If 'If' Weren't So Iffy

[thomtg]

Congruent with my other thread :

[...]

I think you are right. Your presentation of the misinterpretations of the philosophy behind mathematics in this context deserves serious study. If we take the Aristotelian-Objectivist approach as the correct line of sight, then Objectivists have much work ahead of us to turn the ship of Math toward the right direction.

Not bad for an amateur, huh. I'm glad you see the merit in my perspective, Thom, and I appreciate your additional comments, linking mathematics, logic, and epistemology (viz., concept-formation).

I first came up with this angle a little over 12 years ago in David Kelley's cyberseminar on propositions. They were wondering what was the ontological interpretation of x to the zero power. I told them it had to do with an operation that was not performed on the unit 1. (If Ba'al agreed with this, he'd still probably want to say: "an operation that performed no times on the unit 1." I have a couple of essays on my web site about zero powers and such, but I won't bother with links at this time. Those interested can prowl around on www.rogerbissell.com and find it easily enough.

Anyway, I'm grateful for this particular discussion, because it has crystallized my thinking more generally about zero as not an operator, but a blocker of operations. [Emphasis added, Thom.]

I agree with you that in order to properly orient mathematics (back) to the real world, a lot of re-interpretation is necessary. Some of that re-interpretation has to do with getting clear about exactly what operations we are or are not performing. So, it's not all metaphysical/ontological, but the fact that the operations of mathematics can be performed on things in the real world means that such homely little examples I used such as the empty room vs. the room with 5 chairs can actually shed light on what operations can and cannot be performed mentally, mathematically or logically or otherwise.

REB

Roger, I think we are on to something.

For quite a while now, I have been studying about the discrepancy between what most mathematical logicians say about logic and what computer programmers do when they write software applications--or what chip designers do when laying out logic circuits. What I have noticed is that the former have a state model of logic while the latter have a process model.

I mean by this, that logicians are concerned with truth tables, mappings, and all possible contingent states of affairs, as if from an omniscient bird's-eye view; while on the other hand, that programmers are concerned mainly with processing logical operations to reach some goal state. The discrepancy lies in the fact that programmers and designers do not follow what logicians have claimed about the nature of logic.

Now the issue isn't whether without the sciences of logic, mathematical logic, and other subsequent sciences, there would not exist modern products of technology; all of which is true and readily conceded. The issue is whether logicians have interpreted modern logic in such a way that make inferences in the real world practical. And plainly, the fact is that programmers and designers (henceforth, technologists) do not take the former's interpretation into consideration when making their programming or designing decisions to do their work.

In a sense, the practical technologists are doing logic without epistemological guidance from modern logicians as to how it is that their jobs turn out to be productive and conforming to reality. They know that logic works wondrously, but they don't know why. They are perplexed because that which is often claimed to be the explanation doesn't match with what they are doing. The mathematical logicians' theory doesn't match with the technologists' practice. There is the disconnect.

I think this is exactly where Objectivist epistemology can enter to help correct centuries-old mistakes in the various sciences, most crucially, the modern science of logic. To begin, "logic" itself, as a fundamental concept of method, must be redefined on an Aristotelian-Objectivist foundation. (ITOE 36) It is as if we are repeating history: Logic is neither subjective (psychologism) nor intrinsic (logicism) but objective.

For the rest of this article, I'm going to show the consequence of an aspect of the state model of logic. I will contrast this model with how technologists actually do logic. And by modus tollens, I falsify the state model and outline indirectly a theory of the process model of logic.

To illustrate what I mean about the discrepancy, consider the well-known paradox of entailment in modern logic. If whenever you have two contradictory premises in an argument, your conclusion validly can be deuces wild. For example, since the dog is on the mat, and it isn't (at the same time in the same respect); therefore, the U.S. economy negotiates one trillion decisions per microsecond.

Since the argument is valid, so say modern logicians, therefore "the U.S. economy negotiates one trillion decisions per microsecond" is proven by virtue of the argument. In fact, on this basis, you can insert anything for the conclusion-part of the argument, and it will come out as perfectly validated.

If you did not independently know the truth or falsity of a statement, and you plug that statement as the conclusion part of the argument, and the whole argument delivers the verdict of, yes, valid; what then do you conclude of its conclusion? Whenever you hear a "therefore" before a statement, and the argument is a valid one, what do you usually conclude? That the conlusion is false? No, if the point of valid reasoning is to have a way of inferring new truths from prior truths, then naturally whenever you hear that an argument is valid, then you would have to conclude naturally that that which you didn't know to be true or false before is now known to be true.

Not so fast, according to the state model of logic. Modern logicians caution that in reason, don't trust your reason. The inserted statement into the conclusion could have been known by some other means to be false. And the whole argument could still come out clean (laundered?) as valid.

Not only that, there is also this truism concerning your premises: Who can say honestly that he doesn't have one or two falsehoods floating in his head? If so, whenever he makes an argument to infer new truths, might he not have contradictory premises? Therefore, might he not have valid reasoning whose conclusion he cannot trust to be true? So, in other words, you can't trust whatever you knew prior, to be true.

This double injunction--don't trust reason, and don't trust knowledge--follows from the paradox of entailment. It comes straight out of bedrock logic in modern times.

Technologists, and the man on the street with his common sense, don't buy any of that. I won't try to "prove" it just yet, but the evidence is everywhere--counterevidence, that is. All the modern gadgets and conveniences of life attest to this.

The state model of logic is littered with paradoxes. Many of them, if not most, have their origins in a conception of truth as being functional. The entailment paradox is no different and is merely the simplest and nearest to the theoretical foundation of the theory. A cornerstone of this theory is a particular conception of the hypothetical proposition, which in mathematical logic is allegedly mapped to a logical operation called the "material implication." The consequence of this mapping, if followed, would make practical technology unworkable. Indeed, it is not followed, because this "material implication" is itself paradoxical.

A hypothetical proposition is nothing more than an everyday English "if" statement. It is a statement asserting a relationship of dependence between two thoughts, between the antecedent and the consequent, to identify some dependency among facts, events, or possibilities. We say something is a fact because of its relationship to another fact. There is no mystery about it. We assert these relations all the time. But in modern logic, there is a paradox in the way its truth is derived. The grammatical form of the hypothetical proposition in the state-model view is treated as a truth function.

This truth function maps "if" to "either the consequent or not the antecedent," reducing the "if" to an "either-or" statement with a nested "not." And this reduction is called "material implication." (The "either-or" itself is mapped primitively to a logical operation called the "disjunction.") If this sounds complicated to understand, requiring a college education or higher, that's so because it is. Modern logic is a logic not suited for human thinking. Man-made electronic machines only seem to follow this modern logic, but they actually do not--a point that I will show presently.

Put yourself in the place of the first-person pronoun in this situation below and follow HP1 through HP6:

A1. I see a dollar on the table right now.

A2. You have four dollars in your hands, and you honor what you promise.

A2. You will add another dollar on the table for each time I evaluate an "if" statement to be true.

HP1. "If the table has a dollar on it, then the table is serving its function as a table."

MI1. Add dollar now if HP1.

HP2. "If the table has 2 dollars now, then I have gained a dollar."

MI2. Add dollar now if HP2.

HP3. "If the table has 3 dollars now, then you have 3 dollars left in your hands."

MI3. Add dollar now if HP3.

HP4. "If the table has 2 dollars now, then I have gained another dollar."

MI4. Add dollar now if HP4.

HP5. "If the table has 3 now dollars now, then you have no money left."

MI5. Add dollar now if HP5.

HP6. "If the table has any money at all now, then you have some money in your hands."

MI6. Add dollar now if HP6.

Before you read on, how many dollars should there be on the table? Common sense says

_ dollars

. According to modern logic, there should be

5 dollars

on the table. It has to be so because of the way modern logic prescribes how truth is to be computed.

Truth function for entailment: f((P1+P2+...+Pn), C) = V
	Premises	Conclusion	F()	Validity
	T			T			=		T
	T			F			=		F
	F			T			=		T
	F			F			=		T

Truth function for material implication: f(A, C) = MI
	Antecedent	Consequent	F()=	Implication
	T			T			=		T
	T			F			=		F
	F			T			=		T
	F			F			=		T

Truth function for MI-reduced: f.either-or(C, f.not(A)) = "either C OR not A"
	Ante.	Conse.	Not(Antedent)	F()	OR(C,not(A))
	T		T				F		=		T
	T		F				F		=		F
	F		T				T		=		T
	F		F				T		=		T

With these truth functions, the following items are supposed to be hypothetical propositions, the paradoxes of the "if" are in PMI3 and PMI4:

PMI1. "If 2+2=4, then 4-2=2." true

PMI2. "If 2+2=4, then 4-2=3." false

PMI3. "If 2+2=5, then 2+2=4." true

PMI4. "If 2+2=5, then 2+2=6." true

The paradox summarizes conventionally as follows: The "if" statement is true both whenever the consequent is evaluated true, no matter the antecedent; and whenever the antecedent is evaluated false, no matter the consequent.

This is why HP4 and HP5 above, by modern prescription, are evaluated true.

The states of affairs of both the antecedent and the consequent are evaluated simultaneously and independently to compute the truth of the compound statement. At the primitive level, it matters not which clause gets evaluated first, consequent or antecedent, so long as both are evaluated. This is by virtue of the so-called commutativity property of the disjunction operation, e.g., PMI1 being translated and evaluated as equivalent to: either "4-2=2" is true or "2+2=4" is not true.

In fact (1), ordinary people, and practical technologists among them, never follow this state model of logic, not because it is inefficient, but because it doesn't work in reality. To see how this is so, follow the statements MI1 through MI6. If you followed the state-model logic, you would have to add 6 dollars to the table (2 dollars of which you don't have) since the clause "Add dollar now" is evaluated simultaneously and independently of any other clause, and is evaluated true since you are an honorable person.

In fact (2), the consequent "Add dollar now" comes into effect, and to be evaluated, only when the antecedent is true. And when the consequent is evaluated, by your putting a dollar on the table, then the whole compound statement, e.g., MI1, is true. Only if (and I emphasize "only"), and to repeat, only if you fail in that instance to put a dollar on the table can the whole compound statement be false. "Add dollar now" is always carried out, true, whenever it comes into effect for evaluation; your integrity ensured it.

Thus, it should be apparent that there are two different conceptions of logic: a state model in accordance with truth functions and tables, and a process model in accordance with practical life. On the one hand, you are told to evaluate both antecedents and consequents to compute truth. On the other, you think and act otherwise. Logicians would evaluate MI1 through MI6 as true. From the commonsensical point of view, the standard by which people and technological machines operate, MI1, MI2, and MI6 are true. Those are the "if" statements that actually do add another dollar on the table.

Being human, in order to evaluate the iffiness of "if" statements, one has to have accepted some particular standard or another for their evaluations. But how is one to evaluate which "if"-model has the better "if" without resorting to some ontological interpretation about the nature of hypothetical relations, or without resorting to some epistemological conception of a human consciousness that must grasp such relations. The irony is that the state model of logic presupposes a God-like consciousness contemplating eternal relations. If only we were so. Humans don't argue from contradictions if we can help it, and we don't evaluate "if" from disjunctions.

Why don't technologists and the common man buy into modern logic? Because they implicitly hold on to a this-worldly view of logic, which has up until now been left undefended. It is the implicit view that logic must be a practical art, an instrumental means for discovering the facts of reality. And to the extent that this view is in the minds of technologists and that it hasn't been corroded or corrupted or replaced outright by the opposite view; to that extent has modernity been hobbling along.

This quotation from Ayn Rand neatly summarizes what has happened in the science of logic.

Today's frantic development in the field of technology has a quality reminiscent of the days preceding the economic crash of 1929: riding on the momentum of the past, on the unacknowledged remnants of an Aristotelian epistemology, it is a hectic, feverish expansion, heedless of the fact that its theoretical account is long since overdrawn--that in the field of scientific theory, unable to integrate or interpret their own data, scientists are abetting the resurgence of a primitive mysticism. [AR CTUI 11]

[BaalChatzaf]

You have conveniently forgotten the main rule of inference: modus ponens.

If p -> q (material implication) one can infer q only if p is true. So to argue soundly one must not only have the implication but the truth of the premise. Such truth is determined elsewhere and otherwise.

Natural Inference which is based on propositional logic (which you have denigrated) and first order predicate log correctly models who mathematicians prove theorems.

See http://en.wikipedia.org/wiki/Natural_deduction

Rand spent a great deal energy defecating upon mathematics and logic. She bashed physics 50 years ago and it has been chugging on, just the same. In the mean time in the real world, mathematics which is correctly modeled by Natural Deduction has supported physics which both in its theoretical and applied form has vastly increased our knowledge of the material world. All this in spite of Rand telling us how awful math, physics and logic are and how we only make progress by really ignoring math, physics and logic.

Do you really, really buy this? I sure do not.

Ba'al Chatzaf

Edited May 20, 2009 by BaalChatzaf

[Reidy]

I confess to not having read your post in detail, but I think you're saying:

A division of labor prevails. Philosophers ponder what logic and its operations mean. Programmers and circuit designers know what they need to know to get the job done, and they do it. (Much of that knowledge was discovered by effete, hatred-eaten mystics like Frege, Russell and Tarski around the turn of the twentieth century.) Similar divisions prevail between philosophers of law and working lawyers, applied scientists and philosophers of science, workers in the field of X and philosophers of X generally.

This does not reflect on anybody's character or motives, badly or well.

[Laure]

Thom, you haven't shown how the practical logic as practiced by a programmer or circuit designer contradicts in any way the state model of logic that you're railing against.

"If p then q" is truth-value equivalent to "q or not p", and I can prove it.

When can "if p then q" be false? Only in the case that q and not p.

There are four possibilities: p and q, p and not q, q and not p, and not q and not p.

"If p then q" is false for one of those cases,

"If p then q" is either true or false,

so...

for the other three cases, "if p then q" is true.

The other three cases are: p and q, q and not p, and not q and not p, which reduces to:

q or not p.

What part of that do you dispute?

I think your whole line of thinking here is based on the same kind of misapprehension as the novice in a programming class, seeing the statement x = y+1 and saying "no it isn't!" instead of realizing that in a program, x = y+1 is not an algebraic assertion, but an operation: it means "set x equal to y plus 1". Likewise, in a computer program, we wouldn't typically evaluate the truth-value of "if p then q"; we evaluate the truth-value of p, and if it's true, we DO the ACTION q.

[BaalChatzaf]

Thom, you haven't shown how the practical logic as practiced by a programmer or circuit designer contradicts in any way the state model of logic that you're railing against.

"If p then q" is truth-value equivalent to "q or not p", and I can prove it.

When can "if p then q" be false? Only in the case that q and not p.

There are four possibilities: p and q, p and not q, q and not p, and not q and not p.

"If p then q" is false for one of those cases,

"If p then q" is either true or false,

so...

for the other three cases, "if p then q" is true.

The other three cases are: p and q, q and not p, and not q and not p, which reduces to:

q or not p.

What part of that do you dispute?

I think your whole line of thinking here is based on the same kind of misapprehension as the novice in a programming class, seeing the statement x = y+1 and saying "no it isn't!" instead of realizing that in a program, x = y+1 is not an algebraic assertion, but an operation: it means "set x equal to y plus 1". Likewise, in a computer program, we wouldn't typically evaluate the truth-value of "if p then q"; we evaluate the truth-value of p, and if it's true, we DO the ACTION q.

That is on point. The "if" of programing is a conditional execution of some procedure.

Formally If p then do q else do r. This is not material implication. This is the execution of procedure q if p is true else the execution of r if p is not true. Confusing the conditional if of programming with material implication is conflation and semantic confusion.

Material implication is one half of what is necessary to infer a proposition. The other half is to assert the premise and apply modus ponens. That is how conclusions involving material implication are drawn.

Ba'al Chatzaf

[Dragonfly]

Much of that knowledge was discovered by effete, hatred-eaten mystics like Frege, Russell and Tarski around the turn of the twentieth century.

And what is the evidence that Frege, Russel and Tarski were 'effete, hatred-eaten mystics'?

[Reidy]

Read Peikoff or any of the people around him.

[BaalChatzaf]

Read Peikoff or any of the people around him.

That is not evidence, that is bigotry.

I had courses with Tarski. He was neither effete nor a mystic.

Ba'al Chatzaf

[Dragonfly]

Read Peikoff or any of the people around him.

You can't be serious. Peikoff's rants about scientists are complete nonsense. A few examples can be found here and here.

[Reidy]

What a couple of pseudo-Objectivist snarling wimps you are. You will never be friends with Peter Schwartz, much less James Valliant, at this rate.

So you want to name-drop, do you? I heard Tarski lecture once. Didn't understand a word, and I think he was speaking English. Attended class sessions by Church and Montague but decided not to enroll.

[Michael Stuart Kelly]

Pete,

LOL...

You're a hoot.

I saw the from a ways off and wondered where it was going to arrive.

And people accuse orthodox Objectivists of not having a sense of humor (nor being able to detect it)...

Michael

[thomtg]

I confess to not having read your post in detail, but I think you're saying:

A division of labor prevails. Philosophers ponder what logic and its operations mean. Programmers and circuit designers know what they need to know to get the job done, and they do it. (Much of that knowledge was discovered by effete, hatred-eaten mystics like Frege, Russell and Tarski around the turn of the twentieth century.) Similar divisions prevail between philosophers of law and working lawyers, applied scientists and philosophers of science, workers in the field of X and philosophers of X generally.

This does not reflect on anybody's character or motives, badly or well.

Peter,

Though short, your post summarizes a structural point of my article. I also take your parenthetical comment about "effete, hatred-eaten mystics" exactly for what it is in the context: an attempt at humor. Thanks for your understanding.

[thomtg]

Thom, you haven't shown how the practical logic as practiced by a programmer or circuit designer contradicts in any way the state model of logic that you're railing against.

"If p then q" is truth-value equivalent to "q or not p", and I can prove it.

When can "if p then q" be false? Only in the case that q and not p.

There are four possibilities: p and q, p and not q, q and not p, and not q and not p.

"If p then q" is false for one of those cases,

"If p then q" is either true or false,

so...

for the other three cases, "if p then q" is true.

The other three cases are: p and q, q and not p, and not q and not p, which reduces to:

q or not p.

What part of that do you dispute?

I think your whole line of thinking here is based on the same kind of misapprehension as the novice in a programming class, seeing the statement x = y+1 and saying "no it isn't!" instead of realizing that in a program, x = y+1 is not an algebraic assertion, but an operation: it means "set x equal to y plus 1". Likewise, in a computer program, we wouldn't typically evaluate the truth-value of "if p then q"; we evaluate the truth-value of p, and if it's true, we DO the ACTION q.

Laure,

Thank you for your interest in following my train of thought. But I will disagree with you from the outset. I have shown the discrepancy in two modes. What I have asked of the interested reader has been to follow this train of thought to see for himself. One side has paradoxes. The other does not. There is considerable amount of introspection involved in this task. After all, we are dealing with the iffiness of hypotheticals.

Isn't it paradoxical that logic teaches that whenever there is a seeming problem, one should go back and question the premises, and yet when logic itself has paradoxes (as interpreted by the state model logic) few want to examine its premises? This is all I am trying to suggest.

So, if you wish to engage in debugging the problem, either with my train of thought or with logic itself, I welcome the open dialog, provided you grant me the principle of charity. I have presented my case the best I could, given the constraints.

If you are willing to start afresh, may I ask, without looking at the spoilers, how many dollars do you find there on the table after reading HP1 through HP6? I then ask that you introspect the process of how you arrive at that total. Is HP1 true or false, ..., HP6 true or false? I then ask that you use the exact evaluative method just introspected to evaluate the instructions MI1 through MI6. What do you find?

[BaalChatzaf]

Though short, your post summarizes a structural point of my article. I also take your parenthetical comment about "effete, hatred-eaten mystics" exactly for what it is in the context: an attempt at humor. Thanks for your understanding.

I took at as bigotry.

I have actually talked to Imam Leonard about logic and mathematics. He is an ingnoramus and a bigot.

Ba'al Chatzaf

[Laure]

Humans don't argue from contradictions if we can help it, and we don't evaluate "if" from disjunctions.

Gosh, I guess I'm not human. I'm feeling more like Ba'al every day.

In answer to your last post, Thom, I got "5". I admit some of them were a little tricky, so I had to be careful to express all the "if p then q" statements as "q or not p". Humans often read an "if" and wrongly assume that it means "if and only if".

I don't really know what you mean by evaluating the instructions MI1 through MI6. Do you mean performing them? Or determining their truth-value? I'm going to write you up a little computer program that will run your example, but I cannot put an "if-then-do" into an "if" statement as the expression to be evaluated; it's a syntax error. I could make a function called AddADollarIfTrue, and pass in the truth-value of the H statement above it. Then I could decide if I want that function to return "true" in all cases or only if I've added a dollar; it's up to how I want to code it.

You haven't answered whether you dispute that "if p then q" is equivalent in truth-value to "q or not p", and if so, what you think the problem is.

[BaalChatzaf]

Thom T G him say:

If only we were so. Humans don't argue from contradictions if we can help it, and we don't evaluate "if" from disjunctions.

I answer:

My wallet is either on my desk or on the table near my bed. (I go look) Gee. It is not on the table near my bed. Therefor it is on my desk.

The form of the inference is thus: p or q, not p therefore q. It is as common as dust. Humans reason this way quite often. Thom T G knoweth not wheeof he speaks. Not all that unusual for an O'ist (unfortunately).

Ba'al Chatzaf

[Laure]

OK, Thom, here's how I coded it. It's possible that I made a mistake somewhere, but here's what I have. (I made no attempt to make it elegant, because I want to show everything as explicitly as possible.)

#include <stdio.h>

void main()
{
	int Table[7] = {1,0,0,0,0,0,0}; // Table starts with 1 dollar
	int Hand[7] = {4,0,0,0,0,0,0}; // Hand starts with 4 dollars

	if (Table[0]!=1 || &Table)  // Note"  "||" means OR
	{
		Table[1] = Table[0] + 1;
		Hand[1] = Hand[0] - 1;
		printf("Step 1 TRUE, Table=%d, Hand=%d\n", Table[1], Hand[1]);
	}
	else
	{
		Table[1] = Table[0];
		Hand[1] = Hand[0];
		printf("Step 1 FALSE, Table=%d, Hand=%d\n", Table[1], Hand[1]);
	}

	if (Table[1]!=2 || Table[1] == Table[0] + 1)
	{
		Table[2] = Table[1] + 1;
		Hand[2] = Hand[1] - 1;
		printf("Step 2 TRUE, Table=%d, Hand=%d\n", Table[2], Hand[2]);
	}
	else
	{
		Table[2] = Table[1];
		Hand[2] = Hand[1];
		printf("Step 2 FALSE, Table=%d, Hand=%d\n", Table[2], Hand[2]);
	}

	if (Table[2] != 3 || Hand[2] == 3)
	{
		Table[3] = Table[2] + 1;
		Hand[3] = Hand[2] - 1;
		printf("Step 3 TRUE, Table=%d, Hand=%d\n", Table[3], Hand[3]);
	}
	else
	{
		Table[3] = Table[2];
		Hand[3] = Hand[2];
		printf("Step 3 FALSE, Table=%d, Hand=%d\n", Table[3], Hand[3]);
	}

	if (Table[3]!=2 ||Table[3] == Table[2] + 1)
	{
		Table[4] = Table[3] + 1;
		Hand[4] = Hand[3] - 1;
		printf("Step 4 TRUE, Table=%d, Hand=%d\n", Table[4], Hand[4]);
	}
	else
	{
		Table[4] = Table[3];
		Hand[4] = Hand[3];
		printf("Step 4 FALSE, Table=%d, Hand=%d\n", Table[4], Hand[4]);
	}

	if (Table[4] != 3 || Hand[4] == 0)
	{
		Table[5] = Table[4] + 1;
		Hand[5] = Hand[4] - 1;
		printf("Step 5 TRUE, Table=%d, Hand=%d\n", Table[5], Hand[5]);
	}
	else
	{
		Table[5] = Table[4];
		Hand[5] = Hand[4];
		printf("Step 5 FALSE, Table=%d, Hand=%d\n", Table[5], Hand[5]);
	}

	if (Table[5] <= 0 || Hand[5] > 0)
	{
		Table[6] = Table[5] + 1;
		Hand[6] = Hand[5] - 1;
		printf("Step 6 TRUE, Table=%d, Hand=%d\n", Table[6], Hand[6]);
	}
	else
	{
		Table[6] = Table[5];
		Hand[6] = Hand[5];
		printf("Step 6 FALSE, Table=%d, Hand=%d\n", Table[6], Hand[6]);
	}
}

My output is:

Step 1 TRUE, Table=2, Hand=3

Step 2 TRUE, Table=3, Hand=2

Step 3 FALSE, Table=3, Hand=2

Step 4 TRUE, Table=4, Hand=1

Step 5 TRUE, Table=5, Hand=0

Step 6 FALSE, Table=5, Hand=0

Edited May 22, 2009 by Laure

[Dragonfly]

Though short, your post summarizes a structural point of my article. I also take your parenthetical comment about "effete, hatred-eaten mystics" exactly for what it is in the context: an attempt at humor.

I don't see the humor when such things are in fact said by well-known Objectivists like Peikoff. It would be humor if you parody such statements in such a way that even Peikoff would see that they're ridiculous. A nice example is Jonathan's post #104 in the conspiracies thread *) in which he parodies the conspiracy theories by inventing an even more absurd theory, although that's hardly possible! - after all, it's difficult to surpass the absurdity in the theory that the planes that flew into the twin towers were flown by remote control and all that stuff about hijackers has just been invented by the tv companies, no doubt under CIA control! Why should you seriously discuss such absurdities, this is the right moment for Mencken's "one horse-laugh is worth ten thousand syllogisms".

*) for some reason linking to other posts on this forum no longer works.

[thomtg]

Laure,

Thank you very much for the two replies. Your diligence in following this topic, on the nature of hypothetical statements, reminds me of an original Star Trek episode, Episode 305 "Is There in Truth no Beauty?," in which a telepathic doctor, Doctor Miranda Jones, said to Spock before the mind-meld: "Now, Spock, this is to the death, ...!"

Let me address your beginning and ending first.

Humans don't argue from contradictions if we can help it, and we don't evaluate "if" from disjunctions.

Gosh, I guess I'm not human. [...]

You may want to withdraw that judgment, Laure, when we are done with our introspective journey.

[...]

You haven't answered whether you dispute that "if p then q" is equivalent in truth-value to "q or not p", and if so, what you think the problem is.

The hypothetico-propositional form "if p then q" is a legitimate propositional structure that is distinct from other propositional forms for factual identification in human cognition. That it has been claimed by modern logic to be equivalent truth-functionally to the alternate-propositional form "q or not p" is the issue under dispute. I am disputing it. (And you will too if our journey is successful.) The fact to be demonstrated is, technologists, and everyone else for that matter, treat them distinctly. Treating if's as if they were material implications gives rise to paradoxes, making them impractical. I have stated nothing here that I haven't already stated in my original post.

OK, Thom, here's how I coded it. It's possible that I made a mistake somewhere, but here's what I have. (I made no attempt to make it elegant, because I want to show everything as explicitly as possible.)

[...]

Your program works great and concretizes perfectly the discrepancy in modern logic between theory and practice. But we are getting ahead of ourselves. I recommend you use and annotate the program as you introspect in the next three iterations.

In answer to your last post, Thom, I got "5". I admit some of them were a little tricky, so I had to be careful to express all the "if p then q" statements as "q or not p". Humans often read an "if" and wrongly assume that it means "if and only if".

To your answer, I agree that "5" is the answer modern logic would prescribe. The truths of hypothetical statements are to be computed truth functionally, in accordance with the truth table of "material implication." I included the same answer as yours in the original post in the form of a hidden spoiler tag. I also included therein the truth table of "material implication" and its mapping from a compound "disjunction."

[...]

I don't really know what you mean by evaluating the instructions MI1 through MI6. Do you mean performing them? Or determining their truth-value? [...]

Yes, I do mean that you introspect to determine their truth values.

You agree, do you not, that MIx (1 through 6) are instructions in the form of hypothetical statements? These are statements addressed to someone with honor and integrity. Rational human beings think and act on their judgments. These are hypothetical statements to be thought of and acted on. Therefore, these hypothetical statements can be evaluated logically in the same and exact way as the hypothetical statements HPx (1 through 6). You thought and acted per instruction A3 [a typo] to yield your answers, did you not? How did you do it? Introspect what you did.

Laure1. If HP1 then I say "true."

Laure2. If HP2 then I say "true."

Laure3. If HP3 then I say "true."

Laure4. If HP4 then I say "true."

Laure5. If HP5 then I say "true."

Laure6. If HP6 then I say "true."

Laure7. If MI1 then I say "true."

Laure8. If MI2 then I say "true."

Laure9. If MI3 then I say "true."

LaureA. If MI4 then I say "true."

LaureB. If MI5 then I say "true."

LaureC. If MI6 then I say "true."

Secondly, considering the MIx instructions again, would you agree that in comprehending them cognitively as hypothetical statements, they are the same as: "Adding a dollar now is true if HPx is true"? The other person, too, thought and acted per instruction A3. I am asking you now to introspect what he did.

Other1. If MI1 then I say "true."

Other2. If MI2 then I say "true."

Other3. If MI3 then I say "true."

Other4. If MI4 then I say "true."

Other5. If MI5 then I say "true."

Other6. If MI6 then I say "true."

Other7. If HP1 then I say "true."

Other8. If HP2 then I say "true."

Other9. If HP3 then I say "true."

OtherA. If HP4 then I say "true."

OtherB. If HP5 then I say "true."

OtherC. If HP6 then I say "true."

Finally, consider the technological wonders of computer programs, yours in particular. Evaluate your hypothetical statements CS_x (1 through 6).

#include <stdio.h>

void main()
{
	int Table[7] = {1,0,0,0,0,0,0}; // Table starts with 1 dollar
	int Hand[7] = {4,0,0,0,0,0,0}; // Hand starts with 4 dollars

	if (Table[0]!=1 || &Table)  // Note"  "||" means OR				 // CS_1. "If HP1 then {..., I say 'true'}."
	{
		Table[1] = Table[0] + 1;
		Hand[1] = Hand[0] - 1;
		printf("Step 1 TRUE, Table=%d, Hand=%d\n", Table[1], Hand[1]);
	}
	else																// CSn1. "If not HP1 then {...}."
	{
		Table[1] = Table[0];
		Hand[1] = Hand[0];
		printf("Step 1 FALSE, Table=%d, Hand=%d\n", Table[1], Hand[1]);
	}

	if (Table[1]!=2 || Table[1] == Table[0] + 1)						// CS_2. "If HP2 then {..., I say 'true'}."
	{
		Table[2] = Table[1] + 1;
		Hand[2] = Hand[1] - 1;
		printf("Step 2 TRUE, Table=%d, Hand=%d\n", Table[2], Hand[2]);
	}
	else																// CSn2. "If not HP2 then {...}."
	{
		Table[2] = Table[1];
		Hand[2] = Hand[1];
		printf("Step 2 FALSE, Table=%d, Hand=%d\n", Table[2], Hand[2]);
	}

	if (Table[2] != 3 || Hand[2] == 3)								  // CS_3. "If HP3 then {..., I say 'true'}."
	{
		Table[3] = Table[2] + 1;
		Hand[3] = Hand[2] - 1;
		printf("Step 3 TRUE, Table=%d, Hand=%d\n", Table[3], Hand[3]);
	}
	else																// CSn3. "If not HP3 then {...}."
	{
		Table[3] = Table[2];
		Hand[3] = Hand[2];
		printf("Step 3 FALSE, Table=%d, Hand=%d\n", Table[3], Hand[3]);
	}

	if (Table[3]!=2 ||Table[3] == Table[2] + 1)						 // CS_4. "If HP4 then {..., I say 'true'}."
	{
		Table[4] = Table[3] + 1;
		Hand[4] = Hand[3] - 1;
		printf("Step 4 TRUE, Table=%d, Hand=%d\n", Table[4], Hand[4]);
	}
	else																// CS_n4. "If not HP4 then {...}."
	{
		Table[4] = Table[3];
		Hand[4] = Hand[3];
		printf("Step 4 FALSE, Table=%d, Hand=%d\n", Table[4], Hand[4]);
	}

	if (Table[4] != 3 || Hand[4] == 0)								  // CS_5. "If HP5 then {..., I say 'true'}."
	{
		Table[5] = Table[4] + 1;
		Hand[5] = Hand[4] - 1;
		printf("Step 5 TRUE, Table=%d, Hand=%d\n", Table[5], Hand[5]);
	}
	else																// CSn5. "If not HP5 then {...}."
	{
		Table[5] = Table[4];
		Hand[5] = Hand[4];
		printf("Step 5 FALSE, Table=%d, Hand=%d\n", Table[5], Hand[5]);
	}

	if (Table[5] <= 0 || Hand[5] > 0)								   // CS_6. "If HP6 then {..., I say 'true'}."
	{
		Table[6] = Table[5] + 1;
		Hand[6] = Hand[5] - 1;
		printf("Step 6 TRUE, Table=%d, Hand=%d\n", Table[6], Hand[6]);
	}
	else																// CSn6. "If not HP6 then {...}."
	{
		Table[6] = Table[5];
		Hand[6] = Hand[5];
		printf("Step 6 FALSE, Table=%d, Hand=%d\n", Table[6], Hand[6]);
	}
}

[Laure]

Thom, I'm sorry, but I think there's something I don't understand. I don't know the point you are trying to make about these "M" statements. You say, "MI1. Add dollar now if HP1." I interpret that as "if HP1 is true, then add a dollar". Is that correct? That is an imperative statement, not a material implication. If you want to phrase it as a material implication, you could say, "if HP1 was true then a dollar was just added", and you assume that your person with the money is "obedient", then the "M" statements are always true, because it's always the case that either a dollar was added or the HPn was false.

So, what is it that I'm not understanding? I really think we may be able to get to the point where one or the other of us says, "Oh, OK, I get it."

[thomtg]

Thom, I'm sorry, but I think there's something I don't understand. I don't know the point you are trying to make about these "M" statements. You say, "MI1. Add dollar now if HP1." I interpret that as "if HP1 is true, then add a dollar". Is that correct? That is an imperative statement, not a material implication. If you want to phrase it as a material implication, you could say, "if HP1 was true then a dollar was just added", and you assume that your person with the money is "obedient", then the "M" statements are always true, because it's always the case that either a dollar was added or the HPn was false.

So, what is it that I'm not understanding? I really think we may be able to get to the point where one or the other of us says, "Oh, OK, I get it."

Laure,

Thanks for the vote of confidence that we may at some point come to a meeting of minds, one way or the other. I do believe you are beginning to see a discrepancy of some sort. If so, this calls for extra effort at introspection to determine the cause.

An imperative statement (e.g., Kant's categorical imperative) is a statement issuing a command. For example, "(You) turn off the TV and do your homework." It is an order for action. If it is carried out, the action becomes a man-made fact; if not, its absence becomes a fact.

An if-then statement is not an imperative statement. It is a hypothetical statement potentially conveying a hypothetical proposition, one that is either true or false. It is a statement asserting a relationship between two subthoughts, i.e., the relationship that the truth of the antecedent is sufficient to guarantee the truth of the consequent.

A principle of good shopkeeping is, "If you break it, you buy it." This is a hypothetical statement. It asserts a relation of dependence of the presence of one fact on another. It is not a command.

All Objectivist moral principles (unlike those in a command ethics, such as Kant's Golden rule, or Christian morality) are principles in the form of a hypothetical statement. One pratical principle, for example, is "If you want to succeed in a career, (you) work hard at your job." It is never, "Work hard at your job." The relationship here is one of ends to means. A moral principle has a reason; and if the reason is true, it is sufficient to guarantee the truth of the moral consequence. Here is a more abstract principle: "If reason is your sole means of acquiring knowledge, then you should refrain from deluding and corrupting reason by faking reality" (the virtue of honesty). The relationship here is one of value to virtue. (See DK "Ruled--Or Principled".)

MI1 or "(You) add a dollar now, if HP1 is true" is surely a hypothetical statement. It asserts the thought that there is a relationship between the fact of HP1 and the fact of another dollar on the table, i.e., the relationship that the truth of the antecedent ("HP1 is true") is sufficient to guarantee the truth of the consequent ("Add a dollar now (is true)"). It presumes from the start the implicit moral principle (A2) that "If you are honorable, you will do as you promised." So, MIx (1 through 6) are thoroughly and legitimately hypothetical statements.

But notice what I haven't said. I have not mentioned "material implication." Mathematical logic prescribes that hypothetical propositions be mapped to and computed by the "material implication" connective/operation, which is based on the truth table for a compound "disjunction" with a nested "negation" operation.

My claim is that technologists, so long as they are still unaffected at root by this prescription, continue to treat hypothetical statements as hypothetical, not as truth functional, which makes possible all the technological wonders of the world. The reason is that "material implication," if it were put into actual practice, would not work at all due to an inherent paradox. If logic tells us to question our premises if we find problems, why aren't we doing the same to material implications if we find them paradoxical?

If you agree that there is a paradox with material implication, then I ask that you introspect to see whether in your inferential and programmatical processes do you actually evaluate an "if" as an "if" and not as a "material implication." I believe that a three-pass introspection should yield some interesting results.

It should be apparent to both of us from your coded program that you are employing material implication to evaluate HPx. So, I recommend you continue that mode of evaluation in your first pass of introspection. Then when you switch perspective to the person holding the money, determine how it is that he is acting on his principles. (Rational and honorable people think and act on their judgments.) Use this method in the second pass to evaluate everything again. Then in the final pass, confirm what you the technologist actually did when you coded the program. If all goes well, you should see a discrepancy.

[Laure]

Thom, I still don't see what you're getting at with the "M" statements, but I think I understand your problem with statements such as HP4. Please see the thread I started, "Question on Conditionalizing", and visit the Wikipedia links mentioned there. I think you're a fan of "strict implication", which means that we look at the meaning of the P's and Q's and ask ourself if the P being true would in any way cause the Q to be true, and if not, we consider "P --> Q" to be false, even though the normal symbolic logic we learn in school says that it evaluates to true if P is false. I don't see any problem with conventional symbolic logic. We just need to define our terms so everyone is on the same page. In the link in the other thread on "paradoxes", the author mentions the idea that an argument with false premises can be logically "valid" although it is not logically "sound." I can go along with that, and suspect that you can, too. It's just a matter of defining our terms.

[BaalChatzaf]

This quotation from Ayn Rand neatly summarizes what has happened in the science of logic.

Today's frantic development in the field of technology has a quality reminiscent of the days preceding the economic crash of 1929: riding on the momentum of the past, on the unacknowledged remnants of an Aristotelian epistemology, it is a hectic, feverish expansion, heedless of the fact that its theoretical account is long since overdrawn--that in the field of scientific theory, unable to integrate or interpret their own data, scientists are abetting the resurgence of a primitive mysticism. [AR CTUI 11]

Utter nonsense! Great progress has been made in the theory of computability, an outgrowth of mathematical formal logic. We not only know which problems are not recursively solvable, but we have a theory which measures the complexity of computation for those problems which are solvable. In particular great progress has been made in identify the NP-complete and NP-hard problems.

The current state of mathematical or formal logic has enabled us to identify the limitations of formal systems. This work was initiated by Kurt Go'del in 1930 with his Incompleteness Theorems and Allan Turing with has analysis of solvable and unsolvable problems. The science of what is computable and is not was laid down by Turing, Church and Post in the 1930's.

A further offshoot of the theory of computable has been the development of hard to crack ciphers which are required for commercial and military security. One of the useful developments has been the public key-private key ciphers whose security is based on the hardness of certain problems in number theory, in particular the problem of factoring an integer into its prime factors in an effecient manner. This problem is computationally hard.

Once again Rand displays ignorance of the work that was available to her (if only she could read and understand it) in the late 1950's and 1960's. She did not know whereof she spoke (or wrote) in this regard.

Ba'al Chatzaf

Edited May 28, 2009 by BaalChatzaf

[Brant Gaede]

Ayn Rand had a deep polemical streak.

--Brant

[thomtg]

Thom, I still don't see what you're getting at with the "M" statements, but I think I understand your problem with statements such as HP4. Please see the thread I started, "Question on Conditionalizing", and visit the Wikipedia links mentioned there. I think you're a fan of "strict implication", which means that we look at the meaning of the P's and Q's and ask ourself if the P being true would in any way cause the Q to be true, and if not, we consider "P --> Q" to be false, even though the normal symbolic logic we learn in school says that it evaluates to true if P is false. I don't see any problem with conventional symbolic logic. We just need to define our terms so everyone is on the same page. In the link in the other thread on "paradoxes", the author mentions the idea that an argument with false premises can be logically "valid" although it is not logically "sound." I can go along with that, and suspect that you can, too. It's just a matter of defining our terms.

Laure,

Thank you for referring me to the forum thread on the topic of "Question of Conditionalizing" and its many linked articles therein. I have read these articles before. Although I am not a fan of "strict implication," I can see why you think I am, considering that it is an attempt at cleaning up some paradoxical messes with "material implication" in modern logic. That should say something positive about the discipline, that it has some concern for the imprecision in mapping hypothethical statements to material implications. Since you have a better understanding of why I have a problem with HP4 (and HP5, and HP6 for another reason), how many dollars should there be on the table? (Post #1)

I think that the MIx statements, as understood and acted on by any person in the scenario, falsify material implication and strict implication. By this I mean that Lines 1, 3, and 4 of the truth table shared by both connectives are not what constitute or establish the truths of the MIx statements. Their truths entirely depend on the denial of Line 2 if truth tables have to be referred to at all. In other words, the denial of falsity is not the same as the affirmation of truths with regard to truth tables in dealing with hypothetical statements.

What does it mean to assert a hypothetical proposition by means of a hypothetical statement? First of all, it never means asserting the component propositions. The antecedent and consequent are neither said to be true nor false by themselves. It is their relationship that is being asserted. A Hypothetical assertion is an assertion of a logical relationship. It is the basis of conditional proofs; assuming something true, what may then be true.

Secondly, it means that the assertion can be false only if its consequent (and only its consequent) is contradicted by actual non-hypothetically asserted facts. This is not to say that the consequent in itself, and independent of the antecedent cannot be false. But it is to say that for the relationship to be falsified, the consequent must be contradicted because of its dependence on the antecedent. This fact implies the rejection of both Lines 3 and 4 of the truth table as the basis for asserting the truth of the hypothetical statement when the antecedent is found indepently to be false.

Finally, it means that the assertion can be true only when it can be denied that the consequent contradicted actual non-hypothetically asserted facts. This is not to say the consequent must be found independently to be true, but it is to say that the denial of the consequent in its dependence on the antecedent is a fact. This denial implies the rejection of both Lines 1 and 3 of the truth table as the basis for asserting the truth of the hypothetical statement when the consequent is found independently to be true.

Thus, a hypothetical statement asserts a relationship, and its truth or falsity depends on the existence or absence of this relationship, not on the falsity of the (independent) antecedent, nor on the truth of the (independent) consequent.

Material implication and strict implication rely on Lines 1, 3, and 4. They rely on the independent and commutative evaluations of the antecedent and consequent. And they disregard altogether the actual relationship of dependence that is being asserted in evaluating its truth. Hence, the use of truth functional connectives--both material implication and strict implication--in dealing with hypothetical propositions becomes paradoxical in practice.

Now, my criticism for calling inferences from false premises as "valid" has been noted in the root post. Noteworthy also is the fact that David Kelley never draws out the so-called distinction between a "valid" argument and a "sound" argument in his treatment of logic. (See TAOR Ch. 4). The source for this valid-sound distinction, it seems to me, is from the analytic-synthetic dichotomy. (See ITOE pp. 112-118)