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.

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.

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.

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.

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.

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.

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.

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?

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.

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.

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).

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

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.

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]); } }

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."

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.

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.

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.

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)

## Recommended Posts

## 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 by BaalChatzaf## Link to comment

## Share on other sites

## 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.

## Link to comment

## Share on other sites

## 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.

## Link to comment

## Share on other sites

## BaalChatzaf

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

## Link to comment

## Share on other sites

## Dragonfly

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

## Link to comment

## Share on other sites

## Reidy

Read Peikoff or any of the people around him.

## Link to comment

## Share on other sites

## BaalChatzaf

That is not evidence, that is bigotry.

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

Ba'al Chatzaf

## Link to comment

## Share on other sites

## Dragonfly

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

## Link to comment

## Share on other sites

## Reidy

What a couple of pseudo-Objectivist snarling wimps you are. You will

neverbe 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.

## Link to comment

## Share on other sites

## 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

## Link to comment

## Share on other sites

## thomtg

AuthorPeter,

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.

## Link to comment

## Share on other sites

## thomtg

AuthorLaure,

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

exactevaluative method just introspected to evaluate the instructions MI1 through MI6. What do you find?## Link to comment

## Share on other sites

## BaalChatzaf

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

## Link to comment

## Share on other sites

## Laure

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.

## Link to comment

## Share on other sites

## 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

## Link to comment

## Share on other sites

## 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.)

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 by Laure## Link to comment

## Share on other sites

## Dragonfly

I don't see the humor when such things

arein 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.

## Link to comment

## Share on other sites

## thomtg

AuthorLaure,

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 Trekepisode, 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.

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

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.

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.

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."

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 exactway 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).

## Link to comment

## Share on other sites

## 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."

## Link to comment

## Share on other sites

## thomtg

AuthorLaure,

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.

## Link to comment

## Share on other sites

## 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.

## Link to comment

## Share on other sites

## BaalChatzaf

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 by BaalChatzaf## Link to comment

## Share on other sites

## Brant Gaede

Ayn Rand had a deep polemical streak.

--Brant

## Link to comment

## Share on other sites

## thomtg

AuthorLaure,

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

andstrict 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)

## Link to comment

## Share on other sites

## Create an account or sign in to comment

You need to be a member in order to leave a comment

## Create an account

Sign up for a new account in our community. It's easy!

Register a new account## Sign in

Already have an account? Sign in here.

Sign In Now