FORMAL LOGIC

[OldAbramBrown]

Typically in language, SYNTAX and SEMANTICS are different things.
In modern formal logic, logical validity is determined by syntax (the 20 rules of Inference...).
But the soundness of a proof involves the semantic definitions of the Assumption.

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.

 

[OldAbramBrown]

Post 38 points 8 and 9:

8. Human language can be converted into logic notation.
9. A common source of error, is people do not correctly convert human language into logical notation.

The reason point 9 is true is because insufficient endeavour has been put into this.
Was it earlier in this thread, or a different one I was commenting that a dual choice often means between "indeterminate a" and "indeterminate b" (which may considerably overlap anyway).
Often, God says what He is saying by not saying what He is not saying.
How much background knowledge do we need to infer what was going on at - say Sodom just to pick a humdrum NON hot button instance.
Are some of the oblique clues within NT and OT as well as outside of either of those?
Scrolls with old narratives, or letters that were going to be read by strangers en route, are necessarily brief.

 

[OldAbramBrown]

I would like to see the issues in your post 15, analysed as we go along, when they crop up.

My secondary philosophical sources seem to imply that those who had tried to equate maths and logic didn't quite get there.

Nonetheless maths has to be at any moment either relational, or ontological, and when methods lend to interchanging between modes, it's imperative to keep track of the reality of what we are doing and why, in order than we can "put it back nicely" to double check the sense (which some mathematicians admit they don't want to do, and don't want students to do).

True logic has that in common with true maths.

In regard to English usage I love to continually paraphrase to keep readers checking. If the thread is about doctrine (a k a apologetism or evangelism), to what extent are our heard or read sources in religion like this?

 

[Stephen3141]

It is good that SOMEONE has followed the posts this far.
I am reading the responses.

Don't get discouraged at how the logical notation LOOKS. After you think about it, and use it a bit, it will seem quite understandable. don't give up.

For those who have never seen this sort of logical notation before, you haven't seen modern symbolic logic.
I say this, because many Christians who think that they are very "logical" thinkers, and who think that they use reason all the time, are often shocked to realize that they never really understood what modern formal logic is.

--------------------
Some responses to what people have posted, so far...

Response to "human language and logic aren't the same thing"
No, they aren't.
But, human language can be converted into formal logic statements.

Response:
I do not "equate" formal logic with mathematics.
More precisely, I reference the work of Russell and Whitehead, who demonstrated that it is possible to build modern mathematics out of formal logic building blocks.
This is quite different than "equating" formal logic with the applied logics (mathematics, chemistry, physics, electrical engineering, ...).

Response: to basically "it's impossible to convert human language into logical statements"
It depends on what is meant, by this assertion.
It is easy to assign some phrase in a human language, to a logical symbol.
Even "complex" human language phrases can be assigned to a logical notation symbol.
The limitation of modern formal logic, is that it can assert definitions (not a limitation), but that when asserting Rules (A ==> B), then the relationship encoded must be logical causality (entailment). And many discussions in human language, do not encode a simplistic relationship of entailment.
However, when theologians make assertions that take the form of logical entailments, then their assertions are fair game to be analyzed by formal logic.

Response to: " a major source of error in formal logic, is the wrong conversion of human language into logical notation".
This is an observation that multiple teachers of formal logic, have noticed in their classes.

Perhaps I should clarify about Syntax and Semantics.
In logical notation, syntax and semantics are different things.
The logical validity of a proof depends on the syntax of the logical notation.
The logical soundness of a proof, depends on the accuracy of the semantic definitions in the Assumptions part of the proof.
--------------------

 

[Stephen3141]

On the assertion that the ancient rhetorical “logical fallacies” are not quite the same as the logical rallies of modern logic.


Ancient rhetoric was a combination of persuasive speech, and logic. Unfortunately, what the ancients came to identify as “logical fallacies” sometimes did not meet the modern criteria to be logical fallacies.


This disconnect becomes quite obvious, when someone who takes the old Aristotelian argument forms to be “logic”, and takes the ancient “logical fallacies” to be the same in modern formal logic.


Some of the ancient “logical fallacies” survive in modern logic, but some show that they are more like labels of supposed errors, without a definition of how to decide whether or not the error should be applied to a specific argument. (Appeal to Authority, is one of these.)

--------------------
Example: Appeal to Authority


There is no builtin “decision algorithm” that specifies when Appeal to Authority actually applies to an argument. By decision algorithm I mean a precise list of operation to go through, that will clearly indicate whether or not some argument suffers from this “logical fallacy”.

Different Christian denominations, accept different “authorities” on various subjects. And so, they apply the label of “Appeal to Authority” fallacy to each others arguments. But they are often using different accepted authorities.

“Appeal to Authority” fallacies fall under the modern topic of whether or not the assertions and rules in the Assumptions part of a proof, are accurate. If any of these definitions is inaccurate, then the resulting argument/proof is Unsound.

Probably 60% of the differences in Christian theologies, as they appear in American Christian apologetics sites, can be traced back to different definitions in the Assumptions part of a proof.


I standardly appeal to the authority of the (BDAG) Greek-English Lexicon of the New Testament… with the most recent editor of Danker. And to the Greek New Testament that is put out by the (UBS) United Bible Society, 3rd edition.


I standardly DO NOT appeal to denominational commentaries, or denominational biblical language lexicons.

--------------------
Example: Affirming the Consequent


This is an error in logical syntax. The argument involved is
A ==> B
B
Therefore A


Note that there is no Rule of Inference in modern logic, that affirms that this argument is valid.
Note that in modern vocabulary, this is called “abductive reasoning”.
Although this sort of reasoning is useful to look around for possible causes of B, it falls short of the deductive certainty required by deductive reasoning.


--------------------
Example: Red Herring


A red herring is a distraction that is brought up in rhetoric. It distracts from the truth that there is no connection between the Assumptions in the proof, and the Conclusion. Or that the supposed connection is through a fact/rule that is IRRELEVANT to the proof.

Relevance, is VERY important, in a proof.
Using irrelevant statements in a proof, results in an Unsound proof.

But, detecting which statements in the Assumptions part of a proof are irrelevant, from a Christian point of view, becomes (sometimes) a tricky process.

Those who are biblically illiterate, probably CANNOT produce the proper statements in the Assumptions part of a proof, to represent what is RELEVANT about biblical knowledge, to reaching the desired Conclusion.

 

[Stephen3141]

Observation: I see from the lack of responses, that most readers are not interested in formal logic.

My guess is that those from a Protestant Fundamentalist background, have never read the basic definitions of modern deductive logic. And so, they will look at the 20 rules of inference, as if it were a foreign language. Their standard question would be something like "Why do I need all this, to reason?"

Another problem is emotional. Those who were raised reading commentaries about the ancient rhetorical "logical fallacies", may have a difficult time realizing that these are often not logical fallacies, but rather, broad categories of possible reasoning errors, instead of well-defined logical fallacies.

I would say to those from an anti-intellectual Christian background, that this background does not prepare one to interface with the modern hard sciences. Without this interface, it may seem that the modern sciences are incompatible with Christian reasoning.

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

I would include a general warning to those who think that the nature of reasoning, is that the Holy Spirit will mystically reveal truth to us.

This basic anti-intellectual approach to living, goes against the picture of knowledge and wisdom that we have in the Old Testament. And, how it is necessary for us to work diligently to pursue knowledge and understanding. And, that if we do not put in the hard time of study, we will not make progress toward knowledge and understanding.

Those who are anti-intellectual, also undercut the tolls and methodology of critical Bible study, and critical Bible study tools. Ironically, those who take the approach that the meaning of the Bible is self-evident, seriously cripple their ability to understand the Bible.

I would repeat this warning to the younger American generations, who have substituted emotions, for logic. One can train one's emotions to feel "good" about almost anything -- regardless of whether that belief is true.

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

If I continue to put entries into these thread, I will demonstrate the use of formal logic principles on assertions that are relevant to the Christian faith. This includes analyzing modern arguments, for their logical validity and logical soundness.

(The complication of analyzing modern arguments that you will see on social media, is that many Americans cannot unwrap their emotions from their thinking. And they confuse an argument being politically incorrect, with an argument being invalid or unsound. This makes the logical analysis of arguments, a risky thing to do, on Christian web sites. Many sites will not tolerate the logical analysis of arguments....)

 

[2PhiloVoid]

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

You're right to say that a lot of folks, even Christians, don't refer to or rely upon "formal logic." Probably this is due to the fact that this mode of rational evaluation isn't part and parcel of the contents of the Bible.

Add to this the cultural expections of our modern world to assume a diversified and incomplete set of evidences rather than a unified and comprehensive one along with an accompanying lack of real and agreed upon axioms by which to define our variables and/or our various praxis, we end up with a smorgasboard of interests among people of all kinds that sways folks away from seeing much practical use in applying "formal logic."

If the above is the case of the typical individual who lives in any one Western nation, religion (including Christianity) will not be handled and assessed by way of Formal Logic(s), although we can find a number of people, both religious and anti-religious, who make claims as to be doing so. Personally, I don't think they can succeed in doing so for either ethos (i.e. for either a Christian one or a Non-Christian/Anti-Christian one).

At best, I think all we can do is give reasonable consideration to epistemological comments by Pascal, Kierkegaard and Wittgenstein's later philosophical evaluations about the place of "language games" to help in our understanding of various underlying hermeneutical concerns which come antecedent to the clear and discernible application of formal logic, especially where something like Christianity is concerned. I would also downplay the potency of the use of Symbolic Logic, such as developed by Whitehead and Russell, in attempts made by one person or another to makes sense out of the "thing" we call the Holy Bible.

 

[public hermit]

You're right to say that a lot of folks, even Christians, don't refer to or rely upon "formal logic." Probably this is due to the fact that this mode of rational evaluation isn't part and parcel of the contents of the Bible.

Add to this the cultural expections of our modern world to assume a diversified and incomplete set of evidences rather than a unified and comprehensive one along with an accompanying lack of real and agreed upon axioms by which to define our variables and/or our various praxis, we end up with a smorgasboard of interests among people of all kinds that sways folks away from seeing much practical use in applying "formal logic."

If the above is the case of the typical individual who lives in any one Western nation, religion (including Christianity) will not be handled and assessed by way of Formal Logic(s), although we can find a number of people, both religious and anti-religious, who make claims as to be doing so. Personally, I don't think they can succeed in doing so for either ethos (i.e. for either a Christian one or a Non-Christian/Anti-Christian one).

At best, I think all we can do is give reasonable consideration to epistemological comments by Pascal, Kierkegaard and Wittgenstein's later philosophical evaluations about the place of "language games" to help in our understanding of various underlying hermeneutical concerns which come antecedent to the clear and discernible application of formal logic, especially where something like Christianity is concerned. I would also downplay the potency of the use of Symbolic Logic, such as developed by Whitehead and Russell, in attempts made by one person or another to makes sense out of the "thing" we call the Holy Bible.

As I was being thoroughly indoctrinated in analytic philosophy, it dawned on me that it was too easy. I would come into class, and it didn't matter what we read, Dr. Ray VanArragon would outline it in a few premises and a conclusion. Then, we went to work. But it didn't matter because there were no definitive answers!

It was only after university that I began to read existentialists (and others) and see how flawed the analytical approach is if we assume it's a method to obtain certainty. If the goal is clarity, that makes sense. But there is no certainty; there is only epistemic humility.

 

[Presbyterian Continuist]

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

The work of Francis Schaeffer shows an excellent combination of logic and Christian faith, where it is possible to discover that God is there and He is not silent through the process of logic.

 

[Presbyterian Continuist]

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.)

Some secular scientists have come to the conclusion that our universe is finite, It is four dimensional. It has length, width, height, and time. What this means is that there was a time when the universe began, and a time when it will end. Therefore the reality we live in is four dimensional. It is our space-time reality. What the scientists have concluded is that if our space-time reality is finite, then there must be a great reality outside of our finite universe.

If we accept that our universe was designed and created, then the creator must be an entity outside of our space-time dimension. Because time is created, then whoever is in the reality outside of our universe has created time. This means that the creator exists in a multidimensional reality outside our four dimensional reality.

What this also means is that the creator, in his multi dimensional reality, can create a cosmos in what he perceives is an instant, but for us in our reality could take billions of years. This is not evolution, but designed construction totally controlled by the creator, and not uncontrolled chance.

This view of creation comes about by using logic. Also, logic tells us that because of the intricate design of the universe and the different components of nature, that these things must have been designed and constructed by an intelligence that is infinitely greater and more powerful than our own.

If we decide to call this intelligence God, then this God must be infinitely bigger and more powerful than what we could ever imagine.

 

[PloverWing]

Observation: I see from the lack of responses, that most readers are not interested in formal logic.

It's not necessarily a lack of interest. So far, you've been giving a overview/review of formal logic. That's good for readers whose logic background is rusty or incomplete, but it doesn't really call for the reader to post anything in response.

If I continue to put entries into these thread, I will demonstrate the use of formal logic principles on assertions that are relevant to the Christian faith. This includes analyzing modern arguments, for their logical validity and logical soundness.

I think this will give us more grounds for discussion.

 

[Helmut-WK]

What the scientists have concluded is that if our space-time reality is finite, then there must be a great reality outside of our finite universe.

Unsound.The surface of the earth is finite, but this does not mean there is a limit where the surface ends, and beyond that limit there is an »outside« surface.

When scientist think of a finite universe, they don't think of a limit which seperates inside from outside. For modern science, there is no outside - well, there may be a sort of »outside« in a higher dimension, but this question is independent from the question whether our universe is finite or not.
Mathematics is sometimes counterintuitive, e.g. »curved space« does not imply an additional dimension it is curved in. The 2-dimensional surfave o the earth is curved in a higher dimension (3-dimensional space), But for the universe, general relativity does not assume a higher dimension in which the finite (and therefore »positively curved) 4-dimensionaöl spacetime is curved in.

If we accept that our universe was designed and created, then the creator must be an entity outside of our space-time dimension.

This is no reasoning from science. But given Christian assumptions, it is a valid conclusion.

This view of creation comes about by using logic.

Yes the section between the two last quotes was logical valid (or did I overlook anything).

Also, logic tells us that because of the intricate design of the universe and the different components of nature, that these things must have been designed and constructed by an intelligence that is infinitely greater and more powerful than our own.

Logic tells? Given the proper assumption you arrive on that conclusion, but to a person with different assumptions (a non-christian belief system, e.g. atheism) there may be no logical way to arrive there.

As far as I understand, the topic you stress is off-topic here.

 

[zippy2006]

It's not necessarily a lack of interest. So far, you've been giving a overview/review of formal logic. That's good for readers whose logic background is rusty or incomplete, but it doesn't really call for the reader to post anything in response.

On the other hand, the OP seems to have very little idea of what he is talking about. For example:

Example: Affirming the Consequent


This is an error in logical syntax. The argument involved is
A ==> B
B
Therefore A


Note that there is no Rule of Inference in modern logic, that affirms that this argument is valid.
Note that in modern vocabulary, this is called “abductive reasoning”.
Although this sort of reasoning is useful to look around for possible causes of B, it falls short of the deductive certainty required by deductive reasoning.

 

[Stephen3141]

On the other hand, the OP seems to have very little idea of what he is talking about. For example:

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.

 

[Stephen3141]

OK... It seems that those reading the post have no problems with the basic Rules of Inference, and the Quantification rules.

So, I will start to give some examples of arguments.
For these arguments, I will try to write the arguments in the form of:

1. Human speech (or what I would call logical pseudocode)
2. Logical notation

And, I will bring up observations or criticisms of the arguments, that Christians should be working though (when they see the argument).
I may also include a few passages from Scripture, as a background for the arguments.

***** NOTE: Many of these arguments are going to be DEFECTIVE. It is much more challenging to analyze defective arguments, than to list a whole lot of valid and sound arguments. Please don't get offended at this approach.

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

Example: A problem with proof-texting.

21 But now the righteousness of God has been manifested apart from the law, although the Law and the Prophets bear witness to it— 22 the righteousness of God through faith in Jesus Christ for all who believe. For there is no distinction: 23 for all have sinned and fall short of the glory of God, 24 and are justified by his grace as a gift, through the redemption that is in Christ Jesus, 25 whom God put forward as a propitiation by his blood, to be received by faith.
The Holy Bible: English Standard Version (Wheaton, IL: Crossway Bibles, 2016), Ro 3:21–25.

Especially about verse 23, you may have heard "all means all!"

If we try to build a theological truth about WHO has sinned from verse 23 alone, then we could argue...

1. All human beings have sinned.
2. The man Christ was a human being.
3. Therefore, the man Christ sinned.

Logical Notation...

(Assumptions and Definitions)
1. S: have sinned. definition of what S means
2. H: is a human being. definition of what H means
3. c: is the man Christ. definition of a label that stands for "the man christ"
4. for all x (Hx ==> Sx). definition of a rule that means "all human beings have sinned"
5. Hc assertion that the man Christ is human

To prove: Sc

6. Hc ==> Sc (line 4, Universal Instantiation)
7. Therefore Sc (line 5, line 6, Modus Ponens)
QED

Comments on this argument:

a. Obviously, the logically valid conclusion that the man Christ sinned, is not an orthodox Christian conclusion.
Even though each line in the body of the proof refers to a valid Rule of Inference.

b. So, what is wrong with this proof?

Analysis:

-- The basic problem with this proof, is that the rule in the Assumptions part of the proof, is not complete.
There is an exception to the general rule that is given in verse 23.
We see this exception expressed in a different text of Scripture.

-- because the rule describing which human beings sinned is not complete, it is not Sound.
And so, the proof is Unsound.

14 Since then we have a great high priest who has passed through the heavens, Jesus, the Son of God, let us hold fast our confession. 15 For we do not have a high priest who is unable to sympathize with our weaknesses, but one who in every respect has been tempted as we are, yet without sin.
The Holy Bible: English Standard Version (Wheaton, IL: Crossway Bibles, 2016), Heb 4:14–15.

So, the rule "all human beings have sinned" is not complete. It needs to have an exception coded into it.
You can encode an exception into a rule, by excluding the application of the rule to all the entities with specific IDs.
You can use the syntax "x is not c" or "x !=c" (in a form that computer programmers would recognize to get the rule
"All human beings who are not Christ, have sinned"
for all (x) (Hx AND x != c ==> Sx)

This puts an exclusion guard on the rule, and does not allow it to "fire" when the identity of the object is Christ.

Using this Christian form of the rule [for all (x) (Hx AND x != c ==> Sx)]
there is no way in which you can reach the logical conclusion Sc, using the original Assumptions and this corrected rule.

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

Observation: (don't go ballistic...)

A common problem with proof texting, is that not all RELEVANT passages of Scripture are included in an argument/proof.
This leads to Conclusions, that are Unsound, from a Christian point of view.

Observation:

In this instance, a biblically illiterate Christian would probably produce a logically unsound proof,
about whether or not the man Christ sinned.

There is no inherent "entitlement" that biblically illiterate Christians will come to the correct conclusions, when they base their conclusions on a single verse in the Bible.

Logically Unsound rules or assertions are a major source of dysfunctional Conclusions, with regard to the Christian faith.

 

[Helmut-WK]

A common problem with proof texting, is that not all RELEVANT passages of Scripture are included in an argument/proof.
This leads to Conclusions, that are Unsound, from a Christian point of view.

This is very common in arguments against the Christian faith. But sometimes in arguments from Christians also (I feel any example would start a theological debate, so I do not give one).

 

[Stephen3141]

This is very common in arguments against the Christian faith. But sometimes in arguments from Christians also (I feel any example would start a theological debate, so I do not give one).

This is a really good observation.

Especially in modern America, it seems that memorized slogans are thrown back and forth, that are not really arguments.
And I start to become exhausted, thinking that NO ONE cares about what it true.

But... historically, Christians have promoted discussing what is true, and why.
We should not fall into the American cultural trap that there is no truth, or that we cannot carefully discuss what is true.

The advantage of using formal logic, is that the form of an argument (by this I mean a polite discussion) can be objectively analyzed.
And the advantage of using scholarly reference tools to look at the original language of the New Testament, is that amateur opinions are set aside.

As I pointed out in the introductory posts, America has ditched formal logic, so many people do not recognize what "logic" is.
But, formal logic gives us a structure within which to hold polite discussions ("arguments") about all sorts of topics.
It is a service to people, and Christians, to point out when they are using logically invalid arguments.
It is a service to Christians, to point out when they are applying dysfunctional methods, to interpret the Scriptures.
*** It may be a bit uncomfortable, for some Christians to point out dysfunctional thinking. But, if you can survive the discomfort, you can engage in really helpful intellectual growth in other Christians.

I understand what you are saying.
But the much more formal discussion of all sorts of topics, is desperately needed in modern America.
We cannot let the mindless conspiracy theories present abusive versions of the Christian faith.

 

[Helmut-WK]

I understand what you are saying.
But the much more formal discussion of all sorts of topics, is desperately needed in modern America.
We cannot let the mindless conspiracy theories present abusive versions of the Christian faith.

What I wrote was not meant as an criticism of you. I agree to what you say above. The situation in Europe is not that far from America (AFAIK, but I can't really assess the amount of simliarity and difference between EUR ans USA …).

 

[Stephen3141]

Argument Example: Disjunctive Syllogism

In formal logic, a series of statements (an argument) that has Assumptions part, a Body, and a Conclusion is called a "proof".
If the proof is verified as being logically valid, and logically sound, then it is called a theorem, or a syllogism.

Some syllogisms are included in the Rules of Inference, because they are so useful.
Think of Modus Ponens, Modus Tollens, and Disjunctive Syllogism, for example.

The "forms" of Aristotle are some useful syllogisms, for example.

But there are common abuses of proven syllogisms.

--------------------
Example: an abuse of disjunctive syllogism

Definitions and Assumptions:
1. A: you agree with me (assertion)
2. N: you are a Neo-Nazi. (assertion)
3. rule: either you agree with me, or you are a Neo-Nazi. (asserted rule)

4. You do not agree with me. (fact)
5. Therefore, you are a Neo-Nazi. (line 4, line 3, Disjunctive Syllogism)

You could write this in logical notation as...
1. A: you agree with me (assertion)
2. N: you are a Neo-Nazi. (assertion)
3. rule: for all (x) (Ax OR Nx). (asserted rule)

4. NOT (Ay) (fact, in a general form that y disagrees with me)
5. Therefore, Ny (line 4, line 3, Disjunctive Syllogism)

Note that this is a logically valid argument.
IF we accept the Assumptions as true, then the Conclusion logically follows.

But, how on earth are we supposed to demonstrate that the Assumptions are Sound?

Problems with this Argument:

a. what I believe, is undefined
b. what you believe, that differs with my beliefs, is undefined
c. the rule is BADLY dysfunctional. A rule with logical ORs must list options that cover the entire possibility space. And those options listed, must be able to be determined precisely. This rule, does not define precise options, and the options given do not define exclusive logical cases. Someone may disagree with me, and NOT be a Neo-Nazi.

Observation: This is an example of Disjunctive Syllogism that is Unsound.

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Comments: (put your seatbelts on, and don't get offended...)

1. Note that Disjunctive Syllogism is the form of the Sherlock Holmes' argument, that, if you have eliminated all other options, the remaining one, no matter how improbable, must be TRUE. Unfortunately, this is a POOR argument. Because there are thousands of logical cases that are so improbable, that we would not consider them. And so, we do not rigorously work to exclude them.

2 A common form of this argument is...

Either you agree with the theology of my Christian group, OR you are a heretic.
You disagree with the theology of my group
Therefore, you are a heretic.

There are thousands of possible theologies. A "theology" may include hundreds of positions, on hundreds of topics.
Arbitrarily picking out a few positions on a few topics, and saying that they represent the orthodoxy (or NOT) of an individual, is often an arbitrary operation. If someone accepts a short and historical list of Christian beliefs (such as the Nicene Creed), then this goes a long way to demonstrate that their "theology" is not an arbitrary collection of beliefs.

3. Note that if the options in a logical OR statement are nor relevant to a Conclusion, then the proof is Unsound.
Note that if the options in a logical OR statement are not complete, then the argument becomes Unsound.
Note that if the options in a logical OR statement are not accurate (in some way, they do not reflect our shared reality), then the argument becomes Unsound.
For Christians, the options in an OR statement must be consistent with biblical teaching, and the concept of our shared reality that the Bible presents, else the argument is Unsound.

Note that observation (3) covers all sorts of dysfunctions that exist in modern conspiracy theories.

 

[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.