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.

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Perl GCD

O Perl, how I love you.  Here is a Perl subroutine that finds the greatest common divisor of two integers:

sub gcd
{
  (my $x, my $y ) =
    map abs, grep { $_ } @_;
  $y ? gcd( $y, $x % $y ) : $x
}

and here is  a Perl subroutine that finds the greatest common divisor of any number of integers (including 0 integers, for which it returns undef):

sub gcd
{
  (my $x, my $y, my @others) =
    map abs, grep { $_ } @_;
  $y ? gcd( $y, $x % $y, @others ) : $x
}

note the difference, and how small it is.

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. ∎

Investigating Logical Arguments with somerby.net/mack/logic

I created a web application that evaluates logical arguments.  To provide an example of how it might be used, I'm going to use it to help me do a brief investigation of two modal arguments about the existence of God.  The first is Alvin Plantinga's modal argument for the existence of God.  The second is Juan Manuel Correa's counter-argument against the existence of God (or at least against the soundness of Plantinga's argument).  Statements of both arguments can be found on Wikipedia at http://en.wikipedia.org/wiki/Ontological_argument#Alvin_Plantinga.

First off, I'm going to strip the arguments down to what I think is the essential logic that makes both arguments work.  I don't suppose this will be agreeable to everyone, at least not without some persuasion, but I'll just go with it for now and try to defend myself later if I have to.  So, for Plantinga's argument I will use:

<>G->[]G
<>G
->
G

Which is saying,

• If God exists in some possible world, God exists in every possible world.
• There is a possible world in which God exists.
• Therefore, God exists.

For Correa's counterargument, I will use:

<>G->[]G
<>~G
->
~G

Which is saying,

• If God exists in some possible world, God exists in every possible world.
• There is a possible world in which God does not exist.
• Therefore, God does not exist.

 The web application, which, in this post, I will call "the decider", decides that both arguments are valid.  So what's going on?  There are three premises between the two arguments, and each seems plausible.  I suspect the fact that both arguments are valid in spite of their contradictory conclusions means that the three premises are logically inconsistent with each other.  So I try them out in the decider:

<>G->[]G
<>G
<>~G

It tells me they are impossible, i.e. logically inconsistent with each other.  So we can't believe all of them at once.  Which one should I doubt?

Before I try to answer that question, I want to check that neither argument is trivially valid.  I mean, I want to be sure that neither conclusion follows from its premises simply because those premises contain a contradiction and anything follows from a contradiction.  So I test the premises by themselves in the decider.

<>G->[]G
<>G

Then

<>G->[]G
<>~G

The decider says both pairs are contingent.  So no problem there.   What about the two differing premises?  The decider says

<>G & <>~G

is possible, which it just what I expect, so I could believe both.  Which is what I'm inclined to do.  I can imagine a world with God, and I can imagine a world without God.  Of course, just because I can imagine them doesn't mean that there is not some contradiction hiding in either conception.  But I'm still most inclined to drop Premise #1.  It's obvious that []G and <>~G are incompatible, and it's not clear to me that there ever could be such a thing as a necessary being.  Some logics entail that at least one being must exist; See Proposition 24.52 in Principia Mathematica for one.  Even so, just because you can prove that at least one object exists doesn't mean you can prove that it has any nontrivial properties, like divinity.  And I'm not enamoured of the idea that God has every possible perfection; necessary being being one of those perfections.  Clearly, if a being has every possible perfection, it must smell like freshly-baked chocolate chip cookies at least some of the time.  And play a face-melting guitar solo every time it picks up a guitar.  Unless someone is taking a nap within earshot.  But I'm a Christian, and I'm sure both of these perfections are irrelevant to the Gospel, so I don't care.

So, to conclude, I will assert 1. that God exists in some possible world.  2. There is a possible world where God doesn't exist.  Therefore, Premise #1 is false.  

To summarize my reasoning in symbolic form:

<>G
<>~G
[]( (<>G&P) ->  C )
[]( (<>~G&P) -> ~C )
->
~P

Where  P is premise #1, C is the conclusion of  Plantinga's argument, and G is as defined above.

The decider confirms that this argument is valid.

One more thing I'd like to point out: when considering modal arguments for the existence of God, one should be careful about the statement "It is possible that God exists."  It might be interpreted two different ways.  First, as "there is a possible world in which God exists".  The second, as "I don't know anything that precludes the existence of God."  The first is easy to represent in terms of modal logic, but the second is the more natural interpretation.  In my analysis, I assumed the first.  If I were to choose the second interpretation, my analysis of these two arguments would have to be different.

Linking Ada, C and Haskell Together Statically

So, you want to statically link object code from C, Ada and Haskell into one executable? Probably not. But how could that be done, anyway? Many programming languages provide a means both of exporting subroutines to make them callable from C code and importing C functions so they can call them. You can easily find documentation on how to statically link C to one other language. But what if you want link C with two different languages in the same project? Just compile all of the code separately and link the object files, right? Haha. Not so fast! Programming languages have different run-time systems; they have their own libraries and they may use more than just a stack and a heap and a space for global variables. So if you want to include object code from several languages in a single executable, you'll often have to build run-time code for each application and link it in to the executable as well. In this post, I'm going to demonstrate how you can accomplish this in a project that uses C, Ada and Haskell.

Click here.