Category Archives: Verbs

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.

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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.


That Strange Predicate/Relation IS

The predicate IS has two parameters.  Placing arguments in those parameters produces something like the following Relation:

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

This is a SINGLE relation, one may note, just as INVITES and TO_THE_LEFT_OF are. But while the relations INVITES and TO_THE_LEFT_OF are fairly easy to get one’s mind around, IS is a more difficult case. What is the relation between a property and the thing of which it is the property? Should we say that the property “inheres” in the thing? (Whatever “inheres” means.) Should we follow Plato and think of the relation between thing and property as analogous to the relation between reflection in the mirror and the thing or person reflected? So that the thing is a wholly relational entity wholly dependent upon something more real that exists independently, i.e., the property existing as a Platonic Form? Should we be more Aristotlean and think that, while yes, a given property (e.g. RED, e.g. PRIME) is one thing, not many, it is always already “contracted” (the ‘contracted’ business always makes me think of the old freeze-dried instance coffee commercials … the property gets “sucked” into the thing accompanied by the corresponding sound) ala John Duns Scotus into (but where does the ‘into’ come from? Does this mean ‘inhere’?) the thing so that it never exists independently of the thing? So that it has a “unity less than numerical?” (Source of the ‘unity less than numerical’ thing comes from some writing of Duns Scotus which I do not remember.) Should we think, along with William of Ockham, that it is nonsense to think of a single thing, e.g., the property RED, as existing in several places at the same time, so that we have to think of the red of the car and the red of the rose petal as in fact two different properties, even if they exactly match the same color sample held by the Interior Decorator? (So that ‘Red’ in the Relation above would always have to be marked by a number serving as an index?)

Or maybe the Relation IS is not a real Relation at all, but an artifact of a Word. Given the Word ‘is’, we think there is a corresponding Predicate generating Propositions which, when true, form a Relation. But in reality there is no such Relation. Perhaps?


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.


And One More Thing Before I Move On

It seems to me Kroch’s and Santorini’s rendition of this function:

(1)     [[ invite ]] = { Chris Andrew T ) ,
  David Andrew T ) ,
  Eddie Andrew F ) ,
  Chris Brian F ) ,
  David Brian F ) ,
  Eddie Brian T ) }

is off, since in a function the first entity in a given ordered pair can be mapped to one and only one entity.

Shouldn’t the function be represented this way:

(2)     [[ invite ]] = { Chris { ( Andrew T ) , ( Brian F ) , . . . }
  David Andrew T ) , ( Brian F ) , . . . }
  Eddie Andrew F ) , ( Brian T ) , . . . }
}                  

?

Or am I missing something obvious?