FORMAL LOGIC

[public hermit]

So far, it looks like what you're showing is the difference between a valid argument and a sound one. This is partly why logic is not as helpful as one might hope. Almost anything can be expressed as a valid argument, but it will not tell us if it's true.

1. If PH wears shirts on Tuesdays, then the king of France is bald.
2. PH wears shirts on Tuesdays.
Therefore, the king of France is bald.
(Modus ponens)

I, in fact, do wear shirts on Tuesdays. But France does not have a king, much less a bald one. Moreover, there is no connection between my wearing shirts on a particular day and whether the monarch France does not have is bald. How do I know these things? It's not due to the valid argument form above.

We can turn it around and form an invalid argument where the conclusion is, nonetheless, true.

I'm not saying logic is not helpful, but there is a gap between soundness and validity. How do we fill that gap? Experience? Testimony? Faith? Induction? Basically, it seems to come down to various ways of knowing/believing that often fall short of the certainty that deduction promises.

Your example of the sinlessness of Christ is a good example. The sinlessness of Christ is a truth claim based on faith and the witness of scripture. Being able to express that in the proper form of predicate logic is interesting, but it doesn't get us any closer to the truth.

I mean, didn't Wittgenstein show in the Tractatus that all valid formulas are tautologies? If it all reduces to a = a, then we shouldn't be surprised if it doesn't hook onto the world.

 

[Stephen3141]

Among the things I am pointing out are:

- logic can't be done from statements per se because statements are so complex
- in language, syntax evolves because semantics change, and semantics change because syntax changes. This is one of the things that historical linguistics has shed light on.
- assumptions and premises are to be inferred as much as possible in any matter which is an endeavour that won't end.
- thus logic has to be done from meanings.

All words allude. When we have several allusions we can see meanings. (That is intersectional. The universe is intersectional. You and I are intersectional.)

The passage in your post 25 is:

" ... NOTE: These rules of inference deal with the SYNTAX of a logical proof.
They do not deal with the semantic definitions of the propositions used in a proof.
For a proof to be SOUND, within a Christian worldview, all the Assumptions must have definitions that are orthodox. "

Language syntax and semantics provide clues towards logic syntax and semantics, because that's how nature intends.

I am saying: language syntax-and-semantics are different from logic syntax-and-semantics. If I've not misread, you were making the more limited point that in logic, syntax and semantics are different.

Earlier you had juxtaposed "syntax", "propositions", "christian world view" and "orthodoxy" which I why I'm checking.

I hope we can, as we proceed: examine "apologetics" (which seems to be the trendy name for doctrine, which is the problem everybody has), and "orthodox".

Thank you for the book refs, shall follow up. This (it looks like, and I hope) was the thread we needed all along.

My favourite toe dipper (as leisure reader) was Jevons. I also hold to Newman's insisting we conduct our own degrees of our own inference.

I'm still not sure what you are talking about, when you use the word "language".
I use the phrase "human language" when I am talking about English, French, Latin, etc.
I use the phrase "logical notation" when I am talking about the notation of modern symbolic logic.

Unless I specify otherwise, everything I am saying, is about logical notation.

 

[Stephen3141]

So far, it looks like what you're showing is the difference between a valid argument and a sound one. This is partly why logic is not as helpful as one might hope. Almost anything can be expressed as a valid argument, but it will not tell us if it's true.

1. If PH wears shirts on Tuesdays, then the king of France is bald.
2. PH wears shirts on Tuesdays.
Therefore, the king of France is bald.
(Modus ponens)

I, in fact, do wear shirts on Tuesdays. But France does not have a king, much less a bald one. Moreover, there is no connection between my wearing shirts on a particular day and whether the monarch France does not have is bald. How do I know these things? It's not due to the valid argument form above.

We can turn it around and form an invalid argument where the conclusion is, nonetheless, true.

I'm not saying logic is not helpful, but there is a gap between soundness and validity. How do we fill that gap? Experience? Testimony? Faith? Induction? Basically, it seems to come down to various ways of knowing/believing that often fall short of the certainty that deduction promises.

Your example of the sinlessness of Christ is a good example. The sinlessness of Christ is a truth claim based on faith and the witness of scripture. Being able to express that in the proper form of predicate logic is interesting, but it doesn't get us any closer to the truth.

It is true that probably ANYTHING (that is, any conclusion, or any rule) could be expressed in logical notation. And it is also true, that these expressions could be used in a proof that is logically valid.

But as I have repeatedly stated, a logical proof needs to be shown to be logically VALID, and logically SOUND, for the Conclusion to be demonstrated to be TRUE. this is why I present the Rules of inference (to determine logical validity), but emphasize that all in the Assumptions part of a proof be demonstrated as TRUE, also.

About your example:

W: PH wears shirts on Tuesdays
B: the king of France is bald

If you are asserting that these statements are in a relationship of logical causality, then the following constraints MUST hold:

W ==> B (an asserted entailment)

If W is TRUE, then B must be TRUE (Modus Ponens)
If B is FALSE, then W must be FALSE. (Modus Tollens)

Note that the basic truth table requirement for logical entailment, is not met. So W ==> B cannot be a Sound rule.

If you are asserting

W is TRUE
Therefore W OR ANYTHING. must be TRUE, this is a valid inferring by the rule of inference Addition.
(This is a pretty useless inference, most of the time.)

--------------------

Observation:

B: the king of France is bald

By putting this in your Assumptions part of the proof, you are also asserting that when "B" is used in your proof, then the definition "the king of France is bald" is TRUE.

Given that there is no king of France, this assertion cannot be TRUE.
Therefore, any proof that uses this assertion is Unsound.


Note that the logical "OR" constrains at least one of the connected propositions to be TRUE.
If you assert W OR B, then you must guarantee that B can be TRUE, when A is false.

If W is sometimes FALSE (sometimes you do not wear shirts on Tuesday), then to assert W OR B,
you must guarantee that B is always TRUE, when W is FALSE.

I do not see you meeting the basic logical requirements to assert either
W ==> B
or
W OR B.

--------------------

(The other subtle problem is that the statement "PH wears shirts on Tuesdays" must project to all future Tuesdays.
If you cannot guarantee this, then you should more properly assert
"On some Tuesdays, PH wears shirts"

There are very strict limitations on the "for some" quantification.

--------------------

Throwing Unsound assertions into the Assumptions part of your proof, only results in an Unsound argument.

 

[Stephen3141]

Again, I repeat, for a Christian using arguments/proofs in apologetics, ALL the assertions made in the Assumptions part of a proof must be Sound (in touch with our shared reality, and biblically sound),

AND, the logical form of the argument/proof must be VALID.

(Throwing Unsound definitions into the Assumptions part of a proof, or putting relationships of entailment that are Unsound into the Assumptions part of a proof, results in an Unsound proof.)

For a Conclusion of an argument/proof to be TRUE, all the Assumptions must be demonstrated to be Sound, and each step of the body of the proof must be logically Valid.

(None of this is controversial.)

 

[Stephen3141]

Perhaps a deeper explanation of syntax and semantics, in logical notation, is needed.

Logical notation can only encode syntax.
When definitions of logical signs are defined, in the Assumptions part of a proof, the "semantics" connected to the logical sign, is referred to, but the reference is to human language semantics.

(The same is TRUE in the notation of a computer language. Computers do not understand what any of the symbols/variables in the program really MEAN. It takes some hard thinking about this, to understand it.)

Trivial Example:

H: is a human being (definition of the symbol "H")
S: has sinned. (definition of the symbol "S")
H ==> S (assertion that there is a relationship of entailment, between H and S)

There is no way to recover the human language "all human beings have sinned" from the logical
notation H ==> S.

But, the logical notation appeals to the semantics/meaning referred to by the human language phrases
"is a human being" and
"has sinned".

What the logical notation H ==> S asserts, is that there is a relationship of ENTAILMENT between H and S,
whatever H and S really MEAN.

Conclusion:

In the Assumptions part of a proof, where we assert definitions/meanings that are attached to symbols that we will use in a proof, we are appealing to MEANINGS that are, formally, outside of formal logic.

This is why we must carefully evaluate the definitions that we put in the Assumptions part of an argument/proof, so that they are in keeping with "our shared reality" (which includes compatibility with what the Bible projects as our shared reality).

In other areas of life, if we misrepresent some aspect of our shared reality, it is called "bearing false witness" (lying).
So, if we include definitions in the Assumptions part of our proof that bear false witness, then this will result in an Unsound proof.

 

[Stephen3141]

Another subtlety, that you may not have thought about.

In a proof, if we Assume that the rule is TRUE
A ==> B

this means, IF A is TRUE, THEN B is TRUE
and IF B is FALSE THEN A is FALSE.
Those are big "ifs", so to speak.

Even if this rule is sound, this does not demonstrate that in a certain situation, we can say that A is TRUE.
Or, that B is False. It would take KNOWING that A is TRUE, or B is FALSE, for this rule to "fire".

Example: Modus Ponens
An example of a conditional inference.

A ==> B (assumed to be TRUE). (this must be demonstrated, not just "assumed')

A (in the proof, this is asserted to be TRUE, from whatever evidence...)
Therefore, B. (now we can assert that B is TRUE, by Modus Ponens)

But note in the proof, that (somehow) we could assert that A is TRUE. This is not an assumption.

--------------------

Conclusion:

Knowing "rules" that lay out conditional relationships is nice.
But until we know as a fact, the value of one of these symbols, we cannot use the rule to reason.

It takes knowing some facts about a situation, in order to use sound rules.

 

[Helmut-WK]

When I look into the examples, I conclude that explanations that a proof is not valid do not need formal logic, because they base on the observation that some basic assertion (e.g. If PH wears shirts on Tuesdays, then the king of France is bald) is not sound.

EDIT: Well, the last posts (I didn't see them before I posted) are somewhat different.

 

[Stephen3141]

When I look into the examples, I conclude that explanations that a proof is not valid do not need formal logic, because they base on the observation that some basic assertion (e.g. If PH wears shirts on Tuesdays, then the king of France is bald) is not sound.

EDIT: Well, the last posts (I didn't see them before I posted) are somewhat different.

I'm dealing with the topic of formal logic.
Within the structure of modern formal logic, you need to show that a proof violates the Rules of Inference, to demonstrate that a proof is not valid. (Being logically valid, is different than the Assumptions being Sound.)
"Valid" in formal logic, is a technical term. It's not decided by what someone may or not think is "valid" (in modern English).

 

[Stephen3141]

This thread is about formal logic. But, I approach logic from a Christian worldview. And to do this, I need to say a few things about "our shared reality", and what real evidence is, and how we observe evidence.

Many of these topics are upstream of our decisions about evidence, or what we are going to do with evidence. For the ancient philosophers, and the early Christian apologetics, these were common topics of discussion. For the younger generations in America, public schooling (and Christian catechism) often has almost no introduction to thinking about the primitives related to thinking about our shared reality. For this reason, I am going to pound on some basics related to our shared reality, as the Bible presents that subject.

(I am pointing out the need for talking about the subject of what reality is, because modern Americans, and many Christians, are neck deep in conspiracy theories that present alternate realities, that are NOT justified by the biblical concept of "evidence".)

I will be quoting some copyrighted material with references, as this leaves a trail that can be independently verified. I do not like "shooting from the hip" when talking about important subjects. Reference quoted are:

[Formal] Making Bible Study Formal, Stephen Wuest, 2020, Dorrance Publishing Company]

NOTE: Talking about what is real, and what real evidence is, is controversial.
There is no way to discuss these topics, without controversy.
But, I will struggle to be polite.

--------------------
Concerning Evidence and Belief...

"Studying the Bible, begins to make us conscious of the assertions that the Bible makes about what our shared reality is. The Bible presents propositions.

The obvious question about these biblical propositions is whether or not we believe them.

The un-obvious question is:
What constitutes evidence for/against believing the biblical propositions?

Saying “you’ve just got to believe what the Bible says” bypasses everything that the Bible says about the mind/heart (and its central place in our life, and in the identity of human beings).

When I started studying this section exploring evidence, and justified belief based on evidence, I had a real shock. I thought that it would be terrifically boring, and straightforward. But I have found so much material on this subject, that I don’t even know how to best arrange it. " [Formal, 155]

The Bible’s approach to what constitutes evidence, has a different flavor than the discussions in secular philosophy. It’s good to notice this right up front. The Bible does not agree with the current American culture (that deifies personal judgment). The Bible does not present the PC approach “Evidence is whatever I think it should be,” or “it’s relevant evidence if I declare that it is.”

The Bible presents our lives as a shared, unified reality. And when God presents us evidence in this shared reality, he holds us morally/ethicall responsible to react properly to it. The Bible does not go through the agonizing debates of the secular philosophers (who debate whether or not some types of evidence are accessible to all of us, or whether some types of evidence will persuade each of us enough to believe some proposition).

"The God who is presented in the Bible knows what sort of evidence each of us can experience, and presents us with enough evidence so we could be persuaded of the truths that the Bible presents. The Bible presents God as designing human beings, so that we can perceive and understand the evidence that he presents to us. This is a completely different approach, than our secular American society takes." [Formal, 155-156]

Assertion: The shared reality that the Bible presents, is what God holds us morally/ethically responsible for understanding. This is quite different than saying that there are many philosophical models for "what exists", or "what we can observe", or "what we could consider as evidence."

 

[Stephen3141]

--------------------
Concerning Evidence and Belief...

"Part of formal reasoning is carefully considering what it is that we are reasoning about, and how it is that we are reasoning. These are topics that involve evidence and belief.

There are some general topics about “evidence” that a thinking person needs to deal with. Then there are more specific questions about evidence that those who study the Bible need to deal with.

Some of these topics are:
— what is evidence?
— evidence has worth with regard to a specific proposition
— the Brain-in-a-Vat (BIV) problem (can we be fooled by fake evidence?)
— direct and indirect evidence
— is any belief evidence, or just justified beliefs?
— are there different levels of evidence, and how would your describe them?
— evidence encoded in history
— the availability of shared moral/ethical values
— the conscience: God revealing certain moral/ethical truths directly to us
— nature: God revealing certain truths directly to us
— the mind: God giving us a unity of experience, regarding the intellectual life
— morality/ethics and a fair rule of law
— morality/ethics and human rights
— the partial accessibility of our shared reality
— dealing with a unified reality
— being individually responsible for how we deal with the unified reality
— God as part of the unified reality
— does God give all people enough evidence, to believe the propositions in the Bible?
— do different worldly cultures perceive levels of evidence, with regard to some topics, differently?

We need to explicitly think about what evidence is, how we perceive it, different “levels” of evidence, and sound methods of evaluating evidence.

The philosophical discipline of Epistemology considers what good evidence is, that leads to believing/justifying that some proposition is true. This is a secular concept of how to justify (“show or prove to be right or reasonable” in the Dumbo right-click Mac dictionary) believing something. Christians need to consider these secular philosophical discussion topics, because many of them are excellent and relevant. (Some are not, in my opinion.)" [Formal, 164-166]

Assertion: The Soundness of a logical proof involves the evaluation of whether or not all propositions in the Assumptions part of the proof are Sound (TRUE). This is not controversial. But, Christians must realize that to do this evaluation, from a Christian point of view, involves engaging the biblical presentation of what our shared reality is, and what sound evidence is.

 

[Stephen3141]

--------------------
Concerning Evidence and Belief...

There is a connection between evidence, and knowledge, and understanding, and wisdom…

“One of the basic questions in secular philosophy is whether or not justified belief, can be the same as evidence. (In formal logic, we readily accept that we can prove conclusions and rules, that can be used as inputs to later proofs.)

In Jesus’ teaching, he addresses knowledge and wisdom, and understanding and insight (these are topics in the wisdom literature in the Old Testament). And Jesus is very clear: the more knowledge you can get, the better off you are to get even more. And, those who are slackers toward getting knowledge and wisdom, will position themselves to get less and less.

Jesus is presenting the truth that there is “equal opportunity” to seek for knowledge and wisdom. But how much of it we get, is based on our personal effort and diligence. There is no “equality of outcome” with regard to how much knowledge and wisdom we each get. God does not unconditionally give out knowledge and wisdom and understanding and insight.” [Formal,162]

“Jesus taught in parables (a kind of symbolic story), in order to hide the substance of what he was teaching, from people who were not seeking for truth. In the same way, God today hides knowledge and wisdom from those who are only looking for entertainment.

If you do not actively search for knowledge and wisdom, you will not see the evidence all around you that God has provided, that could lead you to knowledge and truth.

Just because evidence presents to us, we do not automatically understand what it means.” [Formal, 163]

Assertion:
— The Bible presents us with a shared reality. (This is not controversial.)
— The Bible presents our senses as “good enough” to perceive reality, and evidence.
(This is not controversial.)
— The Bible presents the gaining of knowledge, and understanding, and wisdom as states that involve a lot of personal required effort, even though sound evidence of our shared reality is available to all of us. (This is VERY controversial, to the Entitlement generations.)
— Jesus sometimes taught in parables, that hid the truths he was talking about from those who were only interested in being entertained. (This could be controversial with some.)

 

[zippy2006]

Response: There is no Rule of Inference that affirms the logical goodness of abductive reasoning.
That is why "affirming the consequent" is a logical fallacy.

I'm not quite sure why you would disagree with this example of a logical fallacy.

1. Affirming the consequent is not abductive reasoning.
2. The lack of a rule of inference is not evidence of invalidity. It is a standard feature of formal logic that just because something has not yet been proved does not mean it is false or invalid. You have ironically fallen into a fallacy for determining validity.

 

[zippy2006]

So far, it looks like what you're showing is the difference between a valid argument and a sound one. This is partly why logic is not as helpful as one might hope. Almost anything can be expressed as a valid argument, but it will not tell us if it's true.

Logic is a science of validity, not truth. That's not a bug, it's just a fact. The notion that there could be a science of truth per se is based on a misunderstanding of truth.

I mean, didn't Wittgenstein show in the Tractatus that all valid formulas are tautologies? If it all reduces to a = a, then we shouldn't be surprised if it doesn't hook onto the world.

You err in assuming that all premises and rules of inferences are axioms in the formal-system sense. The reason logic has been so important for over 2,000 years is because they are not merely axioms. For example, the law of non-contradiction is not a mere axiom; the modus ponens inference is not a mere axiom; etc.

 

[DamianWarS]

I see a lot of "informal" use of vocabulary that talks about "logic" or "logical" methods. But, I see very few Christians who have thought about modern Symbolic Logic (the basis of the hard sciences), and forms of applied logics that the different scientific disciplines have developed for their specialized (narrow) problem solving. The same language about "critical thinking skills" is common, but few authors agree on what this language means.

Also, I see much confusion in the "Christian" discussion of logic, in other apologetic forums. Specifically, there is a lot of confusion as to the similarities and differences between Aristotelian syllogisms and modern symbolic logic, and ancient rhetorical "fallacies" and modern symbolic logic. Christians need to discuss these topics openly (as a number of references to "logical fallacies" are not really relevant in the modern time).

Also, I see a vacuum of Christian discussion about how moral-ethical systems are constructed, how a Christian would construct a formal ME system, and how it would be related to formal logic rules. In this vacuum, many Christians have not thought carefully about what the applied logics in the hard sciences CAN express, and CANNOT express. One topic they CANNOT express is ME systems. But this should be carefully recognized, and the competencies of the hard sciences should not be slandered, because they are incapable of discussing some topics relevant to Christianity.

I see a vacuum in (lay, modern) Christian thinking about "our shared reality", and the components of that reality. Formal logic has to be a component of our shared reality, and there are moral reasons why this is so. But especially Christian groups that are biased toward being anti-intellectual, do not see formal logic principles as a component in our shared reality. And this needs to be discussed among Christians.

The explicit link between formal logic and morality-ethics, needs to be discussed by Christians.

The distinction between logical validity-invalidity in a proof, and logical soundness-unsoundness in a proof, clearly needs to be discussed.

Where I am coming from:
M.S. in Computer Science, Artificial Intelligence
35 years studying multiple notations for logic -- deductive, approximations logics, non-monotonic logics, and machine algorithms
B.A in Classical Languages and Literature, Ancient Greek
I have held to a high church theology for 40+ years, and consider myself a Christian
I have registered a manuscript called "Logic for Christians" with the Library of Congress in 2022, about 430 pages

My proposal for a forum:

1. I am NOT approaching this topic from a vacuum, or as a free-for-all.
2. I would post major propositions/topics about Christianity AND logic, and invite discussions.
3. Topics that I pose for discussion, would be rooted in an updated fusion of modern Symbolic Logic, and philosophical Moral Theory.
4. I would use a logical "pseudocode" that is acceptable to lay people who have not had a college 200 course in Logic.
5. I would stick to the topic of the structure of formal logic, as opposed to the debate of theological positions, although, I would give examples of theological arguments, as examples of how logical propositions are encoded.
6. I would use some copyrighted quotes, in order to protect the integrity of copyrighted material.

(Even after writing all this, I would expect Christians from an anti-intellectual background to misunderstand what I propose, or to disagree with the topics that I present for discussion. Many Christians have heard a lot of vocabulary about "logic" or "being logical", but are not using modern definitions, and still cannot interface with the modern hard sciences, and other applied logics. I expect these misunderstandings, but will be polite.)

Stephen Wuest. July 28th, 2023

does the statement "there is a God" follow formal logic? If not, then how do we reconcile that in our faith we have aspects that are unreconcilable with this system of logic? We may all agree there is a God but the rub is can we agree on where the line of negotiable and non-negotiable values actually is?

 

[Stephen3141]

Logic is a science of validity, not truth. That's not a bug, it's just a fact. The notion that there could be a science of truth per se is based on a misunderstanding of truth.


You err in assuming that all premises and rules of inferences are axioms in the formal-system sense. The reason logic has been so important for over 2,000 years is because they are not merely axioms. For example, the law of non-contradiction is not a mere axiom; the modus ponens inference is not a mere axiom; etc.

I think that we differ quite a bit, in what we take the meaning of basic vocabulary in formal logic means.

A few comments...

No one in their right mind, would accept that logic ONLY deals with the validity of syntax of notations.
If we do not address the soundness of initial assertions, then coming to a logically valid Conclusion, has nothing to do with the shared reality in which we live. Being concerned about the Soundness of Assumptions, IS a vital part of logic.

When I say that a line in a proof, that is not justified by one of the Rules of Inference, is logically invalid, this is correct. And it is different than realizing that there are probably TRUE Conclusions, that no one has yet found a valid and sound proof for. You are confusing these 2 different concepts.

What use of "axiom" are you disagreeing with? Could you give a reply number?

 

[Stephen3141]

does the statement "there is a God" follow formal logic? If not, then how do we reconcile that in our faith we have aspects that are unreconcilable with this system of logic? We may all agree there is a God but the rub is can we agree on where the line of negotiable and non-negotiable values actually is?

Proofs in modern formal logic are conditional.

IF (the Assumptions section of the proof is Sound)
AND IF (each line in the proof is justified by one of the Rules of Inference)
AND IF the Conclusion can be reached,
THEN the Conclusion follows conditionally from the Assumptions.

I'm not sure that you can prove that "there is a God", using formal logic.

In Christianity and Judaism, it is asserted that the existence of God, the creator, is evident to all people, by observing the power and order of the natural world. But, I don't think that this conclusion is a result of formal logic.

We can use formal logic, to reason that it is "reasonable" that there is a great, transcendent being. In the Christian groups, this is what the Intelligent Design authors argue. But, arguing that there is a great, intelligent mind behind the physical universe, is not quite the same as proving that the God of Abraham, Isaac, and Jacob exists.

This is the same conclusion that Mortimer Adler comes to, in his book How to Prove There is a God. He goes through all the historic proofs of the existence of God, and concludes that "the God of the philosophers" reasonably exists. But, this does not prove that the God of Abraham, Isaac, and Jacob exists.

We can use formal logic to come to many conclusions that suggest that it is reasonable to believe that God exists.
--------------------

The question of what characteristics God SHOULD have, is a difficult one. (Really!)

Different religious groups have some quite different EXPECTATIONS about what characteristics God should have. (Even if, within a specific group, they may not realize that their expectations do not exist in ALL religious groups.)

Among Christian denominations, some elevate one characteristic (trying to maximize it) over others. The result is definitions of the characteristics of God, that do not really match.
For Example, Calvin tries to "maximize" the sovereignty of God. The result is that god must make every decision, and therefore human beings cannot (really) have free will.
For Example, some religious denominations seem to like a "democratic" approach to the community, so they do not like the idea of a God who would give them an authoritative and structured leadership.
For Example, a lot of people think that God must be maximally "loving". But there are probably 100 different definitions of what they mean by "loving", and this results is many different definitions of what God must be like.

I would rather look at the experience of God's ancient People, the Jews, and how God interacted with them. In this interaction through history, we see that God has certain characteristics. But, I do not think that this gives us a complete model for what the characteristics of God are.

 

[zippy2006]

If we do not address the soundness of initial assertions, then coming to a logically valid Conclusion, has nothing to do with the shared reality in which we live. Being concerned about the Soundness of Assumptions, IS a vital part of logic.

1. Assertions and assumptions are not sound or unsound, they are true or false.
2. Coming to a logically valid conclusion has nothing to do with truth. Logic preserves truth, it does not create it.
3. We are concerned with soundness because we use logic as a tool to help us arrive at truth, but judgments of truth are extrinsic to logic. Logic stipulates truth and falsity; it does not make judgments about truth and falsity.
4. An axiom is something that is stipulated to be foundationally true within a formal system.

And it is different than realizing that there are probably TRUE Conclusions, that no one has yet found a valid and sound proof for.

1. Soundness presupposes validity, and therefore "valid and sound proof" is redundant.
2. If there is no proof for a proposition--sound or otherwise--then it is not a conclusion. You are talking about a true proposition, not a true conclusion.

If you do not understand these basic points, then you should not set yourself up as a teacher of logic.

 

[Stephen3141]

More on Evidence and Reality...

--------------------
To sum up some points, so far...

-- Christian apologetics MUST be concerned with our shared reality.
-- The biblical presentation of our shared reality, is underlined by "You shall not bear false witness."
-- God holds us morally-ethically responsible to correctly represent our shared reality.
-- Where morality-ethics enters formal logic, for a Christian, is in the imperative that statements in the Assumptions part of a proof (or argument), are truthfully in touch with our shared reality. This is how the Assumptions part of a proof are demonstrated to be Sound (which is a technical term in formal logic, not a fuzzy concept).
-- Validity (again a technical term in logic, not a fuzzy concept) is guaranteed by justifying each line in the body of a proof (or argument) with one of the 20 Rules of Inference, or Quantification Rules). If this is done, then the reached Conclusion is called Valid. But, if any line in the body of a proof is not justified in this way, then the entire proof becomes Invalid.

This is the relationship between our shared reality, Soundness, and Validity in a proof (or argument), for a Christian.

Note that for a Christian, we accept biblical definitions and truths about a great many topics. And this biblical truth is considered to be part of our shared reality. So, we must include this knowledge in designing a Sound Assumptions section for our proof. (This is VERY different than in a secular course in Symbolic Logic, in a secular university).

For a Christian, we accept (as in the teaching of Paul) that God has revealed to us a moral consciousness (or "conscience"), by which we all know the difference between right and wrong. I am considering that this consciousness is also part of our shared reality (and, as with everything else in our shared reality, we can be truthful about this knowledge, or we can lie about it).

I have also asserted that "valid reasoning methods" are a component of our shared reality. In this thread on modern formal logic, I am using the formulation of deductive logic, in modern Symbolic Logic. But this could be expanded, showing connections to approximation logics, applied logics (such as mathematics, chemistry, electrical engineering, biology, probability theory, etc.) non-monotonic logics, modal logics.... I point out that valid reasoning methods are a part of our shared reality, to underline that Christians have a moral-ethical OUGHT to properly handle valid reasoning methods. And that not properly handling them is a form of LYING. And denying that they exist, undercuts another part of our shared reality, which is the language in Scripture that deals with making valid decisions.

I have listed the modern 20 Rules of Inference, and the Quantification rules.
(But not given many examples, yet.)

Note that philosophical axioms and primitives, are UPSTREAM of formal logic rules.
I have made this point, but some readers, I think, are not convinced of this.

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We need to spend time thinking about what Soundness involves, in formulating the Assumptions part of a proof.
This involves how we view reality, and what evidence is.
This is a volatile subject in modern America, but it has got to be dealt with.
The ancient philosophers (and more modern ones), have MUCH to say on this subject of reality.

So moving on....

 

[Stephen3141]

More on Evidence and Reality...

(c) Stephen Wuest
“In this age of doubt that any truth exists, it is necessary to speak plainly about truth and evidence (toward some proposition being true). That there is truth, and evidence supporting it, has always been believed by Jews and Christians.

I include a few quotes, to make this obvious point.

“Biblical faith is evidence-based.” [Determined, p. 136]

“Again and again John records how people believed because of the evidence that Jesus provided (see for example John 2:11; 3:2; 4:41; 4:53; 6:14).” [Determined, p. 136]

“According to John, the faith Christ requires is anything but blind.” [Determined, p. 137]

For Christians who are more in touch with historical Christian apologetics, I would say that the acceptance that we live in a shared reality, could be rephrased to say that we live within a shared body of evidence.

This shared evidence includes knowledge about the physical universe, knowledge about our shared reality of valid reasoning methods, and our shared moral/ethical reality.

If Paul were here today, he would say that the order and design and beauty of the physical universe is a kind of shared evidence for the proposition that there is a designer/creator, and that he has certain characteristics.

Paul also presents to us that God’s revelation of the conscience (or moral consciousness) is a type of evidence given to all human beings, regardless of how we respond to this shared evidence, and regardless of whether we have effaced or obliterated it through willful abuse.”

[Determined] Determined to Believe? The Sovereignty of God, Freedom, Faith, & Human Responsibility, John C. Lennox, republished by Zondervan, 2017. Originally published by Lion Books, 2017.

 

[Stephen3141]

On Evidence, Reality, and Soundness

"To start, we need to consider the classic problem of a Brain-in-Vat (BIV).

This is a consideration about quality of perceptions/evidence.

Philosophers have used thought experiments (just as Einstein did) to examine questions dealing with what evidence is, and how evidence can (validly) justify our beliefs.

One of these thought experiments was the Brain-in-Vat thought experiment.

Suppose we have a Brain in a Vat, hooked up to all the nutrients that a brain needs to survive (and produce “conciousness,” whatever that is). Suppose that we have an electrical stream of sensory information, zooming into this Brain-in-Vat.

Then,
— could this BIV perceive that it is a BIV?
— could this BIV perceive that it is not a BIV?
— are the perceptions of the BIV valid?
— can the BIV ever get enough evidence to justify belief in any proposition?
— could the BIV ever prove that its perceptions were of a real world?
— could the BIV ever believe that there are other sentient beings?
— could the BIV ever believe that it lives in a shared reality, with other sentient beings?
— could the BIV ever have enough evidence to say “I believe that I exist”?

You may say that considering a Frankenstein like BIV problem is just bad science fiction. But what philosophers for centuries have considered, is the questions that this BIV thought experiment needs to address. And these are the same questions that Christians should be addressing, about the nature of reality.

In response to this thought experiment, one secular philosopher said “I think, therefore I am.” Regardless of how the thinking-consciousness happens, it indicates a real “me”.

When Christians start to address these questions, then they start to think about topics and answers that are apologetics for the Christian faith." [Making Bible Study Formal, Wuest, 176-177]

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If Christians do not think that the discussions by philosophers, on the brain-in-a-vat thought experiment is valid, then consider:

1. A person listens to only one new source, every day
2. The news source feeds them conspiracy theories, all day long
3. They have never listened to logical analysis/criticisms of this news source, or conspiracy theories.

At the final judgment...

1 Will Christ condemn them for living in this artificial "reality"?
2. Will Christ accept their concept of "reality"?
3. Will Christ accept their definition of "bearing true witness about reality"
4. Or, will Christ hold them responsible for lying about our shared reality?

Is this artificial reality that they live in, our shared reality?
Is the "evidence" that they accept from this artificial reality, real evidence?

If they formulate arguments/proofs using the 20 rules of Inference and Quantification rules from modern logic (so that their proof is Valid), will their basic definitions in the Assumptions part of their proof, be Sound?

Is it valid for me to seek out some artificial reality, that I want to be immersed in?
Will Christ hold me morally-ethically responsible for living my life in some artificial reality?
What if this artificial reality is chemically induced. What if I choose to be perpetually high?

What if I feel that I am ENTITLED to project any interpretation of reality that I want,
AND to have everyone respect that interpretation of reality?
Will Christ, at the final judgment, respect whatever interpretation of our shared reality, that I have projected?

Can I use ANY definition of propositions and rules in the Assumptions part of my proof,
and DEMAND that my proof is Sound?
Or, do I have a moral-ethical responsibility to represent our shared reality accurately, in the Assumptions part of my proof?

Can using incomplete/insufficient material in the Assumptions part of my proof, lead to an Unsound proof?
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These are types of questions that Christians should be asking, about our shared reality, Evidence, and the Soundness (or not) of proofs/arguments.