September 2016

Sun Mon Tue Wed Thu Fri Sat
        1 2 3
4 5 6 7 8 9 10
11 12 13 14 15 16 17
18 19 20 21 22 23 24
25 26 27 28 29 30  

Subscribe

« Links Mentioned on the Show | Main | Virtues and the Economy »

April 16, 2012

Comments

The laws of logic are implicit in the idea of a 'statement' or 'proposition' - an utterance meant to be 'true' - to say something about the world.

For example, the law of non-contradiction just repeats that a statement is true or false.

RonH

RonH, you are saying "implicit" as though you dont have to reduce further to explain. Is this a denial of the need for necessary preconditions to even have logic intelligible?

Your statement is assuming innate forms and categories of ideas to sort or compare against sense peceptions, making them useful perceptions so long as they are accurately classified. Otherwise there'd be something like noise bombarding the mind with not even an idea of "distinction" to separate pereceptions.

The statement denies the idea of tabula rasa, which I agree with, but is ungrounded opinion if you just let it sit there.

Brad B,

As soon as we define statement and negation we have the LNC. Watch:

Definition: A statement is a kind of utterance.

It's the kind of utterance that is either true or false, but not both.

Definition: The negation of A, call it not-A, is the statement such that:

1) If A is true then not-A is false.
2) If not-A is true then A is false.

From these definitions (alone!): A and not-A are not both true.

That's the 'law' of non-contradiction.

Did you notice? We didn't even need to say what we mean by true or false!

RonH

So you are saying we need to "define" "statement and negation" prior to having the LNC? How does that defining occur? How do you distinguish between words without LNC? Further, what is "word"? It cant just be utterances, babies do that all day long.

Brad B,

Why would you think the LNC is needed to 'distinguish between words'?

RonH

So you are saying we need to "define" "statement and negation" prior to having the LNC?

Certainly not. I define terms needed to clearly describe this aspect of language . Then I pointed out that the 'LNC' is contained in my definitions.

The apologetic seems to rely on some kind of wow! factor:

Wow! The Laws Of Logic! Where do those come from?

Well, this is where they 'come from'.

I guess now you could switch to:

Wow! Words! Where do those come from.

RonH

Hi RonH, I was only using your words. You said "as soon as", then LNC.

I'm not saying you cant use LNC or laws logic in general, you do use them, you just cant account for them or the preconditions for knowledge, nor do you seem to want to. No big deal, I was just pointing it out again.

Brad, 'as soon as' is just an expression; it carries no claim.

Really, it should not be necessary to say that. It should not be necessary to defend one's use of an expression.

If you 'begin' your description of this area of language with statements and negation, 'as soon as' you finish you have the LNC.

If, instead, you were to 'begin' with the LNC, you'd immediately need 'back up' and define 'statement' and 'negation'.

Given my language-based account of the LNC and given that we use this aspect of language whenever we talk, it is not at all surprising that some people can be stumped by this apologetic. Hence its popularity.

You can't just ignore my account of the LNC. You need to acknowledge it or show it's wrong.

If you acknowledge it and you still want to claim the LNC is spooky you need to give a reason for saying it's spooky in spite of having a mundane language-based explanation.

Otherwise, I'm done with LNC.
You wanna do another 'law of logic'?
You wanna do 'preconditions for knowledge'?

You don't know what I want or don't want so skip that angle.

RonH

As soon as we define statement and negation we have the LNC.
Not so. As soon as you do this you presume and rely upon the LNC.

Daron,

You are now talking about a different 'as soon' - the one that has to do with me actually talking as opposed to the one that has to do with my description of language. See?

RonH

Not so. As soon as you do this you presume and rely upon the LNC.

That's a bit like saying I presume the notion of statement when I define statement. It's true and strange not a problem and not spooky.

Yet another response: I did say the LNC was contained in those definitions, didn't I?

Yet another response: When you try to use the LNC you presume and rely upon the notion of a statement.

Just curious RonH-- in your opinion, did the laws of logic exist prior to an intelligent lifeform's ability to define 'statement' and 'negation'?

"When you try to use the LNC you presume and rely upon the notion of a statement."

Hi RonH, I think it's quite the other way around, you wouldn't know statement from gibberish without using LNC--even informally. The mere employment of it demonstrates its necessity, but that doesn't automatically account for logic laws' existence. Logic laws are intelligence and not randomness, they came from somewhere, [not from nowhere]. I already know you aren't concerned with grounding or justifying knowledge, but nontheless, you stand on the foundation provided by another worldview.

I plugged in the word informally above because I wanted to make a distinction that it doesn't have to be syllogism form to make proper use of LNC, in fact it's done informally thousands of time more often than in a formal way. Assuming in informal use is no harm no foul, but when it comes down to making distinctions to do formal logic or epistemology, there is harm and foul.

Of course not Jesse. They are descriptions of talk.

RonH, I'm confused. Are we talking about the same thing when we refer to the 'laws of logic'?

What I have in mind is a set of objective features of reality, e.g. P is not (not P), that hold true regardless of the existence of language.

Here's the type of thing that seems to follow from my understanding of your position. Suppose we go back to the day before humans developed the capacity for language. (Please humor me and pretend for the moment that language could be a sudden development.) At this time, the law of noncontradiction didn't exist. Also at that time, I presume the earth was orbiting the sun. But since the law of noncontradiction supposedly didn't exist yet, we can conclude it possible that the earth was also not orbiting the sun. (Maybe quantum astronomy can save the day-- the earth was in a superposition of orbiting and not orbiting until an observer caused the waveform to collapse.)

I doubt you would hold something so absurd to be true--so clearly, I must not have understood your point of view. Where have I gone wrong here?

Thanks, Greg. This has always been my position intuitively from an analysis of the bivalent nature of logic in the created order as it rises out of the necessary univalence of an eternal God. Bivalence isn't possible without a univalent foundation and univalence is the characteristic logic of God aside from his creation. Naturally this causes problems in trying to apprehend God aside from a bivalent revelation of Himself to us. That is to say that we need something against which to contrast God in order to perceive Him. So creation is the divine art of God making something that is not Himself, yet requiring Him as a foundation.

Here's an argument:

No abstract objects can exist without a mind.
Necessary truths are abstract objects
Necessary truths exist in every possible mind
Thus, at least one mind exists in every possible world.
The actual world is a possible world.
Thus, the mind that exists in every possible world exists in the actual world.
God, if he exists, is the only mind that can exist in every possible world
Thus, God exists in the actual world.

That's a bit like saying I presume the notion of statement when I define statement. It's true and strange not a problem and not spooky.
Demonstrating that the LNC is true and is necessary is not the same as accounting for it.

BradB says:

I already know you aren't concerned with grounding or justifying knowledge, but nontheless, you stand on the foundation provided by another worldview.
Indeed. And this romanticism died long ago to be replaced by leap-of-faith existentialism and death-by-a-thousand-qualifications verificationism and positivism.

Brad,

You talk about formal logic as if the 'formal' part had something to do with logic's connection to the world. It doesn't.

Formal is for form over content: you could do the whole subject after having replaced true and false with 0 and 1 or with maize and blue. In other words, it's like math.

This combined definition of 'and' and 'not' ends with a generalization of our friend, the LNC.

The meanings of negation and conjunction are summed up in these laws: The negation of a true statement is false; the negation of a false statement is true; a conjunction of statements all of which are true is true; and a conjunction of statements not all of which are true is false.
Do ye see it?

-WV Quine. Methods of Logic 3rd edition, page 10.

If / when we find that (classical) logic doesn't describe the world then we will modify it or switch to another kind of logic to try to get better agreement. Actually, I guess we've already had to do this: quantum logic.

RonH

Jesse,

My point of view is that gravity determines orbits while the LNC does nothing.

My point of view is the LNC doesn't exist in the sense that it would need to exist in order to prevent A and not-A from being true together.

My point of view is that (A and not-A) can't be true by definition and for no other reason.

RonH

Hi RonH, my making a point of formal logic and informal logic has nothing to do with its connection to the world, but is addressing the way people make use of it. Formally, logic is under a higher scrutiny to be valid, justified, and hopefully demonstrably sound.

Informally, logical precision is less scrutinized, prior commitments can be assumed, no one is all too concerned to look closer, but obvious contradictions of premises are too easy to spot and challenge to ignore.

Brad B,

If you use 'formal' to modify 'logic' in your own private way, you will confuse people.

Accepting your meaning just long enough to respond, I would say: OK, sounds like formal gathering as opposed to informal gathering, or formal dress as opposed to informal dress.

RonH

Francis, your argument is invalid. From the first few premises it only follows that in each possible world there exists at least one mind. You assume that this is the same mind from world to world, but that is not necessarily so.

Also, several of your premises are dubious.

Can somebody point me to an explanation of the concept of a 'possible mind'.

RonH

Hi Ben, what about this:

"God, if he exists, is the only mind that can exist in every possible world"

Hi RonH, I believe that Francis is referring possible mind to say "possible rational mind". In other words, any mind you can imagine that is a rational mind is a possible mind. Mind = rational mind.

I suspect some relation to his thinking might be rooted in this. I may be sorely mistaken though.

Brad B,

I think he accidentally typed "possible mind" when he should have typed "possible world." The argument doesn't make any sense unless we read the latter.

Of course it's still invalid anyway. Notice the difference between the following premises:

(1) (for all x)(x is a possible world --> at least one mind exists in x)

(2) (for some y)(y is a mind, and (for all x)(x is a possible world --> y exists in x))

Premise (2) is much stronger than (1). According to (2), there is a particular mind, call it y, fixed across possible worlds, which exists in every possible world. In contrast, (1) only states that every possible world contains some mind, but not necessarily the same mind from world to world.

So for instance, suppose there are exactly two possible worlds W1 and W2, and that they each contain exactly one mind M1 and M2, respectively, where M1 and M2 are distinct. On this model, each of the possible worlds W1 and W2 contains a mind, and so (1) is true. However neither of the minds M1 nor M2 exist in each world, and so (2) is false.

So Francis needs some means of inferring (2) from his previous premises. Perhaps he could add the premise, "if (1) then (2)." That would fix the argument's validity.

However his premises are still dubious. Why should we accept a premise like "if (1) then (2)"? For that matter, why should we accept his other premises? At least three of them are not obviously true. But in absence of a reason to accept the premises, we haven't any reason to accept the conclusion of the argument.

Beckwith's argument here sounds more like a play on Plantinga's ontological argument than on Anselm's. Anyway, I think I'll have to agree with Ben on this one. There's no good reason (written here, at least) to jump from a mind existing in every possible world to the same mind existing in these worlds.

RonH:


Definition: A statement is a kind of utterance.

It's the kind of utterance that is either true or false, but not both.

Definition: The negation of A, call it not-A, is the statement such that:

1) If A is true then not-A is false.
2) If not-A is true then A is false.

From these definitions (alone!): A and not-A are not both true.

How?

Don't you, at a minimum, need this inference to be valid?

1') If A is true then not-A is false.
2') A is true
So
3') Not-A is false.

Haven't you just replaced the Law of Non-Contradiction with another law?

"It's the kind of utterance that is either true or false, but not both."

...so what kind of utterances are both true and false?

WL,

I don't need to add anything to the two definitions.

Suppose you contradict me: you say the definitions do allow both A and not-A to be true.

From the first definition, there are but two cases to consider: 'Case A' and 'Case not-A'.

In Case A, not-A is false by the first part of the second definition.

In Case not-A, A is false by the second part of the second definition.

In neither case (that is, in no case) are both A and not-A true.

That's it. I only use the two definitions.

This really is just a special case of the last part of Quine's combined definition of conjunction and negation.

a conjunction of statements not all of which are true is false

It's the same as saying

the product of binary numbers not all of which is 1 is zero
.

Jesse,

My 'but not both' is not there to allow for 'utterances are both true and false.

It was meant to make it clear that the 'or' in 'either true or false' is exclusive.

http://en.wikipedia.org/wiki/Exclusive_disjunction

not

http://en.wikipedia.org/wiki/Logical_disjunction

RonH

Ben,

I believe you are right about the typo.

But there is about 'possible minds' in the AI world. Take this.

I like your critique too.

The Premise (2) is much stronger than (1). part is nice.

And, you are right to ask for justification of the premises.

Do you think the argument could be improved? Fixed?

I doubt it: even after 'possible minds' are gone, some of the terms seem like they will be very resistant to clarification.

For example, can we really nail down 'abstract objects' and the nature of their 'existence' well enough to use them to prove things about the universe?

I think there is only one field that will accept an argument built on such squishy terms.

RonH

Hi Ben and Jesse, I think Francis' statement:

"God, if he exists, is the only mind that can exist in every possible world"


Closes the gap you seem to think is missing. I'm not saying that you are supposed to accept the premise, but it'd be well within an ordinary Christian understanding of God to accept the full intentions of Francis' words. If God is as He's been revealed to be in the scriptures, He is the only mind that can....

Question the premises, fine. I'm not inclined to argue the premises, but I dont think the arguement is invalid at all.

Hmmm....I probably meant to say "closes the gap you think you see",

not

"closes the gap you seem to think is missing."

"In Case A, not-A is false by the first part of the second definition."

No, sorry. Not so. Not unless you accept the rule that "If A then B, plus A implies B".

Ditto for your claim about the other case.

WL,

Can you point out my mistake?
Quine's?

RonH


Brad, I don't have any problem with the premises (Ben does though). But I don't think that closes the gap; let me explain why I think this.

1) No abstract objects can exist without a mind.
2) Necessary truths are abstract objects
3) Necessary truths exist in every possible mind world
4) Thus, at least one mind exists in every possible world.
5) The actual world is a possible world.

All that follows from this is that at least one mind exists in the actual world. There are no reasons given to suppose that any of these minds are shared between some worlds, let alone to suppose that one mind is common between all worlds. So the next step just doesn't follow:

6) Thus, the mind that exists in every possible world exists in the actual world.
7) God, if he exists, is the only mind that can exist in every possible world
8) Thus, God exists in the actual world.

I do happen to think that one mind (God) exists in every possible world--I just don't see how that follows from the proposed argument. By the way, RonH's comments here are causing me to lean more and more toward accepting the validity of premise 1.

Jesse,

How's that?

RonH

Ron-

You say "In Case A, not-A is false by the first part of the second definition."

Case A is the case where A is true.

The relevant part of your second definition is this:

"If A is true then not-A is false."

What justifies your claim that in Case A, not-A is false.

The definition doesn't.

Only when you add the rule that says that from "If A, then B" and "A" you are allowed to infer "B" are you justified in saying that in case A, not-A is false. (Logicians call this rule Modus Ponens.)

That is to say that, you can get by without the law of contradiction, relying only on definitions plus the rule of Modus Ponens.

Or, to put it another way, using your definitions, you can replace one law of logic (the law of contradiction) with another (Modus Ponens).

You haven't gotten to the core question of why any rules of logic are correct (or even what it means for them to be correct).

Hi Jesse, I see your point, I'm sure my initial reading was more charitable than it should have been. The premise #6 pretty much shuts a door. I cant retro #7 into the conclusion after the fact, so I appreciate your pointing this out the way you did. The premise #7 needed to be prior to #6 for the meaning I have been offering to allow validity.

WL,

Leave out my definitions then. Instead consider Quine's

a conjunction of statements not all of which are true is false

Does this remove your temptation to think I'm depending on Modus Ponens?

More detail later.

RonH

Brad, I'm not sure I'm following you. I hadn't intended to show that (6) shuts the door--rather, (6) does not follow from (1--5).

Also, I don't see how switching (6) and (7) fixes the problem. (8) follows from (6) and (7), regardless of the ordering. However, the argument still suffers in this way: (1--5) show that at least one mind exists in every possible world; (6, 7) assume that at least one of these minds is identical in every possible world.

The only illustration I can think of is the following. Suppose I have a collection of baskets, each with at least one apple. Now I claim that the hyperspace apple, if it exists, is the only apple that can exist simultaneously in all the baskets. Since at least one apple is in every basket, the hyperspace apple exists.

Jesse is correct. Ordering does not matter here, insofar as the conclusion simply does not follow from the premises.

This reminds me of James Anderson's argument last year in Philosophia Christi for the existence of God from the laws of classical logic. He tried to get around the problem by claiming that laws of logic depend on minds because they are thought tokens (i.e. not thought types which can be shared), and that each thought token belongs essentially to the mind which contains it. So if a law of logic exists in every possible world, that means the same host mind exists in every possible world too.

Of course that argument is a little ridiculous. There is no reason to think the laws of logic are thought tokens and not thought types, and although I am sympathetic to the notion that laws of logic exist in every possible world, in the end we have no reason to suppose that's true either.

Thanks for the responses, I was working out a reply, then as it was going along, I understood what you have been saying, so I stopped :~).

Ron-

Quine is there presenting one of the truth conditions of the logical operation called conjunction. In ordinary parlance, he's giving part of a definition for the word "and".

You certainly cannot replace the law of contradiction with that alone.

The truth condition you quote, for example, does not imply that A and not-A cannot both be true.

I guarantee you that Quine's overall system of logic depends on the assumption that some rule or rules of inference are valid, possibly that some axioms are true and a number of definitions. The definitions alone are insufficient to underwrite the rules or the axioms. That's why his system includes them.

Now, I also guarantee you that Quine either proved, or could have proven that for any argument, if his system of rules, axioms and definitions declared that an argument is valid, then that argument is valid and for any valid argument, his system of rules, axioms and definitions will show it to be valid.

This is what's known as providing a soundness and completeness proof for your system. Essentially, he'd be proving that (a) you can trust the results of his system and (b) you can expect results from his system.

What Quine does not consider, at least not formally, in his treatment of logic is what's so great about valid arguments. No doubt he has something to say about how valid arguments preserve truth. That is valid arguments don't let you start out with truth but end in falsehood. But that that's wonderful is, I'm sure, never explained.

WL,

Thanks for motivating me to clean this up.

The law of non-contradiction says that a statement and its negation can't both be true.

Is the LNC true?

To find out, we need only clarify our the terms needed to state the LNC itself. These terms are 'statement', 'negation', and 'conjunction'.

A 'statement', A, is something said that admits of being either true or false.

The 'negation' of A, not-A, is true if and only if A is false.

The 'conjunction' of two statements 'A' and 'B' is 'A and B' is true if and only A is true and B is true.

With these definitions in place, here is what the LNC looks like

not(A and not-A)

Since A is a statement, there are two cases: A is true and A is false.

Case A - Replace A with T(rue)

Step........................# justification

not(A and not-A)......# the LNC
not(T and not-T)......# replace A with T
not(T and F).............# definition of negation
not(F)......................# definition of conjunction
T..............................# definition of negation

So Case A is true

Case not-A - Replace A with F (False)

Step........................# justification

not(A and not-A)......# the LNC
not(F and not-F)......# replace A with F
not(F and T).............# definition of negation
not(F)......................# definition of conjunction
T..............................# definition of negation

So Case not-A is true.

Both cases are true so the LNC is true (by the definition of conjunction).

RonH

Things you are assuming Ron:

1) Dilemma reasoning is valid.

Your over all argument looks like this

Either A or B

If A then C

If B then C

Therefore, C

You did not provide a proof from your definitions that that form of argument can be trusted.

2) Modus Ponens is valid.

This sentence is actually an argument all by itself:

Since A is a statement, there are two cases: A is true and A is false.
Where the sentence
A is a statement
is a premise, and the sentence
There are two cases (for A): (i) A is true and (ii) A is false.
is the conclusion.

By itself, this argument is invalid, but we may assume from you accompanying text that your definitions provide the premise that links the two. This definition:

A 'statement', A, is something said that admits of being either true or false.
Which might also be expressed in this way:
For all X, X is a statement, if and only if there are two cases for X: (i) X is true and (ii) X is false.
Well if that's true for all X, it's true for A, so
A is a statement, if and only if there are two cases for A: (i) A is true and (ii) A is false.
seems to follow from this definition (but more on this in the next section).

You don't provide a means of breaking down of "if and only if" statements, but we may assume that you would have in the end. As such, another way of expressing your definition in the case of A is like this:

If A is a statement, then there are two cases for A: (i) A is true and (ii) A is false.

AND

If there are two cases for A: (i) A is true and (ii) A is false, then A is a statement.

It is clearly the first of these that is going to help you in this argument, because this is valid:
If A is a statement, then there are two cases for A: (i) A is true and (ii) A is false.

A is a statement.

Therefore, there are two cases for A: (i) A is true and (ii) A is false.

It is also an instance of Modus Ponens.

Again, you did not provide a proof from your definitions that that form of argument can be trusted.

3) Universal Instantiation is valid

See the above assumption that if something is the the case for all X, it's true for A. That's how we broke down your definition to allow us to tease out the additional premise needed to make your argument valid.

The validity of the rule:

For all X, f(X)

Therefore, f(A)

is not in your definitions.

4) Universal Generalization is valid

Let's assume that the rest of your argument works great.

What has it shown?

That the claim

For the statement A, it is not the case that (A and not-A)
follows from your definitions?

OK fine.

That's not the law of contradiction.

The law of contradiction is that for each statement, X, it is not the case that (X and not-X).

How do you get there from your conclusion?

Well, you might say that since you made no special assumption about A other than the fact that it is a statement, what you have to say about A should really go for all statements. That is to say that you are assuming the rule of universal generalization:

f(A) (where no special assumptions are made about A)

Therefore, For all X, f(X)

is valid.

This rule, again, is not contained in any of your definitions.

I believe some combination of universal instantiation and universal generalization is also at play when you substitute, in turn, T and F for A.

Logical Missteps Your Argument

I must say that I don't quite get what you are up to in each of the cases. I gather that what you want to show in each of the cases is that LNC (as you put it) is true.

Given that, it's curious that you should appeal to LNC for your second premise in each case.

Furthermore, you make one move in each argument that is definitely not valid. In the Case A stanza, it's the move that takes you from "not(T and F)" to "not(F)". In the Case-Not-A stanza, it's the move from not(F and T) to not(F).

The denial of a conjunction does not allow one to infer the denial of one of the conjuncts. It only allows one to infer that it's not the case that both are true. That at least one is false.

To illustrate the problem with your inference here, note that in Case-A you've inferred the denial of the second conjunct from the denial of the conjunction. But in Case-Not-A, you've inferred the denial of the first conjunct from the denial of the conjunction.

Now, since the order of the conjuncts in a conjunction makes no difference, the one inference is good if the other is. So I'm not complaining about the fact that you chose the first conjunct in one case or the second in the other, per se.

But notice what would have happened had you made the opposite choice in each case. Your final conclusion would have turned out to be F in each case. The fact that your inference allows you to reach opposite conclusions from the same or equivalent premises (given that those premises are consistent) demonstrates that it is invalid.

Concluding Remarks

Note that I am not denying the validity of any of the inferences employed in your argument other than those I mentioned in the last section. In particular, I think that the four rules of inference that you assume and that I have called to your attention are perfectly valid. I use them all the time.

But my point is that these rules of inference are not expressed in your definitions.

In the end, mere definition will not get you to the normative force of logic. Why it is that, given the premises I must accept the conclusion? You need at least some fundamental rules or laws as well.

What makes these laws correct? What does it mean for them to be correct?

The comments to this entry are closed.