Category Archives: Propositions

Aristotle’s Sea Battle Argument

In a rough draft of a blog post at work whose real topic had, of course, precious little to do with Aristotle, I playfully tried to explicate his Sea Battle argument to an audience of techies as follows:

The following statement (call it p) is necessarily true:

Either there will be a sea battle tomorrow [at location l], or there will not be a sea battle tomorrow [at location l].

At least one, or possibly both of the constituents of an OR statement must be true if the statement is true.  If the OR statement happens to be an ‘A OR not A’ statement, at most one of  the constituent statements can be true.  What is more, since ‘A OR not A’ must be true, one of the constituent statements must be true.  So either

There will be a sea battle tomorrow [at location l]

is true, or:

There will not be a sea battle tomorrow [at location l]

is true.

But which one?

Suppose that ‘There will be a sea battle tomorrow [at location l]’ is the constituent proposition that is true.  (Call this constituent proposition c1.)  One may already have been struck by the Aha Erlebniss that the sea battle will not  fail to happen (and in fact cannot fail to happen) tomorrow at location l.  (henceforth ‘at location l‘) will be understood.)  But my mind and my imagination feel the presence of a gap between c1 and ‘the sea battle cannot fail to happen tomorrow.’  When I try to jump from the first to the second, I feel a bit as if I were plunging into a void.  The following thought experiment is an attempt to bridge that void,

Start Of Thought Experiment:   To avoid complications involving indexicals, suppose that today, at time t0, I say:

A sea battle happens at time tn.

From the standpoint t0, tn is a point in time that will roll by tomorrow.  Could my statement stop being true at t0+1 (0+1 < n)?  Don’t be silly — of course not.  Someone’s statement ‘The cat (Sylvester, with CAT_ID 347434395) is on the mat (the medieval Persian mat with MAT_ID 84541) at 12:01 pm, October 31, 2014’ never ceases to be true, assuming it was true at 12:01 pm, October 31, 2014.  Ditto my sea-battle statement.  Could my sea-battle statement suddenly stop being true at to+2?  No, of course not.  And so on for every time point starting from t0 and going up to tn.  My statement will be equally true at tn – 1 as well as at time tn. Throughout, it remains true that the sea battle will happen at tn.  There is no room left for the sea battle NOT to happen at tn.

In fact, what would it mean for that statement suddenly to become not true, at some point between to and tn?  Well, suppose — doubtlessly per impossible — that the chain of one set of causes leading to a set of effects serving as causes for yet another set of effects ceases — say, at tn-1 — to be deterministic.  That chain continues unbroken until, abruptly at tn-1, it becomes a flip of nature’s coin whether the sea battle happen or not.  Then, it seems to me, the truth of ‘A sea battle happens at time tn’ ceases to be defined.  The statement is neither true nor false.  Therefore, the statement would be not true, though it would not be false either.

Or again, suppose that the chain continues unbroken until suddenly, at time tn, we end up with (again, per impossible, I am sure) with a weird quantum Schroedinger’s sea battle:  the sea battle is simultaneously in a state of happening and not happening at tn.    In this case, my intuition is, the truth value of my sea battle statement would be undefined at t0 as well as at tn.  End Of Thought Experiment.

So assuming there is a chain of causes working deterministically from t0 to tn, my sea battle statement is definitely true at t0.  And there is no way that the sea battle will fail to happen at tn.  The chain of deterministic causes (assuming this exists) is what gives sense to the idea that my sea battle statement has a definite truth value at t0 — that is is true (false) at that time-point.

This is Fatalism.  Fatalism is often thought to entail that we have no Free Will.  Aristotle comes to this conclusion, and panics.   (At least according to my explication of this stuff to my fellow geek colleagues.)  “Oh my god!!!!!!….er….I mean….oh my Zeus!!!!!  If there is no Free Will, then that loud sucking sound you hear is my ETHICS going down the drain!!!!!  Quick!!! Quick!!!!! Think of something!!!!!!’  (I have to admit that my translation of the ancient Greek here is a trifle free.)  So to save his ethical theory Aristotle decides to assert that while the total original proposition, p, is necessarily true, the truth value of both of its constituents is undefined.  Neither of its constituents is either true nor false.

But I do not see how this (the constituents’ not having a definite truth value) could be so unless the sea battle’s happening (or failing to happen) tomorrow is a matter of nature’s flipping the coin.  Aristotle cannot be right.

I say ‘Aristotle cannot be right’ in full confidence, as a matter of black and white.  Nonetheless, just a little shade of gray, a tiny sliver of doubt, does enter here.  The laws of nature are supposed to be deterministic on the level of apples and triremes, but non-deterministic on the level of protons and electrons (and for all I know on the level of quarks as well). On the micro level, nature is (if I understand this stuff correctly) constantly tossing a coin.  Although one is not supposed to mention quantum physics in a philosophical discussion unless they (intentional use of ‘they’ as a singular gender-neutral pronoun) have completed at least 8 graduate courses in quantum physics (with no grade lower than a B+ in any of them), I do have to at least wonder quantum weirdness might invade the causal chain leading to the sea battle’s occurring (failing to occur) tomorrow in such a way as to make it only 99.9999999999999999999999999999999999999999999999999999999999% probable, not 100% probable, that the sea battle will happen (fail to happen) tomorrow.  Is this enough to blast away the bridge that leads from the present to the future that lets us say that a statement about the future uttered now is either true or false?  I will leave that as a nagging question leaving in its wake just the tiniest whiff of doubt.

* * * * *

If Aristotle were right, then either p is not in fact an OR statement (it only looks like one), which seems rather counter-intuitive to me), or normal classical logic fails to hold for the future.  Contrary to normal, classical logic, it would not be the case that an OR statement is true if and only if at least one of its constituent statements is true.  This would hold only for statements about the present.

But in that case statements such as ‘If this apple drops from the tree under which I am sitting, this apple will splat onto my head in one second’ (call this the ‘apple if-then statement) will not have a defined truth value.  Reducing to ‘Either this apple does not drop from the tree under which I am sitting, OR this apple will not splat on my head in one second’ (‘if p then q’ is the same as ‘not p or q’),  So the truth value of the total apple if-then statement will be undefined because, being a statement about the future, the truth value of ‘this apple will splat on my head in one second’ is undefined.

So if we restrict normal, classical logic to just the present, the number of interesting statements it rules over will become awfully restricted.  Normal, classical logic will become a parlous affair, just as pitiful as the crowning of John Cantacuzenus  and Irene, Andronicus Asen’s daughter in the waning days of the Byzantine Empire.  As related in C.P. Cavafy’s poem Of Colored Glass:

As they had very little in the way of precious stones

(our wretched dominion’s poverty was great

they wore artificial ones.  A heap of bits of glass,

scarlet, green or blue.

I always end my philosphical/logical posts with an homage to Plato’s SYMPOSIUM, for which purpose I will use Ashton Kutcher (swooning, rapturous sigh) yet one more time:

Ashton_Kutcher_0

Look at those stunningly beautiful brown eyes!!!  How can anyone get any work done with beauty like this walking the earth?

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Semantic Arguments vs. Adjuncts (Revised)

This is a version of the post below, revised so as to try to eliminate a number of confusions.

The Wikipedia article Argument (linguistics) starts its discussion of the argument/adjunct distinction by asserting that an argument is what is demanded by a predicate to complete its meaning, while an adjunct is not so demanded.  For example, if someone asks me “What is Joe eating?” my answer would be drastically incomplete if I replied “eats.”  My answer would still be drastically incomplete if I supplied just one argument, ‘Joe’, to say ‘Joe eats.’  Only when I supply a second argument, say, ‘a fried egg’, would my reply not create a sense of a question ludicrously left hanging and an answer simply not given.  The predicate _eats_ has two parameters ( shown here as ‘_’) demanding two arguments, such as  ‘Joe’ and ‘a fried egg’ for my reply to make any sense.

( This example, of course, is my own; I am offering it (maybe tendentiously?) in order to make drawing certain conclusions more natural. )

‘[I]n the kitchen’, however, is an adjunct, since nothing would be left ludicrously left hanging in the air were I to leave that phrase out of the proposition “Joe eats a fried egg in the kitchen.”  The predicate eats does not have a parameter demanding something like ‘in the kitchen’ as an argument.

This criterion — i.e., what is demanded by a predicate to complete its meaning … henceforth I will call this the ‘demands criterion’ — runs into trouble when one notices that sometimes eats demands two arguments, but sometimes demands just one.  One might say:  “Joe goes into the kitchen.  Joe is ravenous.  Joe sees food.  Joe eats.”  ( Imagine a novelist or short-story writer working in a certain style.)  The argument ‘a fried egg’ is not demanded in this particular piece of discourse.

But if ‘a fried egg’ is an argument, not an adjunct to eats, it would seem one would  have to abandon the ‘what is demanded by a predicate to complete its meaning’ criterion and find another criterion for what is to count as an argument and what is to count as an adjunct.  This a contributor (doubtlessly not the same person who put forward the ‘demands’ criterion) to the Wikipedia article cited above tries to do.

But if one wants to retain the demands criterion, they (I am intentionally using ‘they’ as a genderless singular pronoun) can assert that two different predicates, each with a different number of parameters, may get invoked when someone utters  ‘eats’ in a stretch of discourse.  Sometimes the one-place predicate _ eats is invoked, sometimes the two-place predicate _eats_.   Which predicate one uses is optional, depending upon what they feel is called for by the situation and what they want to do with the predicate.  Sometimes the context forces one to use, for example, the two-placed predicate (for example, in answer to the question ‘Joe is eating what?’; sometimes which predicate one invokes is purely a matter of choice.

If all of the predicates demand a certain argument (for example, ‘Joe’ in ‘Joe eats’), what is so demanded is an argument that is not also an adjunct.  If not all of the predicates demand a given argument (‘fried egg’, ‘in the kitchen’), that argument is an adjunct.  In this way, the demands criterion is rescued.

I picture the relations formed by these predicates as follows:

One-place relation formed by _eats:

EATS
PERSON_EATING
PERSON( NAME(‘Joe’) )
PERSON( NAME(‘Khadija’) )
PERSON( NAME(‘Juan’) )
PERSON( NAME(‘Kha’) )
PERSON( NAME(‘Cliff’) )

Here the key is, of course, PERSON_EATING.  The ellipses ‘…’ indicate all the further tuples needed to make this relation satisfy the Closed World Assumption.  (The Closed World Assumption states that a relation contains all and only those tuples expressing the true propositions generated by completing the predicate with the relevant argument(s).)

Two-place relation formed by _eats_:

EATS
PERSON_EATING FOOD_ITEM_BEING_EATEN
PERSON( NAME(‘Joe’) ) FOOD_ITEM( NAME(‘This fried 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’) )

Here the relation formed by _eats_ is a subtype of the supertype formed by _eats.  That is to say, PERSON_EATING is a unique key in this relation, but it is also a foreign key to the PERSON_EATING attribute of the relation formed by _eats.

This means of, course, that in each tuple there is just one thing that the person is eating.  This constraint would be natural enough if one restricts the now of the present tense eats enough so that only one thing could possibly be getting eaten, for example, the egg one piece of which Joe is now bringing to his mouth via a spoon.  But, of course, if one stretches out this now enough so that our hypothetical author could write:   “Joe goes into the kitchen.  Joe is ravenous.  Joe eats a fried egg, an apple, and a salad,” one could not treat the one-place relation as a subtype of the two-place relation.  I think the solution in this case would be to treat what gets eaten as a meal, a meal comprising one or more items.  The meal then could be treated relationally the way an order and its order-items get treated, the orders going into one relation, and orders and order-items going into another, with the orders and order-items together comprising a unique key.

The predicate _eats_ _ (as in ‘Joe eats the fried egg in the kitchen’) can be treated the same way.  And so on for any number of possible adjuncts that a predicate might accept.

If I can get away with this move, then, an adjunct would be any argument that is 1) accepted by a predicate in which the corresponding relation is a subtype of another relation, and 2) the parameter which takes that argument corresponds to an attribute in the subtype relation which is not a foreign key of the supertype relation.  An adjunct then is one kind of argument.  Non-adjunct arguments (arguments that are just arguments, arguments simpliciter) correspond to a unique key in a supertype relation; adjuncts in turn are arguments not corresponding to any attributes in the subtype relations that are foreign keys to that unique key in the supertype relation.

Notice how this treatment of arguments vs. adjuncts (that is to say, arguments that are just arguments and arguments that are also adjuncts) corresponds to the way “optional (nullable) columns” in SQL tables get turned into actual relations, which cannot contain “null values”:

SQL Table (what is eaten is an optional or “nullable value”):

EATS
PERSON_EATING FOOD_ITEM_BEING_EATEN
Joe  Fried egg
Khadija
Juan
Kha Bowl of Pho
Cliff
 …

Here PERSON_EATING is a not-null column, and FOOD_ITEM_BEING_EATEN is a “nullable” column.

This looks like a single relation with an optional parameter (FOOD_ITEM_BEING_EATEN).  So if one both accepts the demands criterion and takes the  SQL table as their cue, PERSON_EATING would be an argument because it is not optional, i.e., always demanded and FOOD_ITEM_BEING_EATEN would be an adjunct because it is optional.  But then one has no way of accounting for when FOOD_ITEM_BEING_EATEN isn’t optional — for example in answering the question ‘what is Joe eating’?  (Compare with the COMMISSION column in the EMP table of Oracle’s sample SCOTT schema when the employee is a salesman.)  One would either have to try to explain away — an impossible task? — the times when eats surely seems to demand not one, but two arguments, or they would have to give up the demands criterion as the way to distinguish between arguments and adjuncts.

But of course SQL is confused.  The SQL table above is mushing together two different relations, the relation formed by _eats and the relation formed by _eats_.  Disentangle the two relations, and you get a two-fer.  You get rid of the nulls, and you also rescue the demands criterion for distinguishing between arguments simpliciter and arguments that are adjuncts.

When you disentangle the relations, you can see that what is optional, when one is talking about adjuncts, is not the attribute value (e.g., fried egg), but which predicate one invokes when they say eats.  To put it a different way, the attribute value is optional only because the predicate is.

I submit, then, that treating a verb as invoking different predicates whose corresponding relations are involved in subtype/supertype relationships does away with the confusing situation that challenges the demands criterion:  i.e., the initially confusing fact that sometimes an argument seems to be demanded for the verb, and sometimes it seems not to be.

Today’s homage to Plato’s SYMPOSIUM is Channing Tatum (aka Magic Mike) again, as in the previous post.

Channing_Tatum_BlackAndWhite

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


Semantic Arguments Vs. Adjuncts

The Wikipedia article Argument (linguistics) starts its discussion of the argument/adjunct distinction by asserting that an argument is what is demanded by a predicate to complete its meaning, while an adjunct is not so demanded.  For example, if someone asks me “What is Joe eating?” my answer would be drastically incomplete if I replied “eats.”  My answer would still be drastically incomplete if I supplied just one argument, ‘Joe’, to say ‘Joe eats.’  Only when I supply a second argument, say, ‘a fried egg’, would my reply not create a sense of a question ludicrously left hanging and an answer simply not given.  The predicate _eats_ demands two arguments, such as  ‘Joe’ and ‘a fried egg’ for my reply to make any sense.

( This example, of course, is my own; I am offering it (maybe tendentiously?) in order to make drawing certain conclusions more natural. )

‘[I]n the kitchen’, however, is an adjunct, since nothing would be left ludicrously left hanging in the air were I to leave that argument out of the proposition “Joe eats a fried egg in the kitchen.”  The predicate eats does not demand that argument.

This criterion — i.e., what is demanded by a predicate to complete its meaning … henceforth I will call this the ‘demands criterion’ — runs into trouble when one notices that sometimes eats demands two predicates, but sometimes demands just one.  One might say:  “Joe goes into the kitchen.  Joe eats.”  ( Imagine a novelist or short-story writer working in a certain style.)  Although one could just as well say “Joe goes into the kitchen.  Joe eats a fried egg”, the argument ‘a fried egg’ is not demanded in this particular piece of discourse.

So if one wants to maintain that the predicate eats takes two arguments, they would  have to abandon the ‘what is demanded by a predicate to complete its meaning’ criterion and find another criterion for what is to count as an argument and what is to count as an adjunct.  This a contributor (doubtlessly not the same person who put forward the ‘demands’ criterion) to the Wikipedia article cited above tries to do.

But if one wants to retain the demands criterion, they can assert that two different predicates may get invoked, depending upon the context, depending upon the circumstances, when someone utters the word ‘eats’ in a stretch of discourse.  ( I am not clearly distinguishing between predicate and word here; perhaps I don’t necessarily need to just right here.)  When one invokes the predicate in order to answer the question ‘What is Joe eating?’, invoking the predicate creates a proposition, or tuple, in a 2-place relation.  In circumstances in which nothing is left ludicrously hanging in the air when one says ‘Joe eats’, the predicate creates a proposition, or tuple, in a 1-place relation.  There are two different predicates that may get invoked when one utters ‘eats’.  And depending upon which predicate gets invoked, ‘a fried egg’ is either an argument or an adjunct.

Two-place relation (demands what is eaten to complete the meaning):

EATS
PERSON_EATING FOOD_ITEM_BEING_EATEN
PERSON( NAME(‘Joe’) ) FOOD_ITEM( NAME(‘This fried 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’) )
PERSON( NAME(‘Cliff’) ) FOOD_ITEM( NAME(‘This strip of bacon’) )

Here the key is composite, comprising both PERSON_EATING and FOOD_ITEM_BEING_EATEN, since we would may want to answer the question “What is Cliff eating?’ with “Cliff eats a fried egg and Cliff eats a strip of bacon.”

One-place relation (does not demand what is eaten to complete the meaning):

EATS
PERSON_EATING
PERSON( NAME(‘Joe’) )
PERSON( NAME(‘Khadija’) )
PERSON( NAME(‘Juan’) )
PERSON( NAME(‘Kha’) )
PERSON( NAME(‘Cliff’) )

Here the key is, of course, PERSON_EATING.

Sometimes what Joe eats is a ‘core element of the situation’, sometimes it is not.  In a possible world there exists a tribe for whom the amount of  energy pounded into the ground by John’s running is a core element of the situation runs, such that something is left ludicrously hanging in the air when one simply says ‘John runs’ and not (to invent a new syntactic marker, ‘tha’, which expresses ‘the energy absorbed by the ground when John runs”’, just as ‘to’ expresses ‘the place to which John ran’ ) ‘John runs tha 1,000 <<some unit of energy>>’.

When what is eaten is an adjunct, not an argument, one can, I think, treat the attribute PERSON_EATING in the two-place relation as a foreign key dependent upon the  PERSON_EATING attribute in the one-place relation.   would be both a unique key in that relation and a foreign key to the one-place relation.  This kind of design is, of course, how one would avoids “nulls” or “optional values” in a SQL table like the following:

SQL Table (what is eaten is an optional or “nullable value”):

EATS
PERSON_EATING FOOD_ITEM_BEING_EATEN
Joe  Fried egg
Khadija
Juan
Kha Bowl of Pho
Cliff
Cliff

Yes — there is a certain oddness, a certain ugliness, to having Cliff suffer from two “null values”.  Maybe there is something fishy about the SQL idea of a “null value”?  But the SQL table does convey the idea that an adjunct is an optional value, while an argument is required.  After conveying this idea, we can get rid of the SQL table with its dubious nulls and replace it with the two-place relation EATS whose PERSON_EATING attribute is a foreign key to the one-place relation.

Once can of course add other adjuncts by creating new relations.

EATS
PERSON_EATING FOOD_ITEM_BEING_EATEN IN ORDER TO
PERSON( NAME(‘Joe’) ) FOOD_ITEM( NAME(‘This fried egg’) ) REASON( NAME(‘Gain Nutrition’) )
PERSON( NAME(‘Khadija’) ) FOOD_ITEM( NAME(‘This souffle’) ) REASON( NAME(‘Gain Nutrition’) )
PERSON( NAME(‘Juan’) ) FOOD_ITEM( NAME(‘This fajita’) ) REASON( NAME(‘Gain Nutrition’) )
PERSON( NAME(‘Kha’) ) FOOD_ITEM( NAME(‘This bowl of Pho’) ) REASON( NAME(‘Gain Nutrition’) )
PERSON( NAME(‘Cliff’) ) FOOD_ITEM( NAME(‘This plate of Thai food with a 5-star Thai-spicy rating’) ) REASON( NAME(‘Show how macho he is’) )
PERSON( NAME(‘Cliff’) ) FOOD_ITEM( NAME(‘This plate of Thai food with a 5-star Thai-spicy rating’) ) REASON( NAME(‘Show how much pain and suffering he can endure’) )
PERSON( NAME(‘Cliff’) ) FOOD_ITEM( NAME(‘This strip of bacon’) ) REASON( NAME(‘Indulge in a guilty pleasure’) )

Here of course, the key is PERSON_EATING, FOOD_ITEM_BEING_EATEN, and IN_ORDER_TO.

This is the way of treating the argument/adjunct distinction that I prefer at the moment, possibly with no good argument for preferring this way to the alternative. The alternative that is at the back of my mind as I write this is something like the following:  there is only one predicate eats, which is a two-place relation.  Or rather, there is only one primary, non-derived predicate eats.  In those cases in which the what-is-eaten argument is optional (so we are giving up the demands criterion for what is to count as an argument), we are projecting on the relation EATS on the PERSON_EATING attribute, to generate propositions such as “Joe eats something.”

EATS(1)
PERSON_EATING SOME ATTRIBUTE
PERSON( NAME(‘Joe’) ) Some thing or things
PERSON( NAME(‘Khadija’) ) Some thing or things
PERSON( NAME(‘Juan’) ) Some thing or things
PERSON( NAME(‘Kha’) ) Some thing or things
PERSON( NAME(‘Cliff’) ) Some thing or things

Here I envisage the demi-urge performing the needed projection by ignoring the FOOD_ITEM_EATEN attribute (perhaps even forgetting there is such an attribute in the relation), then, in order to avoid duplicates (we don’t want our demi-urge to be seeing double!), collapsing what had been two appearances of Cliff into just a single appearance.

The picture of relations above may be pretty (forget the picture of the SQL table — that is definitely not pretty…nothing connected to SQL ever is), but even prettier is  Channing Tatum aka Magic Mike, who is today’s homage to Plato’s SYMPOSIUM:

Channing_Tatum_237

Notwithstanding all of my rapturous sighs at the moment, my sole interest in Magic Mike is, of course, as a stepping stone first, to the Relational Algebra, and then, ultimately, to the Platonic Form of Beauty.


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?

 


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.


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.


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.