Improvements to somerby.net/mack/logic

 After a spate of good input from commenters, I've improved somerby.net/mack/logic in several ways:

• Instead of showing just one counterexample, clicking the "Counterexample" button more than once causes the application to cycle through different counterexamples for the current statement.
• There is a new button: the "Example" button, which is the opposite of the "Counterexample" button.
• Propositions (nullary predicates) can now be lower-case letters as well as upper-case letters.
• There is now a "therefore" operator -- ',' -- apostrophe comma apostrophe, for representing logical arguments.

A Cancel Button for somerby.net/mack/logic

Good news: somerby.net/mack/logic now has a cancel button.  If a decision is taking too long, you can cancel it and continue working.  You don't have to close the browser anymore.

Bad news: somerby.net/mack/logic is not working in Internet Explorer, or at least it isn't working in Internet Explorer 11 on my computer.  I'll try to fix it soon.

It works for me in Chrome and Firefox.  I'd appreciate it if anyone tells me whether or not its working for them in other browsers.  Just leave a comment on this post.

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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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.

somerby.net/mack/logic

Here's a web application that decides simple statements in a fragment of first-order symbolic logic: somerby.net/mack/logic. It's based on that .NET console application I was calling "Monadic Predicate Logic".

Monadic Predicate Logic

Suppose we restricted our thoughts to the concepts of  something, everything, nothing, logical connectives like and, or, if...then or not, and monadic predicates, which are attributions of a property to an individual object, like "the sky is blue" or "tomorrow is Tuesday". (In the first statement, the sky is the object and blueness is its property.  In the second statement, tomorrow is the object and being a Tuesday is its property.  That's pretty restrictive, but it's enough to capture the logic behind such truths as the argument:

Socrates is a man.
Man is mortal.
Therefore, Socrates is mortal.

Which can be restated in a way that makes its logic more obvious:

If Socrates is a man and if something is a man then it is mortal, then Socrates is mortal.

The validity of any proposition that can be constructed from only the above-mentioned concepts is decidable.  By "validity" I mean the proposition is true independent of anything else.  "If something is both large and round, then it is large" is true whether or not anything large and/or round actually exists.  The above-mentioned proof of Socrates' mortality is valid, too.  By "proposition" I mean the meaning of a declarative, true-or-false statement.  By "decidable" I mean someone could write a computer program that would the determine any one of these proposition's validity, and it would work for any proposition.

So why have I been explaining all this?  Because I wrote a computer program that does just that.  It's not the first to do so, and it only works for propositions that contain five or fewer predicates, but I spend a fair amount working on it, so I'm going to share it, anyway.
 
The Application

The application runs from the command line.  It takes one or more text files as command-line arguments.  When you execute it, it will read each file and try to interpret it as a statement of monadic predicate logic.  If it can, it will decide whether the contents are necessarily true (valid), possibly true (valid only if additional premises were added), or self-contradictory (necessarily false).

So, if I had an input file that contains the following text:

// All men are mortal.
x,Hx->Mx
// Socrates is a man.
Hs
// Therefore,
->
// Socrates is mortal.
Ms

I would test it through the command prompt like this:

The Language

mpl.exe recognizes a particular language of symbolic logic.  The language uses only ASCII characters available on American English keyboard.  Its elements resemble either elements of ordinary symbolic logic or C-style bitwise operators.  My intent is for the language to be unoriginal enough for someone who knows symbolic logic to be able to use it without much trouble.  I'm too lazy to give a precise definition of the language, so I'll give a brief synopsis.  If anyone has questions, I'll be happy to answer them.

The elements of the language are as follows:

• Ax - a predicate, A, predicated on one variable x.  Predicates can be any character from A to Z.  Variables can be any character from a to z.  These are what represents attributions of a property to an object.  mpl.exe will handle statements with an unlimited number of these, so long as they use no more than five different predicates.
• (,) - grouping by parentheses, just like in algebra.
• .,:,:., etc. - old school grouping, like in Mathematical Logic.
• x,... - universal quantification, i.e. "for all x, ... is true".  The generalization extends to the end of the group it is in, e.g. "x,Ax&y,By->Cx" means the same as "(x,Ax&(y,By->Cx))", but not the same as "(x,Ax)&(y,By->Cx)".
• 3x,... - existential quantification, i.e. "there is an x such that ... is true".  It also extends to the end of the group it is in.
• ~... - negation.
• ...&.. - logical AND.
• ...|... - logical OR.
• ...->... - material conditional, i.e. IF...THEN...
• ...<=>... - logical equivalence. &,|,-> and <=> all have the same precedence and are left-associative.
• // ... - comments; anything after them is ignored by mpl.exe.

Basically, mpl.exe expects an input file to contain a single statement.  The contents of a file might look like this:

((x,Hx->Mx)&Hs)->Ms

This is the basic categorical syllogism.  You can add comments by adding "//"  and text behind it:

((x,Hx->Mx)&Hs)->Ms //the basic categorical syllogism

mpl.exe will ignore anything behind  "//" .  If a line only contains a comment, mpl.exe will ignore it completely:

// the basic categorical syllogism
((x,Hx->Mx)&Hs)->Ms

A statement can be spread across multiple lines.  mpl.exe treats separate lines as if they were separate operand in a logical conjunction, so

Ax|Bx

Bx->Cx

Cx

means the same as

(Ax|Bx)&(Bx->Cx)&(Cx)

If a binary operator is on a line by itself, every line before it is conjoined and treated as its left operand; everything after it is conjoined and treated as its right operand.  So

x,H->Mx
Hs
->
Ms

means the same as

((x,Hx->Mx)&Hs)->Ms

This allows us to symbolize logical arguments and test their validity.  Think of the first two lines in the text above as the premises of an argument and the last line as the conclusion.  As with computer programs, comments can be helpful:

// All men are mortal.
x,Hx->Mx
// Socrates is a man.
Hs
// Therefore,
->
// Socrates is mortal.
Ms

The Algorithm

I won't say much about the algorithm right now, but I will say it is based on "possible worlds" semantics for logic.  I will also say that it runs in O(2^n^n) time, where n is the number of predicates mentioned in a statement; it does not scale well.  On my computer, mpl.exe finishes instantaneously for statements with three or fewer predicates and finishes in about a second for a statement with four predicates.  I haven't tried it, but I estimate it would take about a day to finish evaluating a statement with five predicates.  I didn't bother to implement mpl.exe in such a way that it would support more than five predicates, considering how long it would take to do that.

The Download

Click here to download the application.

The Future


I plan on extending mpl.exe to handle identity and modal operators for possibility and necessity.  Maybe soon.

Somebody's Wrong

1. Fact: I say someone is wrong about something.


2. It follows then that someone is wrong about something. This is proven indirectly:
   
   1. Assume everyone is correct about everything. If so, then the statement I have made in (1) is wrong.
   2. Since the statement I made in (1) is wrong, someone is wrong about something.
   3. (2.1) and (2.2) contradict each other.
   
   Therefore, someone is wrong about something.


3. Furthermore, that wrong idea is not what I said in statement (1). Proof of this is as follows:
   
   1. (2) proves that someone is wrong about something.
   2. Since someone is wrong about something, my statement that someone is wrong about something is true and thus not wrong.

Therefore, there is at least one wrong idea held by at least one person, and it is not my statement that someone is wrong about something.

So who is wrong?

Judgments and Judgment

I often read Scott Adams blog. A lot of his ideas are just crazy, but every once in a while he makes an observation I totally agree with. In a recent post he mentions that kids decide what they want to be when they grow up based on the few careers they are aware of, without any knowledge of the thousands of occupations that are out there. It's so true, but it's so rarely mentioned by anyone. But I do take issue with Adams saying that a person must believe another person's judgment is inferior if they do not believe in the same religion. I would have posted a comment on his blog, but it already had over 200 comments by the time I read the post, so what's the point?

When Adams' uses the word "judgment", he isn't talking about a particular instance of a person making a judgment, rather a person's general ability to discern truth and falsehood. It might be called intelligence, reason or rationality as well. His concept of judgment is missing something important: the fact that rational judgments are made on the basis of the information a person has available to them. It may be rational for me to think that Christianity is true after reading the likes of C.S. Lewis and N.T. Wright. I can't avoid the conclusion that Christian morality works in practice, after seeing it practiced consistently by much of my friends and family. But I wouldn't expect a person whose only knowledge of Christianity comes from watching snippets of televangelists' preaching, and who has never seen anyone practice what Jesus or Paul taught, to believe Christianity any more than any other ideology they know practically nothing about.

Many of our judgments of fact are based on probability, inductive reasoning, memory, and testimony; none of which are infallible. So, in principle, any of these judgments may be nullified on the basis of new information. Only perfect deductive reasoning from absolutely sure premises avoids this possibility. By way of example, let's consider a person named Jay who is at work and ask ourselves whether a rational person should think Jay ate cereal for breakfast this morning. If all the rational person knows about Jay is that Jay is a live person who eats food, the rational person won't believe anything regarding Jay's current status a person who has eaten cereal for breakfast today. If the rational person knows that Jay is an American, and 49% of Americans eat cereal for breakfast on any given day, they will believe that Jay, as likely as not, ate cereal for breakfast. If the rational person learns from a reliable source that Jay eats cereal every morning for breakfast, they will believe Jay almost certainly ate cereal for breakfast this morning. But if they then find out that Jay is extremely allergic to wheat and the only cereal that Jay encountered this morning was bran flakes, they will believe he did not eat cereal for breakfast this morning, as evidenced by the fact that Jay is at work and not in the ER. Each step of the way, the rational opinion is superseded because of new information.

Good judgments are a function of both "how good [a person is] at determining truth from nonsense" and the information the person has. If it were otherwise, I would have to conclude that Isaac Newton was a crappy scientist for not believing in the Theory of General Relativity. If a politician, or anyone, disagrees with me on religious matters, I can't be obliged to think they are stupid, unless I know their reasoning is flawed. That's my judgment on the matter.

Past Events and Probability

I once read an article that tried to invalidate a favorite premise of people who argue in favor of Intelligent Design, specifically the extreme improbability of life spontaneously arising from non-life. The article started with the assertion that the probability of any past event is necessarily 1, therefore Intelligent Design partisans were committing a dark and dangerous fallacy by insisting that of life existence of life was highly improbable if it arose through purely natural means. I won't deny that Intelligent Design partisans can and do commit dark and dangerous fallacies, but I don't think that believing the stated premise is one of them. Instead, I doubt the validity of the notion that we should always consider the probability of a known past event to be 1 when reasoning about past events. I do so because such an assumption to deduce some apparently false conclusions. Since we should not be able to deduce false conclusions from any true premise, we should not make it a rule that the probability of any known past event is 1, modus tollens.

Here goes my argument. Let's call the assertion under dispute "Proposition 1" and state it as follows:

1. If an event occurred in the past, then the probability that the event occurred is 1.
   

At first glance, this seems reasonable, even intuitively true. It seems internally consistent, and can I think of a way to derive a contradiction from it (not that I've thought that much about it). But that's not what I'm arguing.

Now for Propositions 2 and 3:

2. 1 is greater than 1/2
3. If an event has a probability of 1, then the probability that the event occurred is greater than 1/2.

Proposition 2 should be axiomatic enough for anyone, and Proposition 3 is just an instance of Proposition 2. Sorry to bore you with such triviality, but I need Proposition 3 so I can use it in a Hypothetical Syllogism with Proposition 1 to infer Proposition 4:

4. If an event occurred in the past, then the probability that the event occurred is greater than 1/2.

To state it more simply, if it happened, then it probably happened. This does not seem to pose a problem; if George Washington crossed the Delaware, then we wouldn't be amiss to say that George Washington probably crossed the Delaware, right?

Things only become interesting when we consider a particular case of the contrapositive of Proposition 4:

5. If the probability of a past event occurring is less than or equal to 1/2, then the event did not occur.
6. If the probability of a past event occurring is less than 1/2, then the event did not occur.

(Proposition 6 is just Proposition 5 limited to the particular case where the probability of the event is not equal to 1/2). Proposition 6 is my basis for rejecting Proposition 1. To state it simply, if something probably didn't happen, then it didn't happen, or, in other words, improbable events never occur. In a sense, this is believable. If something happened, then it can't not have happened, right? Maybe so. I can't disprove the law of cause and effect. But I can say that trouble ensues when we try to apply Proposition 6 to a situation where something "probably would have" or "probably shouldn't have" happened.

Take an example: Let's say I find out that my brother the college student failed an exam. I know he's a smart guy, and the class he failed it for wasn't anything difficult like thermodynamics, and most of the professors at the college he goes to are competent teachers and test writers. Therefore, he probably didn't study for the test. If I accept Proposition 6, then I must conclude that he simply did not study for the test.

Here is the problem: I should not be able to draw this conclusion based on the information I have. It is still possible that he did study, but he got a bad professor or he just wasn't understanding the material or perhaps something else. However, Proposition 6 follows necessarily from Proposition 1, Proposition 1 makes this conclusion not only possible, but logically necessary!

Therefore, it would be an obvious fallacy to apply Proposition 1 to reasoning about the probabilities of past events. I am not saying that I've reduced Proposition 1 to absurdity. I'm just saying that it is not a valid rule to be used for reasoning about the probability of past events, which is exactly what was attempted in the aforementioned article regarding intelligent design.