FORMAL LOGIC

[Stephen3141]

Topics coming up soon, but not as quickly as asparagus cooks...

1. What deductive logic REALLY is
2. The "20 rules of Inference"
3. The First Order Quantification rules
4. The relationship between modern deductive logic, and Aristotle's "forms"
5. An evaluation of the ancient rhetorical "logical fallacies".

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[Stephen3141]

Topic: What is Formal Logic

I will take a different path, to explain what formal (deductive) logic is.

"Formal logic deals with logical causality, (sometimes called entailment).

We are very familiar with types of causality that are applications of logical causality, although few people would recognize the phrase “logical causality.”

For example:
4 / 2 yields 2 ( the mathematical notation of an operation)
if we encode this in formal logic:
1. A: “4/2” (the head of the rule)
2. B: “2” (the tail of the rule)
3. A —> B (the generic rule)

we can say in formal logic:
when A is true, then B will always be true (Modus Ponens)
when B is false, then A will always be false (Modus Tollens)" [Making Bible Study Formal, Wuest, 92-3]

Although formal logic deals with logical causality, this is the most general form of causality.
Formal logic does not explain HOW something causes something else.

The causality in formal logic is descriptive. That is, we OBSERVE configurations that, if they meet certain rigorous configurations, indicate logical causality. The configurations that MUST be met, are Modus Ponens AND Modus Tollens. These are syllogisms that are recognized from ancient times.

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

(As an aside, note that the modern scientific disciplines DISAGREE about what "causality" is.
I focus on the concept of causation, because it has been central to the philosophical discussion of logic.
Also, modern American society is caught up in conspiracy theories, and THIS indicates that many
modern Americans have no idea that real causation is.

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Oxford University Press, 2012. This is an ex-library book and may have the usual library/used-book markings inside.This book has soft covers. Clean from markings. In good all round condition. Please note the Image in this listing is a stock photo…

www.biblio.com

"Philosophers have been interested in the nature of causation for as long as there has been philosophy.
[The Oxford Handbook of Causation, 1]

"Despite the attention, there is still very little agreement on the most central question concerning causation [in the hard sciences]: what is it? Is it a matter of the instantiation of regularities or laws, or counterfactual dependence, or manipulability, to transfer of energy, for example?" [The Oxford Handbook of Causation, 1]

I assert that the modern scientific disciplines cannot agree on what "causation" is, because they are interested in using types of applied logics, which are much more specific and tailored to narrow disciplines. The hard sciences are interested in answering the "how" questions, which go much further than the formal logic question of "what" causes what.)

 

[Stephen3141]

Topic: The 20 Rules of Inference

The "20 Rules of Inference" and the "Quantification Rules" (for reasoning about groups of objects, are the core of modern deductive logic.

You should know a few things about these modern deductive logic rules...

1. Different logical notations list slightly different sets of logical rules, but the sum of what they cover, is the same.
2 The 20 Rules of Inference are a combination of axioms, and a few theorems (small proofs), that form a convenient notation to use to justify lines in a logic proof.
3 There are many "partial logics" that have been defined, that only include a subset of the 20 Rules of Inference. These are for solving specialized and narrow types of problems. Boolean logic is one of these partial systems. These partial systems are not Complete, in that they cannot express ALL the types of propositions that we would want to express, and are not guaranteed to be able to produce proofs for all the TRUE conclusions that we would want to produce.
4 The 20 Rules of Inference define how the logical operators AND, OR, and NOT work, among quite a few other operators.

I like the form of the 20 Rules that Irving Copi listed, in the endpapers of his book "Symbolic Logic, 5th edition".
(Don't freak out at the notation... I will explain what it means)

I will see if a cut and paste approach will work, with this web site. (It didn't.)
I will try to paste from a PDF file, and correct the result...

[Copi, back cover] [Making Bible Study Formal, Wuest, 378]

Modus Ponens: (MP) Modus Tollens: (MT)
p ==> q p ==> q
p ~q
∴q Therefore, not p

Hypothetical Syllogism: (HS)
p ==> q
q ==> r
∴p ==> r

Constructive Dilemma: (CD)
(p ==> q) ^ (r ==> s)
p v r
∴q v s

Simplification: (simp.)
p ^q
∴p

Disjunctive Syllogism: (DS)
p v q
~p
∴q

Destructive Dilemma: (DD)
(p ==> q) ^ (r ==> s)
~q v ~s
∴-p v ~r

Conjunction: (conj.)
p
q
∴p ^ q

Addition (add.) Rule of Conditional Proof: (CP) to prove p ==> q
p p (assume)
Therefore, p ^ anything (prove q)
Therefore p ==> q

RULES OF REPLACEMENT

DeMorgan’s Theorem: ~(p ^ q) ≡ (~p v ~q)
~(p v q) ≡ (~p ^ ~q)


Commutation: (comm.) p v q ≡ q v p
p ^ q ≡ q ^ p


Association: (assoc.) [(p v q) v r] ≡ [p v (q v r)]
[(p ^ q) ^ r] ≡ [p ^ (q ^ r)]

Distribution: (dist.) [p ^ (q v r)] ≡ [(p ^ q) v (p ^ r)]
[p v (q ^ r)] ≡ [(p v q) ^ (p v r)]


Double Negation: (DN) p ≡ ~~p

Transposition: (trans.) (p ==> q) ≡ (~q ==> ~p)

Material Implication: (MI) (p ==> q) ≡ (~p v q)

Material Equivalence: (equiv.) (p ≡ q) ≡ [(p ==>q) ^ (q ==> p)]
(p ≡ q) ≡ [(p ^ q) v (~p ^ ~q)]

Exportation: (exp.) [(p ^ q) ==> r] ≡ [p ==> (q ==> r)]

Tautology: (taut.) p ≡ p v p
p ≡ p ^ p

 

[Stephen3141]

NOTE: the notation (above) used in the 20 Rules of Inference is this...

AND ^ (both of the propositions this connects, must be TRUE)
OR v (at least 1 of the propositions this connects, must be TRUE)
NOT ~ (this means that a proposition is FALSE...)

material implication ==>
logical equivalence (the 3 stack of pancakes)

When I use these operators from now on, I will whatever is easier to write on a regular keyboard.

 

[Stephen3141]

Observations about the 20 Rules of Inference...

... Note that ALL of Aristotle's "forms" or syllogisms can be proven using this system of notation.

... Note that to be logically valid, EVERY line in the body of a proof must be shown to be TRUE, by one of these rules.
Any line in the body of a proof that CANNOT be shown to be TRUE by one of these rules, is logically invalid.

... This system of logical notation is Complete. Formally, ALL TRUE propositions expressed with this notation, can be proven to be TRUE using this notation.

... Note that some of the Rules of Inference are Replacement rules. These allow the rewriting of logical propositions into formats that are easier to work with, in specific proofs.

... 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.

 

[Stephen3141]

Thinking about the 20 Rules of Inference... (I would be curious what Christian think of these rules.)

For those who have not used the modern Rules of Inference, you should take some time and think about specific examples of each of the rules. Then, they will begin to make a lot of sense.

Note that the definition of a Material Implication, is important.
p ==> q is logically equivalent to (NOT p OR q)

This means that if we claim that p logically causes q, then we MUST be able to prove that at least one of the following is TRUE: NOT p, or q.
This rule says that logical causality may be expressed with the operands of NOT and OR.
----------

Note that not all the logical operators match what we think of as English definitions.
For example, we could prove that
1. A AND B (assume)
2. A (simplification, 1)
3. B (simplification, 1)
Therefore A OR B (addition, 2)

So, using the method of Conditional Proof...
(A AND B ==> (A OR B)

What this practically says, is that the OR condition is a type of subset of the AND condition.
Note that we cannot come up with a valid proof of
(A OR B) ==> (A AND B because A AND B is potentially a superset of A OR B.

Note that the logical OR operand, is NOT an exclusive OR!

Note that in a logical proof, we number all the lines.
Whenever a new line is produced in the proof body, then we list the rule of inference used, and the input line numbers for the operation.

Note that in this type of proof, where A and B are not given any definition, it is the FORM of the statements that are being used, and it is the FORM of the conclusion that is being proven to be logically valid. This sort of proof cannot address instances of this form, where A and B are given definitions -- that is, this sort of proof cannot address whether an instance of this proof is SOUND.

For example: to add definitions to the variables...
A: I am saved by faith (assume)
B: I am saved by what I do. (assume)
1. A AND B (assume)
2. A (simplification, 1)
3. B (simplification, 1)
Therefore A OR B (addition, 2) ("I am saved by faith OR I am saved by something I do")

From this example, you can see that MANY of the discussions we have over theology, involve differences in basic definitions in the Assumptions part of proofs. These are arguments over whether or not someone's basic definitions are SOUND.
This is why I emphasize, over and over again, going back to the original language of the Bible.

 

[tonychanyt]

Thinking about the 20 Rules of Inference... (I would be curious what Christian think of these rules.)

For an application of these rules, see "Whoever is not with me is against me" vs "Whoever is not against you is for you"

 

[OldAbramBrown]

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

Response about ... deductive logic.
---------------------------------

Set theory deal s with relation while category theory deals with ontology. We need both - absolutely - always.

 

[OldAbramBrown]

.... Deductive logic is 2-valued.

Practically, I like to preserve the knowledge that a "variable"/proposition has an unknown value, when it does.
---------------------------------

One of the two is indeterminate (a word that needs to be rtrremembered), or infinitely multivalued in itself. That's why nearly all "either-or" argy bargy is a complete non starter.

In ontology as pondered by the likes of Aristotle and Kant, there being one of something can imply there being more, and two is a more restricted case than three or more are, hence three or more is the archetypal case.

 

[OldAbramBrown]

... moral-ethical OUGHT 1

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.) ... 2

Separate points:

1 - Kant's second formulation I sum up in one word: Respect.

2 - Husserl insisted on this. Heidegger and William James departed from it. Walter Hopp is a good current proponent of Husserl.

To dumb down, and to despise the intelligence of children is to steal the bread of life from their mouths.

Likewise smug followers of religious leaders under Bismarck departed from this principle: the system that Nietzsche deplored. (Nietzsche as poet didn't understand what kind of "superman" we could have. Be cautious of commentators.) He copied a character of old who looked for an honest man in a market at noon.

The world suffers under the corner-cutting and button pushing system of Bismarck and William James (copied from Jesuit probabilists - who according to Roland Barthes were among the original mechanisers of religion and hence everything that the secular world copied from religion once sanctified).

 

[OldAbramBrown]

Stephen 3141, as a lifelong practical linguist, and of late a leisured hermeneut (doing which has improved my life no end), I insist on the syntax of meaning, which contrasts with your formulation.

 

[OldAbramBrown]

Sciences (including non-physical ones) start from how and work towards what, while mathematics starts from what and works towards how.

I'm a seer, not a doer without reason.

 

[Stephen3141]

Stephen 3141, as a lifelong practical linguist, and of late a leisured hermeneut (doing which has improved my life no end), I insist on the syntax of meaning, which contrasts with your formulation.

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.

 

[Stephen3141]

Be patient... we are coming to the end of the presentation of modern deductive logic.

Still to do:

1. Explain the difference between simple propositional logic, and quantified logic
2. Explain some how modern deductive logic relates to the rhetorical "logical fallacies"

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

Propositional Logic, and its relationship to Quantified Logic.

When we use the notation
"If A then B" this is a form of logical "pseudocode". It is logic, written in human language.
A ==> B this is unquantified propositional logic notation
for all x (Ax ==> Bx) this is the same statement, using first order Quantified logic notation.

Note that in all these notations, the letters A and B can stand for any properly formed logical proposition.
Note that whenever anyone uses the Quantification "for all" or "there exists", then they are using quantified logic.

First Order Quantified logic is normally used in formal logic.
"Quantification" does not refer to numeric quantification, but to a type of group relationship.

"Quantification" deals with the possible relationship between groups, that Venn diagrams can show (for example).

In ancient logic, there are the standard "quantifications" A, E, I, and O. (From affirms, and nego.)
Example: if we're talking about dogs and tails, then this is what the quantifications mean...

A: All dogs have tails
E: no dogs have tails
I: some dogs have tails
O: some dogs have tails

In propositional logic, ALL statements are assumed to have global force.
Example:
D: the individual is a dog
T: the individual has a tail

So in propositional logic we can express
D ==> T "all dogs have tails"
D ==> NOT T "all dogs do not have tails"

Propositional logic is VERY limited in what it can express.

Which is why we standardly use first order Quantified logic. This notation breaks apart the abstract "individual" from the characteristic(s) that it may have. So we can express...

D: is a dog
T: has a tail

for all x (Dx ==> Tx) "all individuals that are a dog, have tails" or "all dogs have tails"
for all x (Dx ==> NOT Tx) "no dogs have tails"
there exists x (Dx AND Tx). "some dogs have tails" "some" means at least one
there exists x (Dx AND NOT Tx) "some dogs do not have tails"

--------------------
Note that in formal logic, an "individual" is not a physical object, or a conceptual object.
An individual is a unique placeholder, that can have any number of characteristics applied to it.
So, the "individual" is itself a unique ID, whether or not it is given a label ID.

 

[Stephen3141]

The Quantification Rules: (sorry about the formatting of this section...)

The (first-order) quantification rules constrain how we may add a quantification to a logical expression, or take away a quantification from a logical expression.

First-order refers to quantifying the individuals that have a certain set of characteristics.

Second-order quantification refers to expressions that quantify BOTH the individuals that have some characteristic, AND the characteristics that the individuals can have.

Generalization refers to adding a quantification to an expression.

Instantiation refers to removing a quantification from an expression.



There are 2 types of Quantification:

1. Universal: this means that ALL individuals are being referred to

2. Existential: this means that “some” individuals (at least one) are being referred to.



So using the dog and tail example…

D: characteristic of “is a dog”

T: characteristics of “has a tail”




Example Expression:

D ==> T. In propositional logic, this asserts that “all dogs have tails”

Universal Generalization: UG


For all x (Dx ==> Tx). This applies the universal quantification to the expression

“all dogs have tails”

Existential Generalization: EG

There exist x (Dx ==> Tx) “some dogs have tails”

This applies the existential quantification to the expression

Universal Instantiation: UI

This operation takes a universally quantified statement, and extracts from it an expression that deals with a specific individual.

So, taking a universally quantified expression…

For all x (Dx ==> Tx). This applies the universal quantification to the expression

“all dogs have tails”


We can extract a statement about “Woofy”, a specific dog, symbolized by “w”

Dx ==> Tw “Woofy has a tail”

We can assert this, because there is the (above) rule

“all dogs have tails” so we can reason

“Woofy has a tail”


Now the more difficult quantification…

Existential Instantiation: EI


Given a existentially quantified statement…

There exist x (Dx ==> Tx) “some dogs have tails”


We can remove the quantification…

Therefore we can say “some dog has a tail” but WE CANNOT SAY WHICH ONE.

We can give the individual a new label, such as “a”, but the individual remains

“without loss of generality”.

Therefore, Da AND Ta

Note that the label for this individual, CANNOT be the same as any individual that has appeared previously in the proof.

When we do a quantification operation in a proof, we have to list what the input expression line is, and what operation (UG, UI, EG, EI) we are doing.

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

So, with the 20 Rules of Inference, and the Quantification Rules,

We have enough to do modern deductive logic.

 

[Stephen3141]

Some of the topics that we should discuss…

1 How modern deductive logic, is related to the ancient “logical fallacies"
2 Common errors in logical operations
3 Common errors in definitions
4 Common errors in social media arguments.

I would be curious what topics you-all would like to discuss, related to modern deductive logic.
(That includes questions about the 20 rules of inference, and quantification rules.)

 

[Stephen3141]

Test Questions Group 1:

1. “Formal Logic” typically deals with deductive logic.
2. Deductive logic uses propositions that evaluate to TRUE or FALSE.
3. Deductive logic is related to “normalized” logics, that have systems of weights.
4. “Completeness” is a characteristic that means that a logical system could potentially express a proof for all TRUE propositions.
5. All logical systems are complete.

6. All logical systems can express all the ideas that are important to Christianity.
7. Modern Mathematics can be expressed using deductive logic building blocks.
8. Modern mathematics can express moral-ethical concepts.
9. Mathematics uses numerical quantification.
10. Deductive logic has Quantifications that are characteristics of group membership.

Answers:

1. TRUE
2 TRUE
3. TRUE
4. TRUE
5. FALSE

6. FALSE
7. TRUE
8. FALSE
9. TRUE
10. TRUE

 

[Stephen3141]

Posting some more test questions about deductive logic.

Test Questions group 2:

1. “Logic” can be presented in apologetics, as a component of “our shared reality”.
2. We are ALL morally-ethically responsible for the components that are in our shared reality.
3. About 20 rules of inference describe most normal deductive operations.
4. The Quantification Rules in deductive logic govern proper reasoning about groups.
5. The goodness of a logical proof centers around logical Validity, and logical Soundness.

6. Logical Validity deals with the Syntax of logical statements in a proof.
7. Logical Soundness deals with whether or not the definitions and rules in the Assumptions part of a proof, are TRUE.
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.
10. A basic source of disagreement among Christians, is that they are using different definitions of basic concepts and rules in the Assumptions part of a proof.

Answers:

1. TRUE
2. TRUE
3. TRUE
4. TRUE
5. TRUE

6. TRUE
7. TRUE
8. TRUE
9. TRUE
10. TRUE

 

[Stephen3141]

Test Questions Group 3:

1. Propositional logic does not use the “for all” or “there exit some” quantifications.
2. First Order Quantified logic uses the “for all” and “there exists” quantifications.
3. Venn diagrams can be used to illustrate group reasoning in logic.
4. First Order quantified logic “quantifies” individuals.
5. First Order quantified logic is Complete

6. Generalization adds a quantification to a proposition.
7. Instantiation removes a quantification from a proposition.
8. Universal quantification describes a characteristic that applies to ALL members of a group.
9. Existential quantification describes a characteristic that applies to at least one member of a group.
10. There are more restriction on Existential quantification, than on Universal quantification.

Answers:

1. TRUE
2. TRUE
3. TRUE
4. TRUE
5. TRUE

6. TRUE
7. TRUE
8. TRUE
9. TRUE
10. TRUE

 

[Stephen3141]

Posting some more test questions:

Test Question Group 4:

So, you really think that you’ve learned something from the posts, so far, Dood?

Here are some tougher questions.

1. And “individual” in formal logic deals with a human being.
2. An “individual” in formal logic, is a unique ID that is a placeholder for a group of characteristics.
3. All things in formal logic are expressed by applying characteristics to an “individual”
4. A valid and sound logic proof, is also called a theorem.
5. A logical proof has 3 parts: Assumptions, the Body, and the Conclusion.

6. Modern logical fallacies are not the same as the “logical fallacies” of ancient rhetoric.
7. Modern logic can represent all the ancient “forms” that Aristotle used.
8. The lack of critical Bible study skills, results in many proofs that are logically Unsound.
9. A Christian who is biblically illiterate, will produce a lot of Unsound proofs (from a Christian viewpoint).
10. Many common arguments on social media, have serious logical errors.

Answers:

1. FALSE
2. TRUE
3. TRUE
4. TRUE
5. TRUE

6. TRUE
7. TRUE
8. TRUE
9. TRUE
10. TRUE