Does 1+1=3?

Science & Technology Physics and Mathematics
21

We all know it’s wrong. You identified why with many words, Rud identified it with few, that’s all I was pointing out.

It is not rationalism since no rationalist (or at least, no non-amateur rationalist) believes you can divide by zero. Descartes, if you’ll remember, was the foremost rationalist and one of the greatest mathematicians in history.

The person who made it up was not trying to prove a contradiction, I am almost sure of it. The person who came up with it was more likely trying to give a test to make people see what went wrong. In fact, this is how it was delivered to me. Someone wrote the proof and said, “So what’s wrong with it?” The point is to train the reader to become good at spotting invalid moves in mathematics. There are many such “proofs”.

Another less tricky one: 1 = (1)^(1/2) = ([-1][-1])^(1/2) = (-1)^(1/2) * (-1)^(1/2) = -1 ==> 1 = -1. You’re not supposed to be convinced by this, you’re supposed to find the error.

22

I’m sorry, I misunderstood. You’re right about the rationalism thing too, I actually meant to write rationalizing. Sorry, I should’ve payed more attention.

23

I’m sorry, I misunderstood. You’re right about the rationalism thing too, I actually meant to write rationalizing. Sorry, I should’ve payed more attention.

It wouldn’t be rationalizing, because that means trying to justify an emotion using pseudo-logic*. Rationalism means basically having mental functioning without referencing reality. Maybe the error is that one cannot divide by zero, but A + B = B only works for A = Zero, which means B is also Zero, is what I was getting at. You’re saying the mistaken step was the one before that.

Modal logic takes a premise (sometimes abstractly) and assumes it is true and comes to a conclusion. The Maverick Philosopher brought up a syllogism similar to this:

All pigs can fly

George is a pig

Therefore George can fly

Even assuming “pig” means the animal and not a fat slob, it doesn’t make sense because pigs cannot fly, even though it is a valid syllogism. That is the crucial difference between how modal logicians handle logic and how Objectivism handles logic. If logic is the art of non-contradiction, and a statement can contradict the facts, then saying all pigs can fly is a contradiction, and is therefore not logical.

Added on edit: A rationalization would be something like you feel angry, and therefore someone must have done you an injustice. Instead of checking your premises to see if someone did commit an injustice to you. It’s primacy of emotions instead of primacy of the facts and their relation to you.

24

That’s a pretty rough characterization of rationalism which I don’t think any rationalist would acquiesce to.

As for modal logic, not just modal logicians but EVERY logician would agree that that syllogism is false but valid. That’s just from what the word “valid” means. It means that, assuming true premises, the conclusion follows from them. The word valid has no other meaning in logic, and so the syllogism is in fact valid. The word “false” means something different than valid, and here the argument is false because of false premises. That’s not modal logic. That’s standard logic. And the adjective “logical” is not something formally defined in logic–it’s a colloquial term, so I have no idea what you mean by it in this context.

Whether this account is Objectivist logic, I don’t know, I’d be surprised if anyone said there was such a thing as Objectivist logic–but perhaps many Objectivists dislike this concept of validity, which I think would be unfortunate since it is a good distinction to show a person what is wrong with his argument and what he should be careful of in future arguments.

25

Yes, but a rationalist or a full modal logician will not look at the facts of reality to verify his mental processing. The argument I have heard regarding modal logic is that the conclusion must be true if the premises are true and if it is a valid syllogism. However, they don’t think true means in compliance with reality, and they don’t think false means it contradicts the facts of reality. In other words, the statement “All pigs can fly” is certainly not true, but a modal logician leaves it at that and doesn’t realize (at least not fully) that it is not true because it contradicts the facts of reality. That point doesn’t enter his head – it is either true or false to him, but he considers a contradiction to be between two opposite statements and not a false statement contradicting reality. That’s the crucial point.

A die-hard rationalist loves to do mental processing not grounded in the facts of his subject in the premises. Often, he does the mental processing on a definition alone, without taking all of the facts into account; thus he considers his conclusion to be true if there are no false statements or contradictory statements in his premises. As an example, take the following syllogism:

Matter does not have free will

Man is comprised of matter

Therefore man does not have free will

The first premise contradicts the facts of reality if you know that man is comprised of matter and yet he has free will, so the rationalist will not consider the first premise to contradict the facts, and he holds it as true because many things comprised of matter do not have free will and they don’t even have a consciousness – i.e rocks, tables, spoons, grass, dirt, etc. are not aware of anything and have no consciousness, let alone free will. So, a rationalist usually over-generalizes, goes strictly by definitions, and does not consider the facts included or subsumed under the premises.

26

Yes, but a rationalist or a full modal logician will not look at the facts of reality to verify his mental processing.

To which facts do you want a modal logician to refer? When discussing physical modality, they look to natural science, if that’s what you want. But if you’re talking about modal systems that account for some notions of possibility, well, they look to reality just as much as any standard logician does–they think about their axioms and write out a system which attempts to characterize them.

The argument I have heard regarding modal logic is that the conclusion must be true if the premises are true and if it is a valid syllogism.

Again, this isn’t modal logic. This is standard logic. Modal logic trades in boxes and diamonds, and accessibility relations. None of that is involved here. In fact, to be blunt, I’m pretty sure you have no idea what modal logic is. Modal logic is nothing more than standard logic with a non-truth-functional unary operator.

However, they don’t think true means in compliance with reality, and they don’t think false means it contradicts the facts of reality.

That is also a rough characterization to which no modal logician would readily acquiesce.

That point doesn’t enter his head – it is either true or false to him, but he considers a contradiction to be between two opposite statements and not a false statement contradicting reality. That’s the crucial point.

Well strictly speaking a contradiction is a statement which is contrary to itself or another statement–“contra” meaning, roughly, “against” and “diction” meaning, roughly, an expression. So it is not even grammatical to say that some statement contradicts reality or a fact. It is only grammatical to say that some statement contradicts a true statement about reality or a fact. So strictly, yes, a contradiction is only a statement and its negation.

Now a logician would NOT necessarily say that pigs cannot fly isn’t contradictory with true statements about reality.

thus he considers his conclusion to be true if there are no false statements or contradictory statements in his premises.

This is not relevant to rationalism, since many rationalists (like in any other branch of philosophy) will have many different notions of truth.

As an example, take the following syllogism:

Matter does not have free will

Man is comprised of matter

Therefore man does not have free will

This would never carry snuff with any logician, rationalist or not. There is the implicit premise that if something is comprised of another thing, then it carries all of the composite properties (or even some, namely, volition or its lack). This would be extremely contentions for any philosopher, rationalist or not. Ironically, though, rationalists tend to believe in free will more often than many other kinds of philosophers, like empiricists.

The first premise contradicts the facts of reality if you know that man is comprised of matter and yet he has free will, so the rationalist will not consider the first premise to contradict the facts,

Look this is silly. A rationalist may or may not believe that man is comprised of matter and has free will independent of all of this, and the same can be said for any other philosopher. If the rationalist believes it, then he will take the argument to be based on false premises (perhaps by denying the antheme), and if he does not believe it then he may or may not accept the argument. Just because a rationalist believes that our most certain truths are truths completely contained in the mind will not influence this subject in any determinate way.

Rationalism has nothing to do with these invalid proofs. They’re just tests and practices on spotting logical errors.

27

Well strictly speaking a contradiction is a statement which is contrary to itself or another statement–“contra” meaning, roughly, “against” and “diction” meaning, roughly, an expression. So it is not even grammatical to say that some statement contradicts reality or a fact. It is only grammatical to say that some statement contradicts a true statement about reality or a fact. So strictly, yes, a contradiction is only a statement and its negation.

This is a true rationalistic statement. You are arguing from the definition instead of arguing from the facts. Saying that I am not writing this reply contradicts the fact that I am writing this reply. It is not necessary to have two (or more) statements contradicting one another to have a contradiction. If you do not correctly identify a fact of reality and it is involved in one of your premises, then the premise contradicts reality.

I might agree with you that modal logic is standard logic using abstract symbols, but the more abstract the symbol the more likely one is to be rationalist – if you do not hold that truth means in compliance with the facts of reality.

Perhaps you can tell me, however, how else one can decide if a statement is true or false if you do not look at reality? That claim in and of itself smells of rationalism. If you never get down to the observationally self-evident at the root of your premises, then you are a rationalist because you are arguing from floating abstractions instead of arguing from the facts. In other words, you are using “logical operands” on symbols that have not been shown to have anything to do with the facts of reality – nearly the very definition of rationalism.

28

Yes, but a rationalist or a full modal logician will not look at the facts of reality to verify his mental processing.

Modal logic deals with the certainty scale (“possible”, “probable”, “certain”) and the fact that there’s no convenient way to describe the logic of sentences of the type “It is possible that X”, “It is probable that X”, where X is any proposition. It’s part of a slow trend to constructing formal models of reasoning that are psychologically valid (though some developments at the hands of modal logicians, like “possible worlds”, have been nuts).

29

Modal logic deals with the certainty scale (“possible”, “probable”, “certain”) and the fact that there’s no convenient way to describe the logic of sentences of the type “It is possible that X”, “It is probable that X”, where X is any proposition.

Be that as it may, even if you are correct, one cannot decide ranges of probability without references to the facts. Without referencing the facts, you can say or show that anything is possible. If I say something like, “It is possible for Sally to get cancer” you have to refer to the facts and see if it is possible for Sally to get cancer. If Sally is the name of your doll, then it is not possible for Sally to get cancer.

Possible means some of the facts indicate it might be true, while none contradict it; probable means there are even more facts to support the premise, and none against it; certain means you have enough facts to totally support the position. Without a reference to the facts and the right types of facts, probabilities are meaningless.

30

This is a true rationalistic statement. You are arguing from the definition instead of arguing from the facts.

What fact do you want? It is a FACT that this is what a contradiction means. Your saying you’re not writing that sentence contradicts the true sentence (which expresses a fact) that you are writing the sentence. That is what a contradiction is, strictly speaking. Otherwise, you’re just being ungrammatical.

I might agree with you that modal logic is standard logic using abstract symbols, but the more abstract the symbol the more likely one is to be rationalist – if you do not hold that truth means in compliance with the facts of reality.

You misunderstand. I did not say modal logic and standard logic are the same–they are decidedly not the same. What I was saying is that your statements have been about standard logic, and NOT about modal logic.

Perhaps you can tell me, however, how else one can decide if a statement is true or false if you do not look at reality?

I don’t know what you mean by “looking at reality”, but I know that any sentence of the form “P and not-P” is false–I need look to nothing empirical to justify my conclusion. If you tell me that a ibdegab is a ruthens and that an ibdegab is not a ruthens, even though I have no idea what those words mean, I know that it is a false sentence (if, in fact, it is a sentence). You could tell me that everything that is in a superposition has indeterminate causal power, and that there is something which is in a superposition which does not have indeterminate causal power. I don’t know what those words mean, really, but it doesn’t matter. The statement is false.

In other words, you are using “logical operands” on symbols that have not been shown to have anything to do with the facts of reality – nearly the very definition of rationalism.

Now that might be closer to rationalism, but it has nothing to do with the invalid argument above.

Be that as it may, even if you are correct, one cannot decide ranges of probability without references to the facts.

Modal logic doesn’t really deal in precise ranges of probability, that’s left for probability theory. Modal logic deals more in just the “necessary” and the “possible”, qualitatively rather than quantitatively.

31

Be that as it may, even if you are correct, one cannot decide ranges of probability without references to the facts.

No, really all that the logic part does is handle the logical relations of words, that any proposition which is “certain” is therefore “probable”, but not vice versa; anything “probable” is also “possible” but not vice versa. The most productive work in the field is about meanings of modals like “should”, “may”, “must” etc.

32

The facts I am referring to are the facts subsumed by the concepts in the premise. If you want to place a contradiction as a relationship between statements (which I decry) then saying something like, “All pigs can fly,” contradicts the facts subsumed by the concept “pig”, since the concept includes all the facts about pigs, including the fact that they cannot fly since they don’t have wings.

Regarding modal logic as David presented it, it may be true that what is certain is also possible or probable, but my point is that you cannot decide if something is certain without reference to the facts. The only way to show something is certain is by showing that all the facts lead to the conclusion and none contradict the conclusion. Without the facts subsumed by the proposition, you don’t have anything useful, just propositions linked to propositions, while none are linked to reality. The link to reality is more of the Objectivist understanding of logic in general. Logic in Objectivism is not just about valid syllogisms with true statements, it is about confirming that it is true by referencing the proposition to facts of reality – all the facts subsumed by the concepts of the proposition or the premises.

33

If you want to place a contradiction as a relationship between statements (which I decry) then saying something like, “All pigs can fly,” contradicts the facts subsumed by the concept “pig”, since the concept includes all the facts about pigs, including the fact that they cannot fly since they don’t have wings.

This is a point that many logicians fail to understand, because they see premises as arbitrary objects subject to no rules. Thus everything they say comes down to prefacing claims with “if you make certain assumptions, then…”, but they do not typically spell out those assumptions.

I have a standing challenge to any formalist logician, which is to give a self-contained Kleene 1967 style proof that 3+2=5. So far, no takers. Formal logic, as it currently exists, has negligible relevance to the study of logic, and probably does more harm than good.

34

Another illustration is the tropism that “logic lets you go wrong with confidence”. What is really meant here is “the rules of logical deduction (a/k/a formal logic as it exists today) let you go wrong with confidence” The problem is most people do NOT understand “logic” to include validating the premises by checking them against reality. Logic is not thought of as “the art of non contradictory identification” but rather just the formal rules that can be used on garbage to produce more garbage (garbage in, garbage out). If Objectivists use the first definition of “logic” however, what do we call the application of the deductive rules, taken by itself (which is a vital component of logic), which is what most people mean by “logic”?

On a related note, my “logic” textbook from college tended to reduce the rules to abstractions, e.g., “if P then Q; P; therefore Q” which they were certainly free to do within the context of what they thought logic was–where the premises P and Q can be completely arbitrary so why NOT use algebraic symbols–but in doing so it became so abstract, I had to help my classmates by creating “real life” examples so they could see WTF was going on. Of course its one step from this floating (because premises are disconnected from anything, much less reality) abstraction to making the claim that deduction never teaches you anything you didn’t already know, and is therefore useless.

35

If Objectivists use the first definition of “logic” however, what do we call the application of the deductive rules, taken by itself (which is a vital component of logic), which is what most people mean by “logic”?

I suggest referring to it as “symbolic deduction”, since that includes the two essential properties of “logic”. The name is just obscure enough that people will not associate it with logic (which is a well-known concept) and thus make the mistake of assuming that symbolic deduction is logic.

36

The facts I am referring to are the facts subsumed by the concepts in the premise. If you want to place a contradiction as a relationship between statements (which I decry) then saying something like, “All pigs can fly,” contradicts the facts subsumed by the concept “pig”, since the concept includes all the facts about pigs, including the fact that they cannot fly since they don’t have wings.

Again, you’re just being ungrammatical. The words you use do not permit the order you put them it. “contradicts reality” is just unintelligible, as contra-DICTIONS regard only DICTIONS–sentences. So you can only contradict a sentence.

Regarding modal logic as David presented it, it may be true that what is certain is also possible or probable, but my point is that you cannot decide if something is certain without reference to the facts.

I’m still not sure I know what you mean. Take any sentence of the form “P or not-P”. I will always know that it is true, regardless of the content, so long as P is a proposition. I could use the examples above, replacing ‘and’ with ‘or’. I don’t need to check anything empirical to know that such sentences will be true.

Now if your point is that modern logic gives no method for coming to know the truth of a single proposition P, that’s true. That doesn’t mean logicians aren’t interested in finding P’s truth-value, but that they have no formal way of finding it–and nor do Objectivists, so that’s not just a shortcoming of formal logicians. Logicians just say that you can have VALIDITY without reference to the truth-value of the particular propositions, but that in no way implies that you get true conclusions if you don’t have true premises. So I don’t see what all the fuss is about. They look to the facts just as much as any Objectivist, in this regard.

37

Again, you’re just being ungrammatical. The words you use do not permit the order you put them it. “contradicts reality” is just unintelligible, as contra-DICTIONS regard only DICTIONS–sentences. So you can only contradict a sentence.

And you are just being overly-etymological. If what you said were true, then the concept of “contradiction” in symbolic deduction systems would be nonexistent, since they do not involve sentences. There is no sentence involved in “A&^A”.

Take any sentence of the form “P or not-P”. I will always know that it is true, regardless of the content, so long as P is a proposition.

How do you come to have that belief? Do you also believe that any sentence of the form “P and not-P” is always false, and if you do, what causes you to have that belief?

Can you give me a single example of a sentence of the form “P or not-P” which you believe to be “always true”. I am rather sure that you cannot, which undermines your position totally. Perhaps you may want to jigger your logical laws a bit, but for now, let’s stick with the standard laws.

38

I generally don’t mind people talking about “contradicting reality”, but Tommy was making a pointed philosophical argument about it, and so in that context we cannot allow loose speak.

As for A & ~A (who uses a caret, ferreal?), since A is either a propositional constant or a variable, it is a proposition (just not written out in full English) or it ranges over them and so represents the set of contradictions in that form. So in the later case, fair enough, it is not a single contradiction.

And yes, I as I said above, anything of the form “P and not-P” is false. I came to believe it by thinking of the schema, and realizing that nothing–in principle–could ever be true of it. An analogy can be made for 1/x. I just think about the expression and realize that the only value for which it is undefined is zero. I need not check all values of x, nor even investigate this by empirical evidence (how could you?). I just think about the meaning and realize that, contained in the meaning of the expression, is that it cannot be understood for x = 0. Likewise, when I think of the meaning of the schema “P and not-P”, I realize that for no P will the sentence be true.

One example of a sentence of the form “P or not-P” which is always true: “I am a man, or I am not a man.”

39

I generally don’t mind people talking about “contradicting reality”, but Tommy was making a pointed philosophical argument about it, and so in that context we cannot allow loose speak.

Okay, which is why you’re getting cut no slack here.

And yes, I as I said above, anything of the form “P and not-P” is false. I came to believe it by thinking of the schema, and realizing that nothing–in principle–could ever be true of it.

So this was an act of pure irrational faith or divine inspiration? How can you possibly realize that the schema “P&^P” could never be true; why can’t I “realize” that the schema “P&^P” is always true? Can you do the same for “A ja ^A” and “A dahje ^A”?

I just think about the meaning

Where does that come from? I only know how to do that by referring to facts, and you seem to believe that correct symbolic deductions are possible without such appeal to reality. So explain.

One example of a sentence of the form “P or not-P” which is always true: “I am a man, or I am not a man.”

Did you notice that this is not a sentence of the form “P or not-P”? It is a sentence of the form “(S V O) or (S V not O)”? Do you want to try again, or do you understand that symbolic deduction is not about sentences and therefore your claim that “contradiction” had something to do with sentences is false?

At some point, you ought to deal with pesky examples of non-contradictory “A&^A” sentences like “Three men left and three men didn’t leave”. I have more examples, if you need them.

40

Okay, which is why you’re getting cut no slack here.

I’m not complaining, and I’m holding my own just fine.

So this was an act of pure irrational faith or divine inspiration?

You think the act of understanding an expression–having a thought–is an act of faith or inspiration?

why can’t I “realize” that the schema “P&^P” is always true?

Because you can’t realize what’s false. You might think it though. But to see that it’s always false, you merely need understand the expression. For ja and dahje, I can’t understand the expression. I’d have to know what the words mean. That might be an empirical discovery, but once understood, the content of the expression itself is not emipirical–it is the content, something like the thought, of an expression.

Did you notice that this is not a sentence of the form “P or not-P”? It is a sentence of the form “(S V O) or (S V not O)”?

Such sentences are of the form “P or not-P”. Unless you’re being over-technical with the precise wording, in which case I meant that I know the truth of “P or it-is-not-the-case-that-P”, and so I am a man or it is not the case that I am a man.

Do you want to try again, or do you understand that symbolic deduction is not about sentences and therefore your claim that “contradiction” had something to do with sentences is false?

Neither–my first attempt was adequate, but I’ll explain if you’re still not clear on these points.

At some point, you ought to deal with pesky examples of non-contradictory “A&^A” sentences like “Three men left and three men didn’t leave”. I have more examples, if you need them.

Yes, such examples are pesky, which is why I general deal in terms of propositions rather than sentences.