In the absence of a concept of truth, where does a system of deductive logic come from?

[GaryTaylor]

But none of that tells me what is meant by the phrase “truth-condition” - I would think it would mean a condition by which we know that something is true, such as the existence of a proof of the statement in a sound system, or the demonstration that any model (in whatever class of models we are considering) which fails to satisfy it must also satisfy a contradiction. But then if we find a truth condition, we also find truth, and if this is the sense in which the phrase is meant, I still don’t understand the distinction in the post I quoted.

I know, what’s your further thinking?

 

[inocente]

I don’t see a logical construct or attempt to prove an axiom. Its random chance. Your going to the lab with no theory or law. Provable truth or correct is random.

The construct can be anything you like, and whatever construct you choose, you can implement it using switches to test whether or not it is true, and the switches can be whatever you wish. You can run the experiment ten thousand times using various switches, from transistors to railway engines, and as long as you choose reliable switches you’ll get the same results each time. There’s nothing mystic going on.

 

[inocente]

Just because we can create a physical system modeling some aspects of logic and interpret it to match logical deductions does not mean that the meaning of logic is entirely interior. It could, for example, mean that the significance of true and false is in fact built into the physical universe as well.

Not so fast. If all logical deductions can be performed using switches, isn’t the logical deduction that logic is about combining switches? That would explain why computers perform it faster and more reliably than us, as they are built from switches and we are not.

 

[GaryTaylor]

Not so fast. If all logical deductions can be performed using switches, isn’t the logical deduction that logic is about combining switches? That would explain why computers perform it faster and more reliably than us, as they are built from switches and we are not.

There is no mathematical statement which we are solving for true or false, no premise evident to be accepted as true-axiom

The switches are nothing more than a representation of resistance applied on paper. They are meaningless aside from their value. In fact a computer in nothing but A switch.

 

[inocente]

There is no mathematical statement which we are solving for true or false, no premise evident to be accepted as true-axiom

The switches are nothing more than a representation of resistance applied on paper. They are meaningless aside from their value.

I’ve no idea why you keep going back to electrical resistance, make the switches from teddy-bear carrying philosophers if you want, then you won’t have to worry about Ohm’s Law, didn’t we already go around this an hour ago?

In fact a computer in nothing but A switch.

Yes, that’s my point. A logical construct is made from switches and is itself a switch, and can in turn be a component of a larger construct, which is also a switch. Welcome to the wonderful world of computers.

 

[GaryTaylor]

I’ve no idea why you keep going back to electrical resistance, make the switches from teddy-bear carrying philosophers if you want, then you won’t have to worry about Ohm’s Law, didn’t we already go around this an hour ago?

Yes, that’s my point. A logical construct is made from switches and is itself a switch, and can in turn be a component of a larger construct, which is also a switch. Welcome to the wonderful world of computers.

I’m going back to a mathematical statement, doesn’t matter what it is, you chose switches and computers which you just added again.

And my point is that your example makes no sense. There is no mathematical statement which we are solving for true or false, no premise evident to be accepted as true-axiom

The computer and switch you added with a 4-page elaboration on HOW the switch works IN the system. Which is related to what you do not have. There is no mathematical statement which we are solving for true or false, no premise evident to be accepted as true-axiom.

You can create whatever you would like with whatever represented ideal. Philosophers could be P1, P2. and P3. Whatever, but you don’t have a axiom-mathematical statement.

You have a random sequence of events which proves nothing as you never set out to prove a theory to begin with. If you want to call that true, be my guest, I see nothing there.

 

[inocente]

I’m going back to a mathematical statement, doesn’t matter what it is, you chose switches and computers which you just added again.

And my point is that your example makes no sense. There is no mathematical statement which we are solving for true or false, no premise evident to be accepted as true-axiom

The computer and switch you added with a 4-page elaboration on HOW the switch works IN the system. Which is related to what you do not have. There is no mathematical statement which we are solving for true or false, no premise evident to be accepted as true-axiom.

You can create whatever you would like with whatever represented ideal. Philosophers could be P1, P2. and P3. Whatever, but you don’t have a axiom-mathematical statement.

You have a random sequence of events which proves nothing as you never set out to prove a theory to begin with. If you want to call that true, be my guest, I see nothing there.

I really don’t understand what your objection is. My contention is as originally stated, that in Boolean algebra ‘true’ and ‘false’ are labels, which is stated formally in the following SEP article (emphasis mine):

*"A Boolean algebra (BA) is a set A together with binary operations + and · and a unary operation −, and elements 0, 1 of A such that the following laws hold: commutative and associative laws for addition and multiplication, distributive laws both for multiplication over addition and for addition over multiplication, and the following special laws:

x + (x · y) = x
x · (x + y) = x
x + (−x) = 1
x · (−x) = 0

These laws are better understood in terms of the basic example of a BA, consisting of a collection A of subsets of a set X closed under the operations of union, intersection, complementation with respect to X, with members ∅ and X. One can easily derive many elementary laws from these axioms, keeping in mind this example for motivation. Any BA has a natural partial order ≤ defined upon it by saying that x ≤ y if and only if x + y = y. This corresponds in our main example to ⊆. Of special importance is the two-element BA, formed by taking the set X to have just one element. The two-element BA shows the direct connection with elementary logic. The two members, 0 and 1, correspond to falsity and truth respectively."

plato.stanford.edu/entries/boolalg-math/*

 

[Linusthe2nd]

We have a natural concept of truth, God made us to know truth. It is nice to have a step by step guide to determine the truth or falsity of a statment. But if you want to use one it better be the best one available among the dozens that populate the universities these days. And I suggest the one proposed by Aristotle is the best. But such a system does little to tell you if what you observe in the world is true or not. But God made our intellect to know such truths as well the truth of propositions.

Linus2nd

 

[GaryTaylor]

I really don’t understand what your objection is. My contention is as originally stated, that in Boolean algebra ‘true’ and ‘false’ are labels, which is stated formally in the following SEP article (emphasis mine):

*"A Boolean algebra (BA) is a set A together with binary operations + and · and a unary operation −, and elements 0, 1 of A such that the following laws hold: commutative and associative laws for addition and multiplication, distributive laws both for multiplication over addition and for addition over multiplication, and the following special laws:

x + (x · y) = x
x · (x + y) = x
x + (−x) = 1
x · (−x) = 0

These laws are better understood in terms of the basic example of a BA, consisting of a collection A of subsets of a set X closed under the operations of union, intersection, complementation with respect to X, with members ∅ and X. One can easily derive many elementary laws from these axioms, keeping in mind this example for motivation. Any BA has a natural partial order ≤ defined upon it by saying that x ≤ y if and only if x + y = y. This corresponds in our main example to ⊆. Of special importance is the two-element BA, formed by taking the set X to have just one element. The two-element BA shows the direct connection with elementary logic. The two members, 0 and 1, correspond to falsity and truth respectively*."

plato.stanford.edu/entries/boolalg-math/

Your caught up on this false idea of true and false. True and false are switching functions already designated and can have only two values: true (represented by 1) or false (represented by 0). It provides a set of rules. True is still true and false is still false.

The two members, 0 and 1, correspond to falsity and truth respectively

 

[onenow1]

I am completely out of my league here, but my thought is this: The idea of 1+1=2 on paper and in mind is correct. However digging a little deeper about differences in kind as seems to me; There is nothing perfect in human construct that measures precisely the same, there is always a difference ? I will follow with much interest.

God Bless

“Always learning but never grasping completely the ultimate truth that is Jesus.”

 

[PseuTonym]

Not so fast. If all logical deductions can be performed using switches, isn’t the logical deduction that logic is about combining switches? That would explain why computers perform it faster and more reliably than us, as they are built from switches and we are not.

You are talking about a finite computation, which is a very special case of logical deduction. Consider the problem of proving or disproving Fermat’s Last Theorem: for all positive integers x, y, z, and n, if n is greater than 2, then (x to the power of n) + (y to the power of n) is not equal to (z to the power of n).

If Fermat’s Last Theorem were false, then it could be shown to be false via a finite computation.

However, to show that Fermat’s Last Theorem is true requires something more than a finite computation, unless you rely upon the fact that somebody already invented a proof, and you simply want to check that there are no errors in the proof. However, how do we know that the assumptions relied upon in the proof that Andrew Wiles invented are consistent with each other? No finite computation can check for the consistency of a non-trivial system of axioms for number theory.

 

[PseuTonym]

The construct can be anything you like, and whatever construct you choose, you can implement it using switches to test whether or not it is true, and the switches can be whatever you wish. You can run the experiment ten thousand times using various switches, from transistors to railway engines, and as long as you choose reliable switches you’ll get the same results each time. There’s nothing mystic going on.

mystic:
“a person who seeks by contemplation and self-surrender to obtain unity with or absorption into the Deity or the absolute, or who believes in the spiritual apprehension of truths that are beyond the intellect.”

It sounds as though you are using an unusual definition of “mystic”, such as the following:
“a person who believes in the intellectual apprehension of a truth that goes beyond a finite computation.”

 

[Bahman]

A point of view was expressed in another thread, and I think that a separate thread will be required to examine that point of view.

Link to post:
forums.catholic-questions.org/showpost.php?p=12773367&postcount=92

Thread title:
Is there a materialist explanation of mathematics?
Link to thread:
forums.catholic-questions.org/showthread.php?t=946610

After a system of deductive logic has been developed, it is possible to ignore what made the development possible, and to try to reduce the potentially difficult-to-analyze concept of a statement being true to the question of how one comes to know or believe that the statement is true. However, I suspect that such an attempt will inevitably fail, as it is founded on self-deception.

To know that a particular statement has been deduced via some assumptions and some system of deductive logic is to know something of no particular significance. For it to be significant, one needs to know or believe that the system of deductive logic that is relied upon does what it is supposed to do: preserve truth. If all of the assumptions are true, and a conclusion is deduced from them, then the conclusion is supposed to also be true.

To relativize truth is to deny the existence of a question until after one has an answer.

I could edit and change what I quoted to get something that I agree with:

#1 “Provable” isn’t absolute, but is relative to the rules of the particular system.

#2 A statement is a theorem if it can be deduced from the axioms.

#3 If all of the axioms are true and there exists at least one valid deduction of the statement from the axioms, then the statement is true.

Request to all readers: please let me know if you see anything controversial there.

Where does a system of deductive logic come from? Consciousness. You experience, decide upon experience and then act.

 

[inocente]

Your caught up on this false idea of true and false. True and false are switching functions already designated and can have only two values: true (represented by 1) or false (represented by 0). It provides a set of rules. True is still true and false is still false.

You have it back to front. Look at the SEP article again. It formally defines the set { 0,1 } and then states “the two members, 0 and 1, correspond to falsity and truth respectively”. 0 and 1 are formally defined, false and true are merely one pair of synonyms. Within the logic unit of a computer, for example, you will not find true or false, only ever zero and non-zero.

So, looking at the title or the thread, in Boolean algebra, the system of deductive logic comes first, and then gives rise to the concept of truth. (Not a very grandiose concept, just a very workmanlike concept, but a concept nevertheless).

 

[inocente]

You are talking about a finite computation, which is a very special case of logical deduction. Consider the problem of proving or disproving Fermat’s Last Theorem: for all positive integers x, y, z, and n, if n is greater than 2, then (x to the power of n) + (y to the power of n) is not equal to (z to the power of n).

If Fermat’s Last Theorem were false, then it could be shown to be false via a finite computation.

However, to show that Fermat’s Last Theorem is true requires something more than a finite computation, unless you rely upon the fact that somebody already invented a proof, and you simply want to check that there are no errors in the proof. However, how do we know that the assumptions relied upon in the proof that Andrew Wiles invented are consistent with each other? No finite computation can check for the consistency of a non-trivial system of axioms for number theory.

Never say never - there is an open challenge to provide a computer verification of Wile’s proof. And of course, whatever logic the verification uses will have to be translated (compiled) into the Boolean algebra used by the computer processor. Just as everything else you can do with a computer, including watching a movie or writing that novel, ultimately rests on the very same processor.

But, leaving that aside, I didn’t need to prove my case for every possible system of logic. One was sufficient. And since we can develop and use and teach Boolean algebra using only switches, or only bits, adding a concept of truth later as an afterthought, job done.

 

[inocente]

mystic:
“a person who seeks by contemplation and self-surrender to obtain unity with or absorption into the Deity or the absolute, or who believes in the spiritual apprehension of truths that are beyond the intellect.”

It sounds as though you are using an unusual definition of “mystic”, such as the following:
“a person who believes in the intellectual apprehension of a truth that goes beyond a finite computation.”

No, any minute someone might reify truth into Truth, at which point it takes on the mystic quality given by your definition.

 

[Iron_Donkey]

Not so fast. If all logical deductions can be performed using switches, isn’t the logical deduction that logic is about combining switches? That would explain why computers perform it faster and more reliably than us, as they are built from switches and we are not.

If all the ways that physical objects interact via gravity can be determined by pencil and paper, does that mean that gravity is all about pencil and paper? If so, does that mean that what gravity does is not really real, in itself, but that instead gravity is the mere way that we subjectively interpret marks on paper? Or is it instead that there are actual rules to gravity that can be represented by marks on paper, so long as we force those marks on paper to follow other rules (and have rules for interpreting them)?

Just because thing A (deductions) can be represented by thing B (switches) does not mean that thing A is really just our way of interpreting thing B or is all about something inherent to B (as you seem to imply in the post before the one I am quoting now).

The reason is in the word “represented”: we are adding structure to the pencils and paper (or switches and electricity) that is not inherently there. The marks on paper are just marks on paper; the positions of switches are just positions of switches. We make the connection between them and what we want them to represent - to a person who is not familiar with this connection, the positions of these switches, or the marks on paper are meaningless. Now, if you add stuff - if you put the switches in the right places, with the right sort of materials between them, then they begin to behave in a way that can be interpreted to match methods of deduction. In the same way, if we put marks on paper in the right place, and require that manipulation of them follow certain rules, these marks begin to behave in a way that can be interpreted to match the behavior of bodies under gravitation.

So no, logical deduction is not just about combining switches. It’s closer to about combining switches in an intelligent way, according to a set of rules - in a way where we force the switches to obey certain rules, and where we look at this pattern of switches, apply the set of rules and then once again using a set of rules exterior to the switches look at them and convert their positions into information about deductions.

The switches are very nearly neutral in the process. You can do the same thing with all kinds of physical materials. Someone actually made a computer in mine craft using stone troughs, oil, and fire - and while minecraft is software and so a combination of switches, there is no reason why such a structure couldn’t be built in reality out of real stone and some sort of flamable oil. You can also do it purely abstractly. Etc.

The part that is constant is that there is a set of rules, by which you force changes in one switch (pencil mark, flaming oil trough, abacus bead, Lego structure) to cause a changes in another, and by which you read the results.

And since the material plays almost no role, I think it would be better to say that the reason we can represent logical deduction in all of these materials is that these materials also follow rules, and that the rules of the materials can be manipulated to force some specific parts of these materials to behave in a way that allows us to create other rules linking states of materials to abstract ideas of true and false. Logic isn’t even about combining switches in an intelligent way, it’s about the intelligent way itself. The switches really don’t matter.

That is, there is actual reason within the materials themselves, not as something we pretend they have, but as something they actually have. If they did not follow rules, they could not be made to follow the rules that we like. But there is also logic within the relationship between the rules that the materials inherently follow that we manipulate and the rules of the abstract system we manipulate them into imitating. The whole system only works because logic and reason are everywhere.

So if you want to say that logic and reason are contained within switches in some way, that is fine. But the things to keep in mind are that a) it takes even more logic and reason to coax out the rules of deduction and b) this in no way says that our ideas of truth and false and proof are merely our projections on switch behavior because, again, that switch behavior wouldn’t exist if logic and truth and falsehood weren’t more than that.

 

[inocente]

If all the ways that physical objects interact via gravity can be determined by pencil and paper, does that mean that gravity is all about pencil and paper? If so, does that mean that what gravity does is not really real, in itself, but that instead gravity is the mere way that we subjectively interpret marks on paper? Or is it instead that there are actual rules to gravity that can be represented by marks on paper, so long as we force those marks on paper to follow other rules (and have rules for interpreting them)?

Just because thing A (deductions) can be represented by thing B (switches) does not mean that thing A is really just our way of interpreting thing B or is all about something inherent to B (as you seem to imply in the post before the one I am quoting now).

The reason is in the word “represented”: we are adding structure to the pencils and paper (or switches and electricity) that is not inherently there. The marks on paper are just marks on paper; the positions of switches are just positions of switches. We make the connection between them and what we want them to represent - to a person who is not familiar with this connection, the positions of these switches, or the marks on paper are meaningless. Now, if you add stuff - if you put the switches in the right places, with the right sort of materials between them, then they begin to behave in a way that can be interpreted to match methods of deduction. In the same way, if we put marks on paper in the right place, and require that manipulation of them follow certain rules, these marks begin to behave in a way that can be interpreted to match the behavior of bodies under gravitation.

So no, logical deduction is not just about combining switches. It’s closer to about combining switches in an intelligent way, according to a set of rules - in a way where we force the switches to obey certain rules, and where we look at this pattern of switches, apply the set of rules and then once again using a set of rules exterior to the switches look at them and convert their positions into information about deductions.

The switches are very nearly neutral in the process. You can do the same thing with all kinds of physical materials. Someone actually made a computer in mine craft using stone troughs, oil, and fire - and while minecraft is software and so a combination of switches, there is no reason why such a structure couldn’t be built in reality out of real stone and some sort of flamable oil. You can also do it purely abstractly. Etc.

The part that is constant is that there is a set of rules, by which you force changes in one switch (pencil mark, flaming oil trough, abacus bead, Lego structure) to cause a changes in another, and by which you read the results.

And since the material plays almost no role, I think it would be better to say that the reason we can represent logical deduction in all of these materials is that these materials also follow rules, and that the rules of the materials can be manipulated to force some specific parts of these materials to behave in a way that allows us to create other rules linking states of materials to abstract ideas of true and false. Logic isn’t even about combining switches in an intelligent way, it’s about the intelligent way itself. The switches really don’t matter.

That is, there is actual reason within the materials themselves, not as something we pretend they have, but as something they actually have. If they did not follow rules, they could not be made to follow the rules that we like. But there is also logic within the relationship between the rules that the materials inherently follow that we manipulate and the rules of the abstract system we manipulate them into imitating. The whole system only works because logic and reason are everywhere.

So if you want to say that logic and reason are contained within switches in some way, that is fine. But the things to keep in mind are that a) it takes even more logic and reason to coax out the rules of deduction and b) this in no way says that our ideas of truth and false and proof are merely our projections on switch behavior because, again, that switch behavior wouldn’t exist if logic and truth and falsehood weren’t more than that.

I’ll be pedantic about your use of the word “force”, perhaps convey or relay would be a better word. Otherwise I agree with most of what you say, including use of any suitable materials in place of electrical components.

When I said “isn’t the logical deduction that logic is about combining switches?”, what I had in mind is that logic, as you say, is about intelligently combining the switches. We wouldn’t be very productive otherwise. But the fact that we can implement logic, and assess the correctness of a logical structure, simply by assembling mundane physical elements is imho worth noting. Engineering, not magic.

Looking at the thread title, I think it remains that we don’t need a concept of truth in order to develop a system of deductive logic. We need to start from concepts such as principles, integrity, properties and equality, but truth not so much, truth can come later, out of the system.

 

[onenow1]

We need to start from concepts such as principles, integrity, properties and equality, but truth not so much, truth can come later, out of the system.

Truth came in the birth of Jesus, we must listen to Him.

God Bless

 

[wmw]

Logic and reason are two different things.

We attempt to reason a few simple truths and call them axioms.

Logical systems with the proper integrity, principles and rules allow us to manipulate these truths to discover greater truths,

As the saying goes, “garbage in, garbage out”, yet, in a more positive vein the converse is correct if truth goes in and the system has integrity then truth comes out.

The “logical system” does not reason, it can only manipulate the truth that we give it and we humans are left to reason an interpretation of the results it gives.

Those who deny man has a rational soul try to confuse the issue by claiming the human reasoners are nothing but “logical systems” also, but then the words themselves would lose their meaning, make no proper distinctions and discussion itself becomes meaningless.