is to the left of somebody

triggers a RESTRICT and PROJECT on the Relation TO_THE_LEFT_OF, which in turn constitutes a selection of a single tuple in that relation, which in turn returns TRUE, which lets us regard the predicate as a function returning TRUE when ‘Charles’ is plugged into the parameter marked by x.

Just so, when the RESTRICT and PROJECT fail to select a tuple, as it does when we substitute ‘John’ for x (John is standing to the right of everyone else, including Genghis Khan), FALSE is returned.

Voila!  We now we have somebody (or, as the case may be, nobody).

It is clear that the predicate:

x is to the left of y

can be treated the same way.

Treating verbs aka predicates relationally this way — that is, as functions implemented by Relations and operations on Relations — has two advantages over simply seeing them as functions in the way described by Kroch and Santorini.  First, we get a semantics for ‘somebody’, ‘something’, etc.  Second, we have a way of conceptualizing in terms of operations of the Relational Algebra the select that occurs when, to use the verb laughs as our example, Luke is selected and the truth value TRUE is returned.  The notion of select is no longer a primitive.

 

Updated on 05/10/2012 to correct an obvious oversight.