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

[PseuTonym]

Look at the SEP article again.

in Boolean algebra, the system of deductive logic comes first

There are two references to logic in the article about Boolean algebras that you linked to.

The first reference is implicit: “A Boolean algebra (BA) is a set A together with …” With that definition, it would be impossible for the article to state any theorems about Boolean algebras because there are many different and conflicting conceptions of sets. So the article is actually presupposing the one most popular kind of set theory, and the usual deductive logic of quantifiers and connectives.

There is a contrast between that logic and what is described in the article as “elementary logic.” What is elementary logic in the article seems to simply involve three connectives: binary “and”, binary “or”, and the one-place connective “not.” The article refers to “the ordinary truth tables” for those connectives.

Now, surely if we are working within a logic of quantifiers and connectives, then we already have a concept of truth tables, and in particular we already have a concept of truth. We recognize the truth tables, because they are basic to a fragment of the deductive logic that we rely upon.

If your analysis were correct, then the article would not have the wording “the ordinary truth tables” for those connectives. No, it would be introducing the idea of a truth table. It would say something like the following: “Now, building on a basis that includes deductive logic of quantifiers and connectives, and that also includes standard set theory, we are ready to introduce two new concepts: truth, and truth tables.”

Notice the question: “where does a system of deductive logic come from?” You did not provide an answer. You simply assumed that we already have a system of deductive logic. Your exact words: “the system of deductive logic comes first.” Where does it come from? How does it arrive? You can talk about the origin of ideas in world history or the learning of ideas by an individual. Explain how – without a concept of truth – truth tables were invented in history, or could have been invented in a fictional, alternative history, or could be studied and understood by an individual student.

 

[inocente]

There are two references to logic in the article about Boolean algebras that you linked to.

The first reference is implicit: “A Boolean algebra (BA) is a set A together with …” With that definition, it would be impossible for the article to state any theorems about Boolean algebras because there are many different and conflicting conceptions of sets. So the article is actually presupposing the one most popular kind of set theory, and the usual deductive logic of quantifiers and connectives.

There is a contrast between that logic and what is described in the article as “elementary logic.” What is elementary logic in the article seems to simply involve three connectives: binary “and”, binary “or”, and the one-place connective “not.” The article refers to “the ordinary truth tables” for those connectives.

Now, surely if we are working within a logic of quantifiers and connectives, then we already have a concept of truth tables, and in particular we already have a concept of truth. We recognize the truth tables, because they are basic to a fragment of the deductive logic that we rely upon.

If your analysis were correct, then the article would not have the wording “the ordinary truth tables” for those connectives. No, it would be introducing the idea of a truth table. It would say something like the following: “Now, building on a basis that includes deductive logic of quantifiers and connectives, and that also includes standard set theory, we are ready to introduce two new concepts: truth, and truth tables.”

Notice the question: “where does a system of deductive logic come from?” You did not provide an answer. You simply assumed that we already have a system of deductive logic. Your exact words: “the system of deductive logic comes first.” Where does it come from? How does it arrive? You can talk about the origin of ideas in world history or the learning of ideas by an individual. Explain how – without a concept of truth – truth tables were invented in history, or could have been invented in a fictional, alternative history, or could be studied and understood by an individual student.

But truth tables are just a way of tabulating what happens. For example, imagine two electrical switches, labeled A and B, wired in series to a battery and lamp. Now experiment to test the various combinations and tabulate the results:

A, B, lamp
off off off
off on off
on off off
on on on

Which is the truth table for logical AND. You can of course write 0 (or false) instead of “off”, and 1 (or true) instead of “on”, or any other symbols according to taste. Similarly, experiment with the switches wired in parallel and we have logical OR.

Wiki says the first known example of a truth table was 1893. It’s a convenient way to express the results, and useful for checking that all possible combinations have been considered, for example with 3 switches the table has 8 rows, with 10 there are 1024 rows (2[sup]10[/sup]).

But I seriously doubt that any of us have truth tables in our heads, and it would be very difficult to explain truth tables to stone age logicians. I think they would be more likely to develop and check a proposition by using a stick to draw lines on the ground, making the lines fork at each decision point, and then following the lines to see where a particular set of decisions took them. Same principles, just a different representation.

 

[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.

Are you claiming that all logical deductions can be performed using switches? Or are you talking about all logical deductions within some very restricted system of logic, a system of logic that is not enough broad enough to allow deductions of non-trivial results in number theory?

 

[PseuTonym]

But truth tables are just a way of tabulating what happens.

In a sense, that is a reasonable statement. Truth tables for binary connectives are a very elementary part of logic, and for that reason are an important part of logic. Truth tables for binary connectives are not enough to get you very far in logic, and it is therefore justifiable to say that they are “just” (i.e. merely) a way of seeing what happens.

However, what are we talking about when we say that we are observing “what happens” and tabulating the results? Observing the behavior of lamps when we have switches wired in series is not the same as observing the truth value of a sentence that is a conjunction of two other sentences. It just happens that the two kinds of observations have an analogous structure.

For example, imagine two electrical switches, labeled A and B, wired in series to a battery and lamp. Now experiment to test the various combinations and tabulate the results:

A, B, lamp
off off off
off on off
on off off
on on on

Which is the truth table for logical AND. You can of course write 0 (or false) instead of “off”, and 1 (or true) instead of “on”, or any other symbols according to taste.

I do not agree that it is merely a matter of taste. You can find many analogies that are structurally perfect.

Here is one example:
-1 * -1 = 1, odd + odd = even

1 * 1 = 1, even + even = even

-1 * 1 = -1, odd + even = odd

1 * -1 = -1, even + odd = odd

It does not follow that it is just a matter of taste in any situation whether we are talking about multiplication or addition. Multiplication and addition are different operations.

More importantly, we cannot observe the above analogy until after we look at the behavior of the elements of the set {even, odd} under addition and the behavior of the elements of the set {-1, 1} under multiplication. So it would be a mistake to say that we have the option of beginning with the concepts of even, odd, and addition and using those concepts to define the numbers -1 and 1. On the contrary, we began with two sets of corresponding facts. If we were unsure about whether or not -1 * -1 = 1, then we would not be able to confirm that the above analogy exists.

 

[BobCatholic]

You can get absolute truths in mathematics. For example it follows from the definition of prime numbers that 17 is a prime number, independent of time, location and the number system you use. Here you can reason deductively.

But if you redefine what prime means, then you can’t say 17 is a prime number. In a relativistic system, definitions of words are subject to change.

 

[inocente]

Are you claiming that all logical deductions can be performed using switches? Or are you talking about all logical deductions within some very restricted system of logic, a system of logic that is not enough broad enough to allow deductions of non-trivial results in number theory?

As a computer is limited to using switches (gates and bits), we can test your question by asking: Is there any logical deduction which cannot be performed on a computer, but can be performed using pencil and paper?

The “pencil and paper” levels the playing field - if we can’t even write down the steps in a deduction then we can’t call it a deduction.

As deduction is about following unambiguous rules, unlike poetry or music for instance, there would seem to be no reason in principle why. Have you an example in mind?

btw there’s a large literature on what can be computed (I’m no expert, just find it interesting), for instance:

en.wikipedia.org/wiki/Computability_theory
en.wikipedia.org/wiki/List_of_complexity_classes

 

[inocente]

In a sense, that is a reasonable statement. Truth tables for binary connectives are a very elementary part of logic, and for that reason are an important part of logic. Truth tables for binary connectives are not enough to get you very far in logic, and it is therefore justifiable to say that they are “just” (i.e. merely) a way of seeing what happens.

However, what are we talking about when we say that we are observing “what happens” and tabulating the results? Observing the behavior of lamps when we have switches wired in series is not the same as observing the truth value of a sentence that is a conjunction of two other sentences. It just happens that the two kinds of observations have an analogous structure.

I don’t believe they are merely analogous. The reason why they both give precisely the same results is because at an abstract level they are the same. Creation is the way it is whether we like it or not.

*I do not agree that it is merely a matter of taste. You can find many analogies that are structurally perfect.

Here is one example:
-1 * -1 = 1, odd + odd = even

1 * 1 = 1, even + even = even

-1 * 1 = -1, odd + even = odd

1 * -1 = -1, even + odd = odd

It does not follow that it is just a matter of taste in any situation whether we are talking about multiplication or addition. Multiplication and addition are different operations.

More importantly, we cannot observe the above analogy until after we look at the behavior of the elements of the set {even, odd} under addition and the behavior of the elements of the set {-1, 1} under multiplication. So it would be a mistake to say that we have the option of beginning with the concepts of even, odd, and addition and using those concepts to define the numbers -1 and 1. On the contrary, we began with two sets of corresponding facts. If we were unsure about whether or not -1 * -1 = 1, then we would not be able to confirm that the above analogy exists.*

When I say matter of taste, I mean it’s up to us what we call the two members of the set. Instead of { 0,1 } we can write { cero, uno } or { red, green } or any other symbols we like. But we must follow the rules of Boolean arithmetic, we can’t change the rules according to taste, and the rules of multiplication are not what you use above. They are:

0 * 0 = 0
0 * 1 = 0
1 * 0 = 0
1 * 1 = 1

So if your taste is to use the symbols { -1,1 } instead, following the above rules gives:

-1 * -1 = -1

Which would be confusing but perfectly legal, since this isn’t ordinary arithmetic, it has its own rules.

We start from only { 0,1 } and the rules of inference. We cannot change the rules, but 0 and 1 are only symbols so we can replace them with even and odd, or false and true, or Fred and Mary instead if we wish, and also choose our own private meanings for those words if we want, since really they remain { 0,1 } whatever we call them.

 

[PseuTonym]

As a computer is limited to using switches (gates and bits), we can test your question by asking: Is there any logical deduction which cannot be performed on a computer, but can be performed using pencil and paper?

A computer is limited to using switches (gates and bits) and software. The software can do a lot. Now you are talking about something quite different from your example of wiring two switches serially between a battery and a lamp and observing that the on and off patterns are analogous to those of an AND gate.

For example, educational software can help people learn about deductive logic and many other subjects. We have to distinguish between what the computer itself is doing and what the software is doing. If somebody were to think that all software they have ever used was built into computers and did not need to be created separately, then the power of computers would indeed seem to be almost magical.

As deduction is about following unambiguous rules, unlike poetry or music for instance, there would seem to be no reason in principle why. Have you an example in mind?

Someone (such as Andrew Wiles) who composes a deductive argument that reaches a non-trivial conclusion (that has not yet been obtained) is working within a framework that imposes rules. Something similar could be said for somebody who composes music, or who plays a good game of golf.

However, I thought that you were talking about checking a deductive argument, and not composing one. In other words, I thought that you were considering using switches or a combination of switches and any software that anybody might create to check the validity of the steps in a deductive argument.

Merely checking a deductive argument seems comparable to checking that somebody is playing golf without violating the rules. It seems comparable to reading a musical score composed by somebody else and simultaneously playing that score on a musical instrument seems to me to be a comparable activity. That would of course be a major challenge in robotics, but human performers who play a musical instrument can handle the challenge of physical coordination.

However, you mentioned music by way of contrast with what you were talking about. I do not see the contrast if we are talking about performing rather than composing, or checking somebody else’s deductions rather than composing deductions of conjectures not previously resolved. It is difficult for me to imagine a person who begins studying to play a musical instrument and then abandons the pursuit specifically because the rules for sight reading written musical scores are too ambiguous. For such a person, I think that we would have a Mr. Bean situation where it would be a major challenge to do something like use an airport without getting arrested.

 

[inocente]

A computer is limited to using switches (gates and bits) and software. The software can do a lot. Now you are talking about something quite different from your example of wiring two switches serially between a battery and a lamp and observing that the on and off patterns are analogous to those of an AND gate.

For example, educational software can help people learn about deductive logic and many other subjects. We have to distinguish between what the computer itself is doing and what the software is doing. If somebody were to think that all software they have ever used was built into computers and did not need to be created separately, then the power of computers would indeed seem to be almost magical.

The software is a set of instructions, and each instruction is held in memory, and memory is a set of switches (bits). Each instruction contains an order code, and each order code corresponds to an electronic circuit in the central processor which is made of switches (gates). Everything the software is and does, without a single exception, reduces to switches.

There is also data, and data also reduces to switches, including music and movies, because the only way to store anything is in the form of bits, and each bit is either 0 or 1.

My entire point is that everything which can be represented in hardware, firmware, software and data must, perforce, be reducible to switches since there’s no other way it can even get into the computer.

By analogy, just as every material thing consists of atoms, everything stored in a computer consists of switches.

*Someone (such as Andrew Wiles) who composes a deductive argument that reaches a non-trivial conclusion (that has not yet been obtained) is working within a framework that imposes rules. Something similar could be said for somebody who composes music, or who plays a good game of golf.

However, I thought that you were talking about checking a deductive argument, and not composing one. In other words, I thought that you were considering using switches or a combination of switches and any software that anybody might create to check the validity of the steps in a deductive argument.

Merely checking a deductive argument seems comparable to checking that somebody is playing golf without violating the rules. It seems comparable to reading a musical score composed by somebody else and simultaneously playing that score on a musical instrument seems to me to be a comparable activity. That would of course be a major challenge in robotics, but human performers who play a musical instrument can handle the challenge of physical coordination.

However, you mentioned music by way of contrast with what you were talking about. I do not see the contrast if we are talking about performing rather than composing, or checking somebody else’s deductions rather than composing deductions of conjectures not previously resolved. It is difficult for me to imagine a person who begins studying to play a musical instrument and then abandons the pursuit specifically because the rules for sight reading written musical scores are too ambiguous. For such a person, I think that we would have a Mr. Bean situation where it would be a major challenge to do something like use an airport without getting arrested.*

I made the contrast because we can don’t expect a piece of music or poetry to be logically correct. It would be meaningless to ask whether Beethoven’s Ninth is logically correct. We can’t even say for certain what is music, for example whether wolf pack calls qualify as music.

Nor is a musical score complete within itself. Computers reading and simultaneously playing a score are not the challenge you imagine, it’s been done for 40 year’s or more, see MIDI, but there’s still the choice of what instruments to used, how long to hold notes, how much vibrato, accelerando and so on. Whereas any piece of logic can be represented within the computer in a self-contained form, as a set of constructs which reduce directly to instructions, and the instructions reduce directly to Boolean arithmetic, and in that form there is no requirement for a concept of truth, since the set only contains 0 and 1.

 

[PseuTonym]

I cannot find anywhere in this thread an answer to the question in the title of the thread.

There is a very simple reason for suspecting that there is no answer. In other words, there is a very simple reason for suspecting that it is not possible to develop a legitimate system of deductive logic unless one has a prior concept of truth.

The simple reason is …

In the absence of a concept of truth, there is no label “true” and no label “false” available to assign to a statement. So we would not be able to say that some statement P is true and some statement Q is false. However, that is our fundamental technique for demonstrating that “Q is a logical consequence of P” is false.

If a proposed system of deductive logic allows us to prove that Q is a logical consequence of P, even though P is true and Q is false, then we have confirmed that our proposed system of deductive logic is defective. Given that symptom, we can look for an underlying flaw within the proposed system of deductive logic and attempt to remove the flaw.

In the absence of a concept of truth, there is no way for us to detect that a proposed system of deductive logic is flawed.

 

[utunumsint]

For Aquinas, for example, the transcendentals of truth and being are convertible. That which has being is true. And vice versa, that which is untrue does not have being. Perhaps, however, we could arrive at a knowledge of being that is so perfect, that the concept of false would simply no longer have any purpose. All that would be known is what is true what exists.

I suppose one could replace the concepts of truth and falsehood with what has being and what does not have being. … but that is just another way of saying the same thing.

My two cents.

[edit] Of course, if we had perfect knowledge, there would be no need for deductive knowledge. Deductive knowledge presupposes a process of coming to know. Which, as you say, does not appear to be possible without considering alternative propositions which turn out to be false.

God bless,
Ut

 

[opusAquinas]

I think Godel’s Incompleteness Theorem is relevant here but I forgot why.

 

[grannymh]

Deductive logic is very straightforward. The standard example of a deductive argument goes like this:

1. All men are mortal
2. Hans is a man
   Conclusion: Hans is mortal.

Could point 2. be an explanation of mortal?

 

[grannymh]

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.

In the Deductive Method of reasoning, it seems common sense that we, in a sense, manipulate and/or explore the truth of an axiom. We tug at the axiom to see what else there is within it.

 

[grannymh]

I cannot find anywhere in this thread an answer to the question in the title of the thread.

There is a very simple reason for suspecting that there is no answer. In other words, there is a very simple reason for suspecting that it is not possible to develop a legitimate system of deductive logic unless one has a prior concept of truth.

The simple reason is …

In the absence of a concept of truth, there is no label “true” and no label “false” available to assign to a statement. So we would not be able to say that some statement P is true and some statement Q is false. However, that is our fundamental technique for demonstrating that “Q is a logical consequence of P” is false.

If a proposed system of deductive logic allows us to prove that Q is a logical consequence of P, even though P is true and Q is false, then we have confirmed that our proposed system of deductive logic is defective. Given that symptom, we can look for an underlying flaw within the proposed system of deductive logic and attempt to remove the flaw.

In the absence of a concept of truth, there is no way for us to detect that a proposed system of deductive logic is flawed.

Yes.
It seems to me that in order to have a Deductive Method of reasoning, there has to be a primary truth–not a concept of truth. There is a difference.

 

[opusAquinas]

Yes.
It seems to me that in order to have a Deductive Method of reasoning, there has to be a primary truth–not a concept of truth. There is a difference.

Right. Now I remember why Godel’s theorem is relevant.

 

[Hans_W]

Could point 2. be an explanation of mortal?

This question came from:

1. All men are mortal
2. Hans is a man
   Conclusion: Hans is mortal.

Answer to your question: No

(1) and (2) are simple premises, or simple statements. They could be true or false. But if they are true, then the conclusion MUST be true as well.

That’s the beauty of a deductive argument. The conclusion must follow. However, a deductive argument doesn’t give you anything really new. The conclusion is basically already contained in the premises.

An inductive argument gives you a new fact, but you don’t get the absolute certainty which you can expect from deduction.

Neither deductive nor inductive reasoning have anything to do with the truth values of the premises.

 

[PseuTonym]

However, a deductive argument doesn’t give you anything really new. The conclusion is basically already contained in the premises.

We can imagine a being who can see all deductive consequences of any given assumptions. It seems that you are looking at things from the point of view of such a being when you say that conclusions are “contained in the premises” and aren’t “really new.”

I presume that you know the rules of chess. Tell me: is there is winning strategy for white? Is there a winning strategy for black? Or, if both players play the best possible moves, is it always a draw?

Presumably, the answer to that question is “contained in the rules of chess” and is not “really new” if you already know the rules of chess.

This question came from:

1. All men are mortal
2. Hans is a man
   Conclusion: Hans is mortal.

That is an extremely trivial deductive argument, and it would be a mistake to generalize from that example and believe that all deductive arguments are trivial. In some cases, there is a proof strategy, but the actual details are handled by a computer, because there is no known proof that is short enough and simple enough for there to be any practical value in writing it down in full. For example, that seems to be the case for all known proofs of the four color theorem.

 

[PseuTonym]

It seems to me that in order to have a Deductive Method of reasoning, there has to be a primary truth–not a concept of truth. There is a difference.

To work within a given system of deductive reasoning and deduce some conclusions, one begins with some assumptions. In other words, to apply a system of deductive reasoning, one needs to possess some statements that are known to be true or at least believed to be true.

However, where does a system of deductive reasoning come from? In order to create a reliable system of deductive reasoning, one needs a concept of truth. One needs to understand what it means to say that a statement is true, and one also needs to be able to recognize that some particular statements are true, and that some particular statements are false.

 

[Hans_W]

I presume that you know the rules of chess. Tell me: is there is winning strategy for white? Is there a winning strategy for black? Or, if both players play the best possible moves, is it always a draw?

Presumably, the answer to that question is “contained in the rules of chess” and is not “really new” if you already know the rules of chess.

And how do you see pure deductive reasoning being applied to a game of chess?