Category Archives: R.M. Chisholm

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?

 

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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.


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.


The Predicate As A Truth Valued Function

So far we have been modeling sentences in which nothing is left unspecified.  Chris invites AndrewLukas laughs.  How could we model, however, sentences such as Chris invited someone, Someone invited Andrew, Someone invited someone, Joe ate something, Someone laughed … sentences in which at least one of the “central participants in a situation” is left unspecified?  We can model these sentences, I think, by applying the Relational Algebra to them — or, more precisely, to the propositions that underlie them.  In this post, I start laying the groundwork for showing how we can use the Relational Algebra to model sentences containing ‘someone’, ‘anyone’, and the like.

Let me begin by outlining the key premise behind Relational Database Theory: 

Predicates generate propositions which are either true or false.  A given Database Relation comprises all and only the true propositions generated by a given predicate.  (This is the Closed World Assumption.)  We can apply various operations of the Relational Algebra to the propositions contained in a Database Relation.

The key premise in Relational Database Theory talks about predicates.  What, then, is a predicate?

What the database theorist C.J. Date calls a predicate is what Kroch and Santorini call, in the primer on Chomskyan linguistics quoted from in the post below (The Verb Considered As A Function) a verb.  Date explains what a predicate is better than I can, so let him speak (LOGIC AND DATABASES THE ROOTS OF RELATIONAL THEORY, Trafford Publishing, Canada, 2007, p. 11):
 

A predicate in logic is a truth valued function.

In other words, a predicate is a function that, when invoked, returns a truth value.  Like all functions, it has a set of parameters; when it’s invoked, arguments are substituted for the parameters; substituting arguments for the parameters effectively converts the predicate into a proposition; and we say the arguments satisfy the predicate if and only if that proposition is true.  For example, the argument the sun satisfies the predicate “x is a star,” while the argument the moon does not. 

Let’s look at another example:

x is further away than y

This predicate involves two parameters, x and y.  Substituting arguments the sun for x and the moon for y yields a true proposition; substituting arguments the moon for x and the sun for y yields a false one. 
 

The key premise mentions Database Relations.  What, then, is a Database Relation?

The concept of a Database Relation is an elaboration on the concept of a Relation as defined in mathematics.  In mathematics, a Relation is defined as the subset of the Cartesian Product of two or more sets.  (What a Cartesian Product is will be obvious from the example.)  For example, in the sets {John, Charles, Cliff, Dan} and {Leon Trotsky, Genghis Khan}, the Cartesian Product is { (John; Leon Trotsky), (John; Genghis Khan), (Charles; Leon Trotsky), (Charles; Genghis Khan), (Cliff; Leon Trotsky), (Cliff; Genghis Khan), (Dan; Leon Trotsky), (Dan; Genghis Khan)}.  If, now, we pick out a subset of this Cartesian Product by seeing who happens to be standing to the left of whom at the moment, we get this Relation:  { (Charles; Genghis Khan), (Cliff; Genghis Khan), (Dan; Leon Trotsky)}. 

In other words, our Relation is what we get when we start with the predicate:

x is standing to the left of y

and plug in values for x from the set {John, Charles, Cliff, Dan} and values for y {Leon Trotsky, Genghis Khan}, throw away all the false propositions that result, and keep all of the true propositions.

Let me go out on a limb, then, and say that a proposition (remember, our key premise mentions propositions) is a tuple, that is to say, an ordered pair (for example, (Charles, Genghis Khan) ) in a Relation.  (Please, pretty please, don’t saw this limb off.) 

This means then that a proposition is a state of affairs ala R.M. Chisholm.  For example, the proposition Charles is standing to the left of Genghis Khan is the state of affairs comprising the flesh and blood Charles standing to the left of the flesh and blood Genghis Khan.  Propositions as states of affairs are the meaning of sentences… But I digress.

Back to Relations. 

A Database Relation, I have said, is an elaboration of a Mathematical Relation.  A Database Relation comprises a Heading consisting of ordered pairs of (Name Of Type; Type) and a Body consisting in a set of ordered pairs (Name Of Type, Value).  A type is a set, for example, the set of integers, the set of words in a given language, the set of people, the set of cities in the world, and so on.  A value of a type is a member of the set identical with that type.  I will leave name undefined. 

A Database Relation is an abstract object;  it is either an object really existing in some Platonic Heaven someplace or it is a fiction, depending upon which theory of abstract objects is the correct one.  Database Relations form the conceptual skeleton of databases concretely implemented by an RDBMS (Relational Database Management System) functioning inside a physical computer, but at least at the moment I am not talking about physical computers and the software they run.  I am talking about the abstract object, something that has the same status as the number 3 or the isoceles triangle. 

Why do I want to talk about Database Relations rather than Mathematical Relations?  It will be easier in the  posts that (hopefully) will follow to illustrate the Relational Algebra operations Projection and Restriction.  I know how to apply these operations to Database Relations; I am not sure how to apply them to Relations simpliciter.   Projection and Restriction are the Relational Algebra operations which, I claim, will give us a model for sentences such as Joe ate something. 

I’ve laid the groundwork for such a model; now let me go on to produce the model.