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

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.

“true” isn’t absolute but relative to the rules of the particular system

A theorem is true if it conforms to the axioms:

Link to post:
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Thread title:
Is there a materialist explanation of mathematics?
Link to thread:
forums.catholic.com/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.

I see mathematics as very different to science. Mathematics is useful and essential to science, but it can stand on its own.

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.

In our knowledge gained through science (our knowledge of nature) everything is based on induction, which doesn’t give you absolute truth.

Not sure if I answered the question.

It is my understanding that there is only one basic deductive logic system. It is also my understanding that the basic purpose of Deductive Method of Reasoning is to deduce truth or derive truth. My resource was long ago Google; however, I did not search out the various complications. I am interested in additional information about the actual Deductive Method of Reasoning.
Thank You. :smiley:

Our minds were created to know truth. The way it goes about it is perfectly natural. I wouldn’t get hung up on logical systems. See the thread " How do we come to know things? " on this forum.

Linus2nd

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.

There is no new “truth” coming out of a deductive argument. The conclusion is already contained in the premises. Also note that one of the premises must be a universal, in our example it is “all men are mortal”. But that’s already an assumption. Who known if all men are mortal (and it doesn’t say anything about women).

Mathematics is the exception here because you start from “axioms” which are the starting points of reasoning (the premises) in mathematics. These are assumed to be true. An example would be “Any point can be connected with any other point through a straight line”.

Logic is a language. The truths that it provides are contained within the framework of said language. Propositional calculus, first-order logic, the various modal logics, and many-valued logics all have their own rules and ‘truth’ in one system might not hold in the others. That’s why for the more advanced logics, we stop looking for “truth” and looking for “truth conditions” as determined by the rules of the language.

In first-order logic, for example, truth conditions depend on a given interpretation where all predicates, singular terms, functions, and the domain is set. The sentence ( ∀x) Wx→Mx with the interpretation Mx= x is a woman Ox= x is mortal, and the domain is organisms, then we seem to have a true statement. (All women are mortal) However, if the interpretation is Wx= x is a prime number Mx= x is an even number and the domain is the natural numbers, we have a false statement. (All prime [natural] numbers are even)

I think that I agree with everything that you wrote, as written, but reading between the lines, I see some things that should be clarified. I will try to clarify those things.

You hint at a connection between absolute truths and deductive reasoning. Do mathematical statements that are true have some special kind of truth? Are there at least two kinds of truth: ordinary and absolute? Is a grounding in deductive logic what ensures that a mathematical statement has not merely ordinary truth, but absolute truth?

I say that plain vanilla truth is good enough. I have no need for any concept of absolute truth.

Now, we know that we cannot begin to prove a conjecture until after somebody thinks of an idea and formulates it. Deductive logic does not provide us with a list of conjectures that we are to prove. Nor does deductive logic itself provide us with everything that we need to prove a conjecture of number theory, unless we are talking about a very trivial conjecture of number theory.

Furthermore, if we are using a particular system of deductive logic, then we have a wonderful tool to use, and we should ask where that tool came from. There is a history of ideas, and the development of logic has taken thousands of years. What made it possible was the human ability to recognize some simple truths. Before we can recognize that statement T is a logical consequence of statement S, we need to have the concept of truth. First we recognize that statement S is true. If we cannot do that, then we are like a baby who can crawl, but cannot sit, stand, or walk. In that situation, we are unlikely to successfully run.

Now, I suspect that before we can say more we may need to confront a problem with language, specifically the words “believe” and “know.” The word “believe” has a variety of uses. It might signal that I am aware that others disbelieve. Or it might signal that I am discussing something unfamiliar to me, and I am ready to be told that there is no controversy and it is universally believed, and I am also ready to drop my belief when I learn more.

We can make a list of statements. If I say that I believe the statement at the beginning of the list, then I anticipate that people will contradict me, and tell me that I know it, and that I do not merely believe it. If I say that I know the statement at the end of the list, then I anticipate that people will contradict me, and tell me that I believe it, and that I do not know it. The problem is that there is no way to know where the line is drawn.

We need a word like “believe” minus the suggestion of not knowing. We want to be able to say that we believe something, and neither claim that we know it, nor claim that we do not know it.

Yes, basically. Formal proof systems are ways at arriving at new statements from old statements, and do not necessarily have to give new statements that are true. The property of only transforming true statements to true statements is called soundness, and systems that aren’t sound (w.r.t. whatever model(s) we are interested in) are generally not interesting. The property of having everything that is true be provable (completeness) is also nice to have, but generally more difficult.

[quote=Rhubarb]Logic is a language. The truths that it provides are contained within the framework of said language. Propositional calculus, first-order logic, the various modal logics, and many-valued logics all have their own rules and ‘truth’ in one system might not hold in the others. That’s why for the more advanced logics, we stop looking for “truth” and looking for “truth conditions” as determined by the rules of the language.

In first-order logic, for example, truth conditions depend on a given interpretation where all predicates, singular terms, functions, and the domain is set. The sentence ( ∀x) Wx→Mx with the interpretation Mx= x is a woman Ox= x is mortal, and the domain is organisms, then we seem to have a true statement. (All women are mortal) However, if the interpretation is Wx= x is a prime number Mx= x is an even number and the domain is the natural numbers, we have a false statement. (All prime [natural] numbers are even)
[/quote]

However, (∀x) (Wx→Mx) is not provable from PC itself, so while the truth of that statement depends on the model (or meaning we assign to W, M, etc, which I believe is what you are calling language), this doesn’t say anything about the proof system - the proof system doesn’t care about any of that sort of thing at all. A PC-valid statement would be something like “(Wx→Mx) OR (Mx→Wx)”, which will be true in any model of PC whatsoever (despite looking rather funny, the joys of the conditional).

So the actual rules of logic will still only give you true statements. If you want to restrict to certain types of models, you can throw in more axioms, then restrict to models in which those axioms hold, etc, but the same thing will reply. So logic itself is independent of our interpretations of the symbols and such.

I’m not sure what you mean by truth conditions vs. truth. I mean, I know we do things like represent truth by a function that sends statements to either 1 or 0 depending on stuff, but that’s just a fancy way of saying the statement is true or false based on other things. I am also aware of multi-valued logics, but I would hesitate to call those more “advanced.” More complicated, yes. But kind of silly.

I’m not sure what you mean by truth conditions vs. truth.

True mathematical sentence. We are talking, true-correct- “Provable”.

numbersleuth.org/trends/goedels-theorem-for-dummies/

Sorry, I don’t see the distinction being made between looking for “truth-conditions” and looking for “truth” in that link either. In a formal system, a statement is provable or not (we may not know which, but it is one or the other). In a model, a statement is true or false (leaving aside multi-valued logics). If we’re working on the proof theory side, we look for proofs. If the proof system is sound and we have found a proof, then that means that the statement is true in all models for which the proof system is sound. It is also true that the non-existence of a proof of a statement does not always imply the falsehood of a statement, a la Godel, but that gives a distinction between provability and truth. (Ex: one can construct models of ZF in which the axiom of choice holds, and also in which it does not - in some of those models it is true, in others it is false, but neither it nor its negation are provable from ZF.)

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 do think it is good to have a sense of what is an actual truth vs a conceptual definition.

In dramatic tests in the past 50 years of the Theory of Relativity we are reasonably sure that there are no straight lines between two points; rather, space is curved. Yet, Einstein used the mathematical system who’s axioms posit the very opposite of curved space.

How can he do it? He must first have a system that conceptualizes what a true straight line is as understood by mathematics. With this he can describe the proposed actual truth of his theory and develop the equations to describe it. This leaves us a comparison from the proposed actual truth to the conceptual definitions of our mathematics.

In this I hope I’ve cleared up why systems of logic don’t necessarily have to be actually true to be a useful tool conceptually and may also be very useful because of precise definitions and proofs of internal consistency.

This does not mean that we don’t have a “concept of truth” only that it deviates from some simple mathematical axioms. Nor does it mean that there isn’t an actual truth, but apart from revelation from God absolutely knowing what the actual truth is can be hard to conceptualize.

I think you should read this article on radio active decay by Edward Feser. It deals with how we know the truth about the truth value of some things science says about reality and what actually the truth is at the deeper level of metaphysics.
edwardfeser.blogspot.com/2014/12/causality-and-radioactive-decay.html

Linus2nd .

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.

Lot of this is perception, I can mathematically deduce through equation a truth which proves itself mathematically, such as with a law. So it is of significance if I can apply it in the lab, recalculate for tolerance variance, and the math still proves. In this sense it does preserve truth. In theory this is applicable now in practical experience. There is no reason to believe it wouldn’t in this case remain what is true to this paradigm. However, in practical experience this may not be all -together true. There may still be need for recalculation due to tolerance variance.

Logic is performed using only switches in Boolean algabra. ‘True’ is merely a label for a closed switch, ‘false’ a label for an open switch. A logical OR is a set of switches wired in parallel, an AND is a set of switches wired in series, and so on.

Any logical construct can then be tested empirically by wiring a battery and a lamp to our switches.

Since the logic is performed by a bunch of switches, this shows that any significance or meaning we attach to ‘true’ and ‘false’ depends on us.

Btw this is of course how computers do logic, and by extension, arithmetic - computer.howstuffworks.com/boolean.htm.

Since the logic is performed by a bunch of switches, this shows that any significance or meaning we attach to ‘true’ and ‘false’ depends on us.

But what we call truth about the law is the logic, that doesn’t change. E=IXR remains intact and we call it truth? Devices are added into a circuit with their value and the equation in mind. Resistance is what we are talking about. You are calculating total resistance of all devices or switches etc.

In the next section, we’ll look at how a full adder is implemented into a circuit.

Not sure what point you’re making. What’s Ohm’s Law got to do with it?

Your talking circuits, The computer runs AC or DC, the law applies.

logic is performed by a bunch of switches

The logic is the law, not the work of the devices, the law is ohms law-“true”. The work of the switch is the resistance-ohms, open and close contacts. Its part of the entire equation.

Don’t see how it’s relevant, but OK, instead of electrical switches, let’s make our logical construct from a bunch of philosophers. Pass a teddy-bear to the first philosopher, and if we told her that she represents a closed switch she passes it on to the next philosopher in the construct, but if she represents an open switch she hangs on to the teddy-bear. And so on for each philosopher-switch in the construct.

If we eventually get the teddy-bear back from the last philosopher in the construct, the construct is true. Otherwise, if the teddy-bear stops being passed on, the construct is false.

Or we can use even more exotic alternatives if you wish, I don’t think Dr Bool would mind.

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.

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.

Which is just another way of saying that the universe makes sense. Which can’t be empirically proven, obviously, (few things can) but it’s the sort of assumption that most people make, and appears to match up pretty well with out experience.

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