FORMAL LOGIC

[Stephen3141]

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

 

[Akita Suggagaki]

I encourage you to go for it but keep in mind that most people here do not have your background.

 

[d taylor]

-
​

What does "consider yourself a christain" mean. Do you think you are a christian because you go to church.

 

[Stephen3141]

A beginning of a list of topics to discuss:

1. Our shared reality
2. Components in our shared reality
3. Valid reasoning methods are part of our shared reality
4. The moral-ethical connection to our shared reality
5. ....

 

[Stephen3141]

I encourage you to go for it but keep in mind that most people here do not have your background.

I agree. I am trying to communicate with lay Christians.

 

[Akita Suggagaki]

I agree. I am trying to communicate with lay Christians.

Fire away. let's see how it goes. How will you proceed? with Question or statement?

 

[Stephen3141]

Coming at formal logic, from a Christian point of view, I think, requires starting (informally) with the topic of "our shared reality".

And, because, we have not gotten into any formal discussion of what logic is, this will be an informal inductive approach.

[Given an approach that starts with these topics...

A beginning of a list of topics to discuss:

1. Our shared reality
2. Components in our shared reality
3. Valid reasoning methods are part of our shared reality
4. The moral-ethical connection to our shared reality
5. ....]

"The basis for all scientific disciplines, is the assumption that we share the same physical reality. The basis for philosophical moral theory is that we share the same reality (including moral/ethical reality). The basis for social justice, is that we share the same reality. The basis of property rights and representational rights as citizens, is that we all share the same reality. The basis for human rights, is that we share the same reality, and we share the ability to talk about concepts (such as freedom , and justice, and respect) that we all share in a conceptual world.

Some of the types of reality that we share are:
— physical reality (including time)
— moral/ethical reality
— the reality of a valid method of reasoning
— free will, so that we can choose how we use our mind."
[Making Bible Study Formal: , Re-Introducing the Intellectual Disciplines Into Bible Study, 76]

Informally, as Christians, we can argue that we live in a "shared reality".
Bypassing all the philosophical discussions of what exists (Ontology), what is TRUE (Epistemology), how we can know what is TRUE, how we perceive this shared reality... the Bible present human beings as living in a single universe (physical reality).

The Bible presents human beings as having a conscience/moral consciousness (in Paul's language), and being aware of what sin is. This is what I mean, when I talk of moral/ethical reality. As Christians, we also recognize that God gave us the Scriptures (I will avoid questions of how God did this -- they are off topic). We will all be judged at the final judgment, by this same divine moral-ethical law. So, from a Christian worldview, we live in a shared moral-ethical reality.

I will speak from the position that human beings have a mind, and can make decisions. Being able to make multiple decisions, from the same starting material, is required for us to use valid and sound logic, or invalid or unsound logic. For those who think that we do not have this freedom, there is no reason to read this thread about formal logic. (I am not going to engage in theological discussions of whether or not we have "free will". I consider these to be off topic.)

Assertion: Valid reasoning methods, are part of our shared reality.

(I would be curious to hear your thoughts, on this assertion.)
(I will try to flesh out some arguments for the Assertion, in a later post.)

 

[PloverWing]

I enjoy formal logic, so I'd be interested in the discussions you're proposing. I suspect that there will be limits to how much of religious experience can be formalized using symbolic logic, but let's see how far it can take us.

(Are we posting our résumés? BS Mathematics, PhD Computer Science. The propositional calculus and the first-order predicate calculus are the formal systems most comfortable for me, as I use them frequently at work. Other forms of logic are rustier for me, but I should be able to brush up if you want to use some other kind of formal logic system.)

As to the assertion: I think I'd agree to taking this version as one of our starting assumptions:

"Valid reasoning methods are a formalization of part of our shared reality."

That is, the symbols we use are arbitrary, but they seem to be capturing something that's deeply a part of shared human intuition about what's true and false.

 

[public hermit]

This sounds fun. I will have to go with @PloverWing adjustment of the initial assertion. I usually think of validity in terms of argument form.

 

[Akita Suggagaki]

It seems that "invalid or unsound logic formulations" are just as much a part of our shared reality. Or valid and sound logic formulations are not really shared by all. In fact, I wonder how some people think the way they do. Our shared reality seems to be shrinking as our unshared realities seem to be growing.

 

[tonychanyt]

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

Thanks for sharing.

Check out A Denomination-Free, Disciplined, Logical, and Probabilistic Approach to Biblical Hermeneutics. Tell your friends

 

[Stephen3141]

I enjoy formal logic, so I'd be interested in the discussions you're proposing. I suspect that there will be limits to how much of religious experience can be formalized using symbolic logic, but let's see how far it can take us.

(Are we posting our résumés? BS Mathematics, PhD Computer Science. The propositional calculus and the first-order predicate calculus are the formal systems most comfortable for me, as I use them frequently at work. Other forms of logic are rustier for me, but I should be able to brush up if you want to use some other kind of formal logic system.)

As to the assertion: I think I'd agree to taking this version as one of our starting assumptions:

"Valid reasoning methods are a formalization of part of our shared reality."

That is, the symbols we use are arbitrary, but they seem to be capturing something that's deeply a part of shared human intuition about what's true and false.

I enjoy formal logic also.

I posted my education, because many people who discuss logic, don't really know what they are talking about. I'm not trying to impress people.

My first few posts are just setting up where formal logic belongs, I think, in our experience.
Addressing "our shared reality" is, I think, a way to ground the discussion of logic.
I take "valid reasoning methods" to include all sorts of stuff, with formal logic as a subset.

 

[Stephen3141]

This sounds fun. I will have to go with @PloverWing adjustment of the initial assertion. I usually think of validity in terms of argument form.

Right. the logical validity of an argument/proof, involves the syntax of the statements.

Of course, for a Christian, we're concerned with writing arguments/proofs that match reality also, and so are also logically sound. And this involves definitions/meanings of everything in our Assumptions part of the proof (definitions, and rules).

although we can play around with "mental experiments" like Einstein, as Christians we want our arguments/proofs to be in touch with this world, as opposed to some other alternate "play world".

 

[Stephen3141]

(Coming at formal logic, from a Christian point of view, I think, requires starting (informally) with the topic of "our shared reality".
And, because, we have not gotten into any formal discussion of what logic is, this will be an informal inductive approach.)

My next initial assertion:
Christians (and Jews) hold that there is a Moral-Ethical OUGHT to properly represent our shared reality.
The informal argument would be something like this...

We live in a shared reality (this is what the Bible presents)
We are commanded to bear true witness (not to lie)
What is it that is the object of this telling the truth, but our shared reality?

The command not to lie is VERY important for Christians, as the New Testament warns us that there will be no liars in the kingdom of heaven.

6 And he said to me, “It is done! I am the Alpha and the Omega, the beginning and the end. To the thirsty I will give from the spring of the water of life without payment. 7 The one who conquers will have this heritage, and I will be his God and he will be my son. 8 But as for the cowardly, the faithless, the detestable, as for murderers, the sexually immoral, sorcerers, idolaters, and all liars, their portion will be in the lake that burns with fire and sulfur, which is the second death.”
The Holy Bible: English Standard Version (Wheaton, IL: Crossway Bibles, 2016), Re 21:6–8.
(Ouch!)

8 Now we know that the law is good, if one uses it lawfully, 9 understanding this, that the law is not laid down for the just but for the lawless and disobedient, for the ungodly and sinners, for the unholy and profane, for those who strike their fathers and mothers, for murderers, 10 the sexually immoral, men who practice homosexuality, enslavers, liars, perjurers, and whatever else is contrary to sound doctrine, 11 in accordance with the gospel of the glory of the blessed God with which I have been entrusted.
The Holy Bible: English Standard Version (Wheaton, IL: Crossway Bibles, 2016), 1 Ti 1:8–11.

Specific Assertions:

-- we are not to misrepresent our shared reality
-- part of our shared reality is valid reasoning methods
-- part of our shared reality is the objects of valid reasoning

So we have a moral-ethical OUGHT to properly use valid reasoning methods to think about objects in our shared reality.
Because using invalid reasoning methods, involves misrepresenting parts of our shared reality. This is a form of lying.

An assertion that follows these:

-- IF valid reasoning methods are part of our shared reality, THEN
choosing to be anti-intellectual is misrepresenting a component in our shared reality.
That is, choosing to be anti-intellectual is a form of lying about our shared reality.


(NOTE:

Some denominations hold to a very mystical view of knowledge, in which "the Holy Spirit will reveal truth to me".
While this IS a function of the Holy Spirit, note that holding that THIS is the only way to discover truth,
does not match the way in which the Scriptures present the many ways in which we can discover truth.

And, if we hold to a theology in which the Holy Spirit must be the only method to know truth,
THEN we annul all the language in Scripture that commands us to think, consider, decide, judge properly.
This type of theology is an abuse of the plain meaning of many texts in Scripture, as the biblical authors used this language.

There is no necessary contradiction between a function of the Holy Spirit as a revealer of truth,
AND our moral-ethical OUGHT to recognize valid methods of reasoning as part of our shared reality,
AND our moral-ethical OUGHT to properly use reasoning methods to represent our shared reality.)

This is how I would ground formal logic, in a Christian worldview.

 

[Stephen3141]

More introductory assertions about formal logic…

Assertion 1: There are many types of “logics”.
Assertion 2: Traditionally, “formal logic” means deductive logic.
Assertion 3: Relationships between deductive logic, and many other forms of logic,
can be demonstrated.
Assertion 4: The hard scientific disciplines often use “applied logics” that are developed
on the foundation of deductive logic.

NOTE: Deductive logic is the focus of this thread.

Assertion 5: Deductive logic is typically considered to be “2-valued” (TRUE/FALSE) logic,
as each proposition can evaluate to TRUE or FALSE. But, practically, it is
more practical to consider that the value of a proposition may also be
UNKNOWN (as in a computer program). So until we can prove that the
value of a proposition is TRUE or FALSE, we can consider that it has the
value of UNKNOWN.
Assertion 6: Human language can be at least as accurate as deductive logic, or the applied logics.


About Assertion 1:

There are many types of logic. Deductive logic, inductive logics, non-monotonic logics, dynamic logics, logics based on statistics, probability theory, modal logics….

About Assertion 3 and 4:

Note that Russell and Whitehead wrote Principia Mathematica about 1903. And they demonstrated that all of modern mathematics can be constructed out of deductive logic, and infinite set theory. So, mathematics (which has numeric quantification) is NOT different than deductive logic.

Note that Computer Science students know that non zero-one statements can be translated into a decision statement, which is essentially deductive logic. So
“The probability of event A is between 12% and 17.45%
Thresholds into TRUE or FALSE.

Note that in applied logics, every line in the body of a proof must threshold into TRUE or FALSE. In this way we see that all sorts of logics are built on 0-1 deductive logic.

About Assertion 6:

I have gone to schools taught in English.
All the textbooks on technical subjects, have been written in English.
Therefore, English can be at least as precise as all the technical subjects that I have studied.

So, although Deductive logic may be seen by some people as being very different from other forms of applied logics, it is not.

 

[PloverWing]

A couple of small points.

Assertion 5: Deductive logic is typically considered to be “2-valued” (TRUE/FALSE) logic,
as each proposition can evaluate to TRUE or FALSE. But, practically, it is
more practical to consider that the value of a proposition may also be
UNKNOWN (as in a computer program). So until we can prove that the
value of a proposition is TRUE or FALSE, we can consider that it has the
value of UNKNOWN.

Can you elaborate on this for me? My experience with 3-valued logics was brief, and a long time ago. My reservation is that "unknown" seems like the wrong name for the intermediate value. Whether a proposition is true is different from whether I, personally, know it to be true.

(Consider, for example, all the proofs by cases we've done over the years. "We don't know whether P is true or false. But if P is true, the theorem follows, and if P is false, the theorem also follows; therefore, the theorem follows." There's a difference here between P's truth value and what I know about P's truth value.)

So, can you specify a little more how UNKNOWN relates to TRUE and FALSE here?

Note that Russell and Whitehead wrote Principia Mathematica about 1903. And they demonstrated that all of modern mathematics can be constructed out of deductive logic, and infinite set theory. So, mathematics (which has numeric quantification) is NOT different than deductive logic.

Remember to take Gödel's incompleteness theorem into account here.

Note that Computer Science students know that non zero-one statements can be translated into a decision statement, which is essentially deductive logic. So
“The probability of event A is between 12% and 17.45%
Thresholds into TRUE or FALSE.

Can you elaborate on this as well? You might be saying: "If probability(A) > threshold, then assign A a truth value of TRUE, else assign A a truth value of FALSE." From there, boolean operators can apply to A. And we might be able to prove things like A & B -> C. But we do have to keep in mind that C's truth value is as unknown as A's. Is that the kind of thing you have in mind?

About Assertion 6:

I have gone to schools taught in English.
All the textbooks on technical subjects, have been written in English.
Therefore, English can be at least as precise as all the technical subjects that I have studied.

English can be precise, but it often isn't, which is one reason that the language of these textbooks is augmented with all kinds of mathematical symbols. But, yes, a great deal can be expressed in human language if we're very careful.

 

[Stephen3141]

These are good questions. I prompted them, because I don't want to talk about a simplistic understanding of logic.

---------------------------------
Response: Is deductive logic 2-values or 3-valued?

Formally, deductive logic is 2-valued, and the values allowed are TRUE/FALSE.

In real life, if you try to teach kids using homework problems, then it is useful to view deductive logic propositions/"variables" as being able to have the "value" "I don't know", when we cannot demonstrate that the value of a proposition or "variable is either TRUE or FALSE.

"I don't know" is what we can say about the value of a proposition or "variable", when we cannot demonstrate what its TRUE/FALSE value is. Although "I don't know" is not historical logicians would call a "value" of deductive logic propositions/"variables", it is a very common attribute ABOUT the value of deductive logic propositions/"Variables". And IT IS VERY USEFUL in logical arguments, to be able to say about some proposition/"Variable" that in the situation, we don't know its value.

When we apply formal logic to theology, or digital circuits, or computer algorithms (and the initialization values of variables), it is VERY useful to know, in any given situation, IF we don't know the value of a proposition or variable.

Example:
Given: A ==> B
B ==> C
B is TRUE

Therefore: what we can announce to the world about the values of the "variables" is:
A is Unknown
B it TRUE
C is TRUE

Although, technically, we can still say that the value of A is actually TRUE XOR False, we do not know what the value is. And it is a further statement about A, if we do say that we know that the value of A is Unknown.

The USEFULNESS of using 3 "values" is that we can create models or rule-fact systems, and can positively say what we can demonstrate from the model/rule-fact system, including which propositions or "variables" we CANNOT deduce the value of.

This ability to look at a logical model (including theological models), and being able to say that given some set of inputs, we still cannot determine the value of some proposition or "variable", is a way of saying that the model may incomplete (a very useful thing to know), or that our input knowledge may be incomplete (a very useful thing to know).

At the beginning of a proof, in the "Assumptions" area, we state what assume is TRUE.
At the beginning of a proof, we do not assign the desired Conclusion a value of TRUE XOR FALSE, even though we know that deductive logic is 2-valued.
If in the Assumptions part, we assign each "variable" or proposition that is not given the value of TRUE/FALSE the value of "Unknown", then we can make a complete listing of the values of each proposition/"Variable", at each line in the proof. THIS is a very interesting exercise.

Technically, you're right. Deductive logic is 2-valued.
Practically, I like to preserve the knowledge that a "variable"/proposition has an unknown value, when it does.
---------------------------------

 

[Stephen3141]

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

Response about mathematics not being different than deductive logic...

Deductive logic does not have numeric quantification.
Russell and Whitehead showed how to create number lines, and numeric quantifications out of deductive logic, and infinite set theory.
And so, mathematics is a kind of superset of the (traditional) deductive logic, although mathematics is built out of the components of deductive logic.

Because mathematics has a degree of numeric quantification, mathematics is NOT Complete.
Deductive logic (at least first order calculus), without the mathematical extensions, is Complete.
So, modern mathematics has some qualities that are different from the first order quantified deductive logic.

But, my point, which may not have been precisely expressed, is that modern mathematics is built out of the components of deductive logic and infinite set theory, and so cannot be said to be something "different" than deductive logic. I made this point, because a lot of people don't realize this. And they (in error) think that "mathematical reasoning" is somehow more precise than deductive logic.
---------------------------------

 

[Stephen3141]

---------------------------------
Response about the assertion that human language is often not used in a precise way...

That is correct.
But an imprecise use of English (or any other human language) does not prove that it CANNOT be used precisely.

And, all the symbols in mathematics can be defined in human language. And generally, have been.

I think that a more useful observation is that we use notations for thinking, that happen to be efficient and clear, to the degree that we need to express certain thoughts, in a proof. We could use deductive language, to do mathematics. But, it would be HORRENDOUSLY inefficient.

Worse, the collapse of language skills among modern Americans, may give them the wrong idea that human language CANNOT be used precisely. We see that the "electronic screen" generations have a tendency to NOT be able to do formal logic, OR engage in philosophical discussions about primitives. We might think that this is because these topics are not relevant or interesting... but I think that the younger generations would need to gain a larger vocabulary to engage in these disciplines.

Increasingly, in Christian apologetics, I have to ask people what they mean by words or phrases that they use. And, the level of imprecision in their language is so bad, that their questions are unanswerable.

Those who do not have a high level of knowledge of human language, CANNOT use formal deductive logic. This is a CRIPPLING limitation on their abilities to think.

 

[Stephen3141]

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

Response to comments about Godel and his proofs...

Godel exploded the belief in the early section of the 20th century, that mathematics (and logic) were a complete system of reasoning, and that axiomatic systems could be created for all of mathematics.

---------------------------------
Response to a comment on Godel.

Godel actually wrote about 13 Incompleteness proofs.

One of the reasons why we continue to get strange assertion about the infinite power of mathematics or logic, is that many modern Americans are IGNORANT of what has been proven about logic (and mathematics).

And, TO UNDERLINE A VERY IMPORTANT CONCEPT, even though a reasoning system (such as first order quantified deductive logic, or different disciplines in mathematics) may be shown to have limitations, that DOES NOT ANNUL what it is proven to be good for.

The arrogance of the logical-mathematical thinkers at the beginning of the 20th century, was that they assumed that mathematics-logic was complete, and that all that was TRUE could be proven, and that all problem domains could be described by axiomatic models.

Simple propositional deductive logic is Complete. This is a technical term, meaning that every TRUE proposition encoded in this type of logical notation could theoretically be proven with a proof. (Maybe no human being to date, has discovered the proof, but theoretically, someone COULD discover one.)

First order quantified logic (which reasons about groups of individuals using "for all" and "there exists") is Complete.
Second order quantified logic (which can reason about both groups of individuals, and groups of characteristics) is considered to be Incomplete. By incomplete, logicians mean that there are concepts that can be described in this logical, for which no proof exists, in this logical notation.

If you read "The Outer Limits of Reason" by Yanofsky, he describes types of questions that mathematics and logic CANNOT address. For all Christians who are aggressive readers, and interested in Christian apologetics, you should read this book.

BUT, but, but, ... because we know that there are types of limitations on logic and mathematics, that does not mean that mathematics and logic do not address ANY topics authoritatively. Nor does it mean that these limitations come into play in most of the ways in which we use mathematics and Logic.

Christians who are anti-intellectual wrongly claim that knowledge can only be gained from a mystic gift from the Holy Spirit. In this way, they often deny the goodness of logic, and the valid methods of reasoning that human beings have demonstrated apply to all sorts of topics. This is like claiming that, because all usual human being have sinned, then God cannot (actually) sanctify us so that we can live a holy life. In reality, these are 2 very different considerations.

We use logic for what it is good for. We affirm what it is good for.
And, we affirm the real mystery that we can know many of the limitations of human logic.
And, we affirm that human logic is valid and sound for reasoning about certain topics.

Gödel's Proof|Paperback

An accessible explanation of Kurt Gödel's groundbreaking work in mathematical logicIn 1931 Kurt Gödel published his fundamental paper, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems." This revolutionary paper challenged certain...

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The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us|Paperback

An exploration of the scientific limits of knowledge that challenges our deep-seated beliefs about our universe, our rationality, and ourselves. Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the...

www.barnesandnoble.com

These are ENORMOUSLY interesting questions.
But they do not undermine the way we use logic to think about the Christian faith.