Another Square of Opposition

In my former post, I showed Terrence Parsons' theory of Aristotle's Square of Opposition in symbolic form.  I also noted that the Square of Opposition holds up under the modern interpretation of the four forms of Term Logic if it is assumed a priori that the subject terms of the forms is non-empty (I am not the first to note this; see section 2.2.2 of iLogic).  So there are two interpretations of the four forms which affirm the Square of Opposition.  But are there others?

I found another interpretation of the four forms of Term Logic that affirms the Square of Opposition.  It's a parallel to Parsons' Square.  Parson constructs the square by taking the modern interpretation of the four forms, bestowing existential import upon Form A and denying existential import to the form on the opposite corner - Form O.  The interpretation I found is constructed by by taking the modern interpretation of the four forms, bestowing existential import upon Form E, and denying existential import to Form I on the opposite corner.  Here is a statement of it in symbolic form:

// Another Square of Opposition

// "All S are P", with no existential import
A <=> (x,Sx->Px)

// "No S are P", with existential import
E <=> ((x,Sx->~Px)&(3x,Sx))

// "Some S are P", with no existential import
// "If there are any S, some of them are P" might be a better way to state it.
I <=> ((3x,Sx)->(3x,Sx&Px))

// "Some S are not P" under the modern interpretation with existential import
// Since it has existential import, there's no need to state is as "Not all S are P".
O <=> ~(x,Sx->Px)

->

// Contraries
~(A&E)

// Contradictories
A ^ O
I ^ E

// Subcontraries
I | O

// Subalterns
A -> I
E -> O

I'd like know if anyone else has thought of it before.

This interpretation, like Parsons' interpretation and the modern interpretation combined with a non-empty subject term, affirms the Logical Hexagon:

// The Logical Hexagon:

// "All S are P", with no existential import
A <=> (x,Sx->Px)

// "No S are P", with existential import
E <=> ((x,Sx->~Px)&(3x,Sx))

// "Some S are P", with no existential import
// "Some S are P, if any S exist" is a better way to state it.
I <=> ((3x,Sx)->(3x,Sx&Px))

// "Some S are not P", existential import
// Since it has existential import, there's no need to state is as "Not all S are P".
O <=> ~(x,Sx->Px)

// The statement U may be interpreted as "Either all S are P or all S are not P."
U <=> ((x,Sx->Px)|(x,Sx->~Px))

// The statement Y may be interpreted as "Some S is P and some S is not P"
Y <=> ((3x,Sx&Px)&(3x,Sx&~Px))

->

// Subalterns: AI, AU, EU, EO, YI, YE
A->I
A->U
E->U
E->O
Y->I
Y->O

// Contraries: AE, EY, YA
~(A&E)
~(E&Y)
~(Y&A)

// Subcontraries: IU, UO, OI
I|U
U|O
O|I

// Contradictories: AO, UY, EI
A^O
U^Y
E^I

A Second Theory of Term Logic

I added Term Logic to somerby.net/mack/logic for fun. While doing the necessary research, I discovered the logical Square of Opposition, which is kind of cool. Terence Parsons wrote an illuminating article on the Square. In it, he argues convincingly for an interpretation of 2-term propositions that affirms the Square of Opposition, and also convincingly that this interpretation is Aristotle's intended interpretation. I like the article so much that I've chosen to use this interpretation in my application, defining the four forms of propositions (SaP, SeP, SiP, SoP) just as he does. Even so, I doubt that this is the only coherent theory of Term Logic held by premodern logicians. Here I shall explain why. You can click on any of the symbolic statements in this post to test them in somerby.net/mack/logic.

When explaining why the interpretation of the O-form as "Some S is not P" did not cause problems for premodern logicians, Parsons dismisses the possibility that they assumed that the S-term was not empty, stating "Explicitly rejecting empty terms was never a mainstream option, even in the nineteenth century". But I'm not so sure. First of all, just because they did not explicitly reject empty terms does not mean they implicitly rejected empty terms. Second, they did not have to reject empty terms altogether to make this interpretation of O-form compatible with the traditional Square of Opposition. They only needed to assume a priori (and perhaps unconsciously) that the S-term was not empty whenever they were making an argument. This isn't a very rigorous thing to do, but it's a natural thing to do. Usually, if we are making assertions about some kind of thing, it's because some such thing exists and we want to say something meaningful about it. Reasoning about unicorns may have its uses, but they are not obvious.

Suppose that some pre-modern philosophers, like Boethius and Peter of Spain, did not interpret the propositional forms as Aristotle intended. Instead, they assumed a priori that the S-term was nonempty, and let the O-form have existential import, just as Boethius seemed to be doing when he translated "Some S is P". Then, instead of the Square of Opposition being this:

// Aristotle's Square of Opposition

A <=> ((x,Sx->Px) & (3x,Sx))
E <=> (x,Sx->~Px)
I <=> 3x,Sx&Px
O <=> ((3x,Sx&~Px)|(~3x,Sx))

->

// Contraries
~(A&E)

// Contradictories
A ^ O
I ^ E

// Subcontraries
I| O

// Subalterns
A -> I
E -> O

they believed the Square of Opposition was this:

// A Hypothetical Alternative to
// Aristotle's Square of Opposition

3x,Sx // Assume a priori that S is not empty.

A <=> (x,Sx->Px)  // (Existential import here would be redundant.)
E <=> (x,Sx->~Px)
I <=> (3x,Sx&Px)
O <=> (3x,Sx&~Px) // Assume O has existential import.

->

// Contraries
~(A&E)

// Contradictories
A ^ O
I ^ E

// Subcontraries
I | O

// Subalterns
A -> I
E -> O

The relationships of the Square hold in this interpretation as well as in Aristotle's.

And then there is the matter of the Principle of Obversion and the Principle of Contraposition. Parsons says that some medieval logicians advocated these principles, though they are both fallacious under Aristotle's interpretation of the four forms. The following is not necessarily true:

// The Principle of Conversion by Contraposition,
// with Aristotle's interpretation
// of the A-form and the O-form
((x,Sx->Px) & (3x,Sx)) <=> ((x,~Px->~Sx) & (3x,~Px))
((3x,Sx&~Px)|(~3x,Sx)) <=> ((3x,~Px&~~Sx)|(~3x,~Px))

This is not necessarily true, either:

// The Principle of Obversion,
// with Aristotle's interpretation
// of the A-form and the O-form

// Every S is P = No S is non-P (SaP <=> Se~P)
((x,Sx->Px) & (3x,Sx)) <=> (x,Sx->~~Px)

// No S is P = Every S is non-P (SeP <=> Sa~P)
(x,Sx->~Px) <=> ((x,Sx->~Px) & (3x,Sx))

// Some S is P = Some S is not non-P (SiP <=> So~P)
(3x,Sx&Px) <=> ((3x,Sx&~~Px)|(~3x,Sx))

//Some S is not P = Some S is non-P (SoP <=> Si~P)
((3x,Sx&~Px)|(~3x,Sx)) <=> (3x,Sx&~Px)

Why did some logicians make these mistakes? And why did other logicians like Peter of Spain endorse them? Maybe to them, they weren't mistakes. Under what we call the modern interpretations of the four forms, these principles are necessarily true.

// The Principle of Conversion by Contraposition,
// with the modern interpretations of the forms:
(x,Sx->Px) <=> (x,~Px->~Sx)
(3x,Sx&~Px) <=> (3x,~Px&~~Sx)

// The Principle of Obversion,
// with the modern interpretations of the forms:

// Every S is P = No S is non-P (SaP <=> Se~P)
(x,Sx->Px) <=> (x,Sx->~~Px)

// No S is P = Every S is non-P (SeP <=> Sa~P)
(x,Sx->~Px) <=> (x,Sx->~Px)

// Some S is P = Some S is not non-P (SiP <=> So~P)
(3x,Sx&Px) <=> (3x,Sx&~~Px)

//Some S is not P = Some S is non-P (SoP <=> Si~P)
(3x,Sx&~Px) <=> (3x,Sx&~Px)

Being necessarily true, they will still, of course, be true under an a priori assumption that the S-term is nonempty. So maybe there was a theory of term logic floating around Medieval Europe that looked like this:

3x,Sx

A <=> (x,Sx->Px)
E <=> (x,Sx->~Px)
I <=> (3x,Sx&Px)
O <=> (3x,Sx&~Px)

->

// Contraries
~(A&E)

// Contradictories
A ^ O
I ^ E

// Subcontraries
I | O

// Subalterns
A -> I
E -> O

If so, then they really did have a coherent theory of Term Logic which affirmed the Principle of Conversion by Contraposition and the Principle of Obversion. I can't be sure, since I haven't looked for evidence to the contrary, e.g. Peter of Spain discussing empty terms in Summulae Logicales Magistri Petri Hispani, but as far as I know, it makes sense. I guess I'll have to read some Medieval logic to find out. It's too bad I don't know Latin.

An Argument Against the Possibility of Transworld Identity of Indiscernibles

I'm going to attempt prove that transworld identity of indiscernables is impossible. Maybe this is unnecessary because either it is obvious to anyone who cares or proving it is easy enough that it hardly needs to be done, but I'm going to do it anyway. I would like to establish its truth because of my modal symbolic logic app, somerby.net/mack/logic. Right now, it does not handle transworld identity in a reasonable way. I need to fix this. My plan is to change the algorithm so it operates under the assumption that transworld identity of indiscernables is impossible, which is (in my opinion) the correct behavior.

Let me explain what I mean by "Transworld Identity of Indiscernibles". Transworld identity is when an object is considered to exist in two different possible worlds. Let me explain that I mean by "possible worlds", too. Abraham Lincoln existed in the real world. In the imaginary world of the movie Abraham Lincoln: Vampire Hunter, Abraham Lincoln also existed and, in addition to his duties as president, hunted vampires. These two Abraham Lincolns are supposed to be the same person somehow; if they weren't, then Abraham Lincoln: Vampire Hunter would not have the question "what if Abraham Lincoln had to save his country from vampires" to drive its plot, and it would just be about a bearded man in a stovepipe hat killing vampires. Or maybe it was. I don't know. I haven't seen the movie. So here's a less fantastical example: consider the statement "if I hadn't been caught in traffic, I would have been at the meeting on time". It's an ordinary, plausible thing to say, and it mentions two possible worlds: the real world, where I was late for a meeting, and a hypothetical world where I was not late for that same meeting. The statement implies that I exist in both possible worlds; there's late me, and then there's punctual me, and they are both supposed to be me somehow. That's transworld identity.

Identity of Indiscernibles is a principle that defines one particular kind of identity. This principle is that two things are identical if and only if it is impossible to distinguish them in any way. It is the kind of identity that is represented by the equals sign (=) in math and in symbolic logic. To state the principle in mathematical terms, x = y if and only if there is no statement φ = "... α ..." such that φ is true when x is substituted for α everywhere in φ and φ is false when y is substituted for α everywhere in φ.

So we have this thing called "Transworld Identity" and we have another thing called "Identity of Indiscernibles". What about Transworld Identity of Indiscernibles? Is it ever true that x = y when x and y are in two different worlds? I say "no", and here is my argument:

Suppose object o₁ exists in possible world w₁ and object o₂ exists in possible world w₂. Suppose further that w₁ and w₂ are different. If they are different, then there must be some quality Q which w₁ has which w₂ does not have; else how are they be different? Supposing all of that, then the statement "o₁ exists in a world with quality Q" is true and the statement "o₂ exists in a world with quality Q" is false. Therefore, by the definition of "identical" given above, o₁ and o₂ are not identical. ∎

Essays!

Here's some recommended reading; several essays, articles & such that I've read at one time or another and found to be especially illuminating or memorable:

The Problem With Music, by Steve Albini
An enlightening look at the relationship between new bands and record labels. I used to want to be a rock musician. This essay confirmed to me that I should be an indie rock musician, if anything. I eventually gave that up because there was no money in it and I didn't have the ability or motivation to write songs, but that's another thing.

I, Pencil, by Leonard E. Read
A testament to the creative power of free market economies.

Fern-Seed and Elephants, by C.S. Lewis
C.S. Lewis, as a literary critic, rips on the methods of some Biblical critics.

The Special Theory of Relativity, by Richard Feynman
This is best explanation of Special Relativity I've ever encountered. I think that is because 1. He starts with Newton's physics, which I understand, and compares it to Special Relativity, and 2.
He uses math. Math is hard and boring and scary, so some explanations try to avoid it, relying on imagination alone, and fail miserably. Maybe because modern physical theories are mathematical statements. Don't fight it! Give in to the algebra. Listen to the calculus.

Parkinson's Law, by C. Northcote Parkinson
On second reading, I think the title is immodest and the article is a bit tongue-in-cheek, but it's nonetheless convincing. I bet Parkinson's Law applies to corporate managers, but it's counteracted by the fact that laying off excess managerial staff is a good way to cut costs.

The Myth of Sisyphus, by Albert Camus
I generally like the famous existentialist writers, even though I disagree with them on some very basic things. Maybe it's because we ask the same questions, though we accept different answers. This essay has popped into my mind many times since I first read it years ago.


G.K. Chesterton

Chesterton gets his own section. He's sophistical at times, but his insight and his humor are priceless. These are a few of my favorites.

• A Sermon on Cheapness
• Cheese
  
• On Certain Modern Writers and the Institution of the Family
• The Modern Martyr
• The Sectarian of Society

The Problem of Who Lois Loves

The other day, I got to thinking about one of the discussion topics in from Intro to Philosophy class I took seven years ago. I don't know why. It's a philosophical problem that goes as follows: If two things are the same, then everything that is true of one of them is true of the other. Clark Kent and Superman are the same person, but Lois Lane loves Superman and Lois Lane doesn't love Clark Kent. What's up with that? This problem seems to belong in the same league as Socrates' House (or is it Plato's Boat?), the omnipotence of God and such, but unlike some of these other introductory-level philosophical problems, I think I actually have an answer to it. Here it is; sorry to bore you if you've already got it figured out.

The concepts of Superman and Clark Kent (especially Lois Lane's concepts) are different, but they both are attached to the same real-world entity. Thus, what is true of Lois Lane's concept of Superman is not always true of her concept of Superman (in fact, they are quite distinct), but they do have one thing in common: Lois connects both of them to the same thing in reality (though she does not know it).

This implies something interesting about love: The act of loving someone - or at least of loving someone in the romantic sense of the word "love" - requires having a particular concept of who that person is. If it didn't, Lois would feel the same way when thinking of Clark Kent as she would Superman.

Lois' love for Superman is ultimately directed at the same object as her indifference to Clark Kent, but since this sort of loving implies conceptualizing its object, and Lois' concepts of Clark Kent and Superman are distinct, it is possible for Lois not to love Clark Kent.

By the way, I found out that the creators of Superman first conceived of him as a human villain. Far out.

Created Meaning

Suppose I woke up one afternoon in a Macy's department store and I had no idea how I got there. After rubbing my face, visiting the restroom, and checking men's slacks just in case they actually have anything in my size, I'm sure I'd begin to ask questions like "Why am I here?", "How did I get here?", and "Did someone put me here, and, if so, then for what purpose?". I was having a hard time answering these questions on my own, but an elderly sales associate told me that I was placed there by Macy himself so that I might begin a lifelong career in retail. Of source, I rejected the associate's explanation; for all I knew, it might have been a self-serving lie on her part, and I had no direct evidence that such a person as Macy existed, much less that this putative Macy cared anything about my career. I was not wearing clothes suitable for an interview, and I woke up at the opposite end of the store from the service desk where the job applications were kept. If I was meant to get a job there, I had no special indication this was so.

My appearance in the department store may have a meaning: Perhaps Macy did indeed bring me there (kidnapper!) to get a job. Perhaps it is a prank my friends are playing on me. Perhaps it is a sign from God that I should not get wasted then go wandering around town, passing out in random locations.

My appearance in the department store may not have a meaning: Perhaps I merely suffered from simultaneous attacks of narcolepsy and somnambulism. Perhaps I got wasted and went wandering around town, but God wasn't trying to show me anything from it. Perhaps, because of some unanticipated consequence of quantum physics, I teleported into that Macy's store while taking an afternoon nap.

What if, I tried to create a meaning for the situation I was in? I don't know why I would want to do this, since I'd much rather know the actual meaning of my appearance in Macy's, if there is one; but suppose I did. If we take the phrase "create a meaning" literally, then it is simply absurd. I cannot create a meaning for my appearance in Macy's. I may discover it, I may imagine it or pretend it, but I can't create meaning after the fact, simply because I cannot change the past nor intentionally cause something that I did not intentionally cause. Is this what existentialists mean when they talk of creating meaning?

Of course, existentialists are smarter than that. But then, what does it mean to create meaning? I suppose if I appeared in Macy's without intending to be there, I could let the visit serve some purpose. I couldn't properly say "I'm here because I need to buy a new belt", but I could say "Since I'm here, I might as well buy a new belt." This makes perfect sense. The thing is, it seems to me that none of the existentialists are as bland as that. Nietzche could have saved a lot of paper if he merely said "Life is meaningless, but, since we're here, we might as well do what we like and not let any old customs or religious dogmas get in our way", but he didn't. Maybe I've just answered my own question. If life is meaningless and we might as well do as we like, then thinking people might as well do the absurd and create meaning, just because they like doing it.

Concept and Reality

Can God create a rock so big he cannot move it? The easy answer would be "yes he can, but if he is omnipotent, he evidently has not yet chosen to do so." But seriously, what if we find some inconsistency in our notion of God? Are we then obliged to conclude by the law of non-contradiction that God does not exist? I say no, if we are clever enough to distinguish between concept and reality.

Such an obligation would depend on a certain proposition, here stated in logicky terms:

For all real entities x, if our concept of x is not logically consistent, then x does not exist.

Important to my argument is the distinction between a real entity and our concept of it. These two things are what St. Anselm would have called "a thing in reality" and "a thing in the mind", respectively. Though in order to think about some object, say a rock, we must have some concept of what that rock is, the rock and our concept are distinct; one is "out there" and the other is in our brain. Of the two, only the thing in our brain can properly be said to be logically consistent or logically absurd. Only propositions can contradict and a rock is not a proposition.

With this understood, I can provide an easy couterexample to the above-mentioned rule: Let us pretend that we have two theories of physics known to us and call them Q and R. Let us pretend that each theory, on its own, is an amazingly accurate and poweful explanation of a great deal of physical phenomena, and the two together account for pretty much everything we know for sure about the physical world. Let us pretend that not a few successful technologies have been developed based on these theories. But then, let us further pretend that these two theories are logically inconsistent with each other, that is to say, if we combine the two, we end up with some contradictions. What then would we do? We might do like Hegelians and find in Q a thesis, then find in R its antithesis, then try to find their synthesis in some theory S. Failing that, we might just choose to live uncomfortably with Q and R's inconsistency; we might resign ourselves to the idea that physical objects are not perfectly comprehensible. But at no point would we doubt the existence of physical objects; we've run into them too many times in our own experience to do that.

Therefore, since we would not allow such a situation to disprove a known reality, we cannot honestly say we affirm the proposition stated above on pain of inconsistency. We could modify it to say:

For all real entities x, if we are not sure x exists and our concept of x is not logically consistent, then x does not exist.

But this modification is not reasonable. By the definition of "real entitiy", the existence of a real entity does not depend on our knowledge of its existence. Therefore, we cannot use this rule to disprove the existence of some external reality, even if we doubt that entity's existence.

Like any other external reality, so it is with God. When a traditional Christian says they believe in God, they mean that the object of their faith is an external reality and not a concept, though of necessity they must have some concept of what God is. If she runs into contradictions when contemplating God, she does not need to become an atheist. She might just choose to consider the highly probable possibility that her understanding of God is not perfect.

The Pagan Urge

While on a trip to Ghana with the Gospel Choir, Professor Jelks said something which was memorable to me. He said to some of us in the group (and I paraphrase to you), 'I woke up this morning and stepped outside my room and saw the sun and I felt like worshipping it. Then I thought to myself, "Hey, I'm a Christian, I don't worship the sun!"'.

In spite of my stodgy Christian self, which tenses whenever it senses anything theologically liberal, I sympathized with him. The sun really is something wonderful, when you think of it. This was especially clear in Ghana, where it shines through the harmattan haze gloriously onto forests, fields and jungles, and rises and sets each day at nearly the same time year-round. I totally agreed and still do agree, that the sun does and should inspire worship. After hearing that observation from a regular person, I started to understand why so many pagan religions worship the sun - and so many other parts of nature, for that matter. I "feel them", in the current idiomatic sense of the phrase. It's not just about the practical benefits the sun brings, or its apparent power, but about an ineffable feeling it and other parts of nature inspire.

It did not take much thought for me to realize that this feeling did not threaten, but rather vindicated Christianity; because Christianity vindicated the feeling. If, in response to these urges, we worship the sun; if we dance it front of it and pray to it and make offerings to it, we express our feelings. But our feelings are absurd when we take into account the fact that the sun is a perpetually exploding ball of gas which is not aware of anything, including our love and adoration for it. Pagan worship makes sense inasmuch as it is a natural response to our feelings. It does not make sense inasmuch as it is an attempt to communicate appreciation to something that does not perceive appreciation. My Christian theism had this problem solved even before I posed it. It only took a moment to realize that I can act out these feelings for the sun by worshipping the God that created it. The same goes for the moon, rivers, trees and every other thing I happen to like. Indeed, I can express this feeling for anything and everything at once by worshipping the creator. God, by definition, desires and appreciates worship. If we sing praises to God, he hears. If we exalt him, he's glad of it. The urge to worship is not only genuine, it's functional. Also, it's efficient. There aren't enough days in the year to have a festival for every aspect of nature that awes us. There aren't enough people to maintain a priesthood for each cult we might create. The Ephesians worshipped Artemis because they only had time and energy for Artemis. Take away their local patriotism and they still would have neglected Zeus and Aphrodite and Ares out of practical necessity. In this respect, a monotheist can accomplish what a polytheist cannot even attempt.