# Category Archives: FALSE

## Aristotle’s Sea Battle Argument

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

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

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

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

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

is true, or:

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

is true.

But which one?

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

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

A sea battle happens at time tn.

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

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

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

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

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

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

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

* * * * *

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

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

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

As they had very little in the way of precious stones

(our wretched dominion’s poverty was great

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

scarlet, green or blue.

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

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

## 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:

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

## The Predicate Returns A Relation

We have seen that the predicate:

x is to the left of y

is mapped to the truth value TRUE when Charles is substituted for x and Genghis Khan is substituted for y.  The Relation TO_THE_LEFT_OF comprises all true propositions and only true propositions that get generated when values are substituted for x and y.  So the predicate is a function whose range is the truth value TRUE for every proposition that is included in the relation, and FALSE for every proposition that is not included in the relation.

I think, however, that we would get a slightly simpler account if we see the predicate as a function returning Relations comprising the single proposition TRUE, or the single proposition FALSE.  In the Relational Algebra, we would get a relation comprising the single tuple (and therefore proposition) TRUE if, after doing the Restriction that gives us:

Charles is to the left of Genghis Khan.

we then projected on the null set of attributes (“columns”).  We would then end up with Chris Date’s TABLE_DEE, that is, the Relation with cardinality 0 (o attributes, that is, 0 “columns”) and a single tuple.  TABLE_DEE is the Relation that corresponds to (I guess I should say ‘is identical with’) the weird classical logic proposition TRUE.  The predicate returns the proposition TRUE wrapped in the Relation TABLE_DEE when the Charles and Genghis Khan substitution is made.

Correspondingly, when John is substituted for x and Genghis Khan is subsituted for y, so that we get:

John is to the left of Genghis Khan.

the Restriction selects no tuple in the Relation TO_THE_LEFT_OF.  We then have a Derived Relation with a cardinality of 2 (i.e., the Relation has 2 “columns”) holding the null set of tuples.   If we then project on the null set of attributes, we end up with a Relation of cardinality 0 comprising 0 tuples.  Chris Date calls this Relation TABLE_DUM, and it holds the tuple, that is to say, the proposition FALSE.  The predicate returns the proposition FALSE wrapped in the Relation TABLE_DUM when the John and Genghis Khan substitution is made.

Thinking of the predicate as returning either TABLE_DEE or TABLE_DUM simplifies things a bit, because it means we never have to leave the Relational Algebra when modeling the predicate.  Everything gets explained in terms of just one set of operations, the operations of the Relational Algebra.