Paradoxes of Material Implication

[Speakpigeon]

If someone is able to come up with some nasty unwanted results - then back to the drawing board.

Well, bad news, as far as I can tell.
First, I don't get the same results at all as you do for ¬(p → q) → p, though it also comes out not valid.
Second, the Modus Ponens comes out not valid as well. If confirmed, that would be terminal.
Here are the details. It's all done automatically but I checked each column of the two tables against your definition for conjunction, negation and implication.
I'll let you sort out whether my result is wrong or whether you need to go back to the drawing board.
Tell me if you find any mistake in my two tables.
You could reasonably define validity as no false value rather than all true values as you seem to do.
That would be closer to the original definition, keep paradox 4 not valid and also rescue the Modus Ponens...
EB

[Garry G]

If someone is able to come up with some nasty unwanted results - then back to the drawing board.

Well, bad news, as far as I can tell.
First, I don't get the same results at all as you do for ¬(p → q) → p, though it also comes out not valid.
Second, the Modus Ponens comes out not valid as well. If confirmed, that would be terminal.
Here are the details. It's all done automatically but I checked each column of the two tables against your definition for conjunction, negation and implication.
I'll let you sort out whether my result is wrong or whether you need to go back to the drawing board.
Tell me if you find any mistake in my two tables.
You could reasonably define validity as no false value rather than all true values as you seem to do.
That would be closer to the original definition, keep paradox 4 not valid and also rescue the Modus Ponens...
EB

First - thank you for having the curiosity and taking the time. It is the kind of response I was hoping for. I have just scanned your tables and they look correct to me. I made mistake on my table. Part of the problem of doing them by hand.

I need to think carefully about what counts as valid and the role of formulae that are never false but have an N or a C in their truth function.

The question of validity is an interesting one for this system. There are three truth values. The table you produced for A ⋀ (A ⇒ B) ⇒ B is one of those truth functions that is never false on any permutation of truth possibilities. This is an interesting status. If the inference is true on every possibility it is reliable and we could say the inference is validated, but I think that is going to get confusing. So for argument sake let us call inferences that are never false "reliable".

The reason the values C and N don't reach the threshold of theorem - I figured - is that if you negate a theorem there is a contradiction. But if you negate the truth function TCNT the result is FNCF. Two permutations are still true. It is not a contradiction. So that is my reasoning why N and C are not sufficient to prove a theorem.

If N,C are accepted then paradox 1 (explosion) becomes a theorem. Keeping T as the only designated value for theorems keeps paradox 1 out. But it also treads a line where the system cannot be said to be exactly paraconsistent because the paradox is never false. I was thing this was quite a neat solution. I regard 1 as more of a puzzle than a bad example of reasoning. If we admit inconsistency into our premises then it is not absurd to think anything might follow. On the other hand it is intolerable that a contradiction proves anything.

All reliable inferences are gong to have the same status as 1. Modus bones for instance.

Another option is to change how False behave under inference. i.e. F->T = T, F => N = T, F => C = T, and F +> F = T. In other words make it perform as it does under standard material implication. This is about the only leeway there is to adjust the implication table. This version validates modus ponens as T on every permutation (not so sure now I made a mistake on the paradox 4 table but I'm sure that is still true). But if we go that route then paradox 4 is never false. I look at this and veered towards the table as presented in the OP because of paradox 4. I think that is clearly a faulty example of how to reason and I wanted an implication that wiped the board of paradox by ensuring there is an F in every truth function (with the exception of paradox 1 which is reliable).

a lot depends on how we define tautology. Is tautology true on every possibility (i.e. T or N or C) or is it the negation of contradiction and thus just T on every possibility. It can be one or the other in this system but it cannot be both. I think these are interesting questions and I tend to see them as real nuances standard logic passes over pretty much unaware these kinds of questions could ever exist because it does not have the apparatus. I tend to think tautology is best defined as the negation of contradiction.

Presently my preference is to allow two classes of formula. Theorems proper are T on every possibility. The laws of thought for example, non contradiction excluded middle, principle of identity and double negation are all T. so maybe all formula T on every possibility should be regarded as laws of thought while there are also reliable formula - those T, N,C and never false on every possibility.

Advice and well meant criticism is always welcome.

[Scott Mayers]

Apologies for the slow reply. I have now caught up with the post I think your are referring to. Been a bit busy today.

That's cool.

As to my own experience on the matter, 'tautologous' expressions are those that have all truth values for each possible assignment; 'inconsistent' if it takes on all false values for each possible assignment; 'contingent' if at least one value of the formula is true AND one is false for all assignments.

If "consistent", the formula is EITHER tautologous OR contingent.
If "non-tautologous", the formula is either contingent or inconsistent.

[From pg 68 of "Beginning Logic" by E.J. Lemmon, 1978 paperback edition.]

Terminology may be getting in the way but I don't disagree your basic point. What you call contingent is not what I am calling contingently-true. We talk at cross purposes. When I say contingently-true I mean any statement that is true but the truth is accidental, or coincidental or temporary and so on. I could have used another phrase like Extraneous. (A minor motivation of this thread is to see what terminology is helpful and what is confusing).

I am happy to adjust terminally wherever it is helpful. I am not wedded to the term contingent. However my analysis of the values {11, 10, 01, 00} mean there must be two opposing versions of truth. I could have neglected to offer further explanation and just called the three values True 1, True 2 and True 1+2. It just so happens there are pretty of adjectives that fit the bill and these are aptly labelled contingent.

The relation of the etymology of "contingent" is 'con-'(with) + 'tangent' (touch), and I believe the past logicians chose it in light of something that is 'touchable' is sometimes and untouchable at other times. So it fits with how you are using it colloquially in the same way. Something that doesn't "have" be be true is one that is either one exclusively but minimally something that has been touched (ie sensed) at least once AND at least untouched at least once. Otherwise they'd be essentially always true or always false.

I studied both the traditional and computational logic (with still ongoing new stuff all the time.)The color venn diagram suffices to treat each place value in a binary-valued system alone without complications. Just assign them as each position with variables in some universal set. That is how computational logic sets up "Karnaugh maps" as an extension to include any number of variables in a diagramming format. [Try setting up normal venn diagrams for say six variables, for instance.] It also uses jargon used to describe any number of variables. The logic only requires four definitions and is extended from the boolean as:

AND: A & B = min(A,B)
OR: A + B = max(A,B)
CYCLE: A￫ᴮ = (A plus B)mod N
COMPLEMENT: ~A = P minus A, where P = N minus 1

Depending on what your need to design some logic for color though requires more information. All you technically need is one of those wheels, assign the background as 000 in the additive (adding white light). Then you have an '8-bit' code for those colors. I'd need to know more to understand if what you want to use that for? How are you relating this to "material implication"?

The material implications are of two in propositional calculus:

As I've already mentioned:

(1) Given P, you can conclude the conditional, if Q then P. [P ⊢ Q → P]

The material 'paradox' is only extant from the fact that the variables of P and Q here can be ANY proposition, even if unrelated by their semantic meaning. For example, from the same reference of Lemmon's: Given "Napoleon was French", then we can conclude that, If "the moon was blue", then "Napolean was French".

I answered this in my other reply.

(2) If given not-P, then you can conclude the conditional, if P then Q. [-P ⊢ P → Q]

Example: If given that "Napoleon was not Chinese", "If "Napoleon WAS Chinese" then "the moon is blue."

Bertrand Russell first noted this and explained that the confusion is only due to assuming the equivalency of an "if...then...." statement with the implication concept. This is similar to the confusion of causation to be treated by implication as "if....AND then....". The meaning with causation requires to be expressed in the opposite way by implication. If A causes B, then the implication is B implies A.

I think Russell has to say that because material implication does not follow some very basic intuitions. Looked at the other way, it is the material implication operator that is the problem.

try this example as a counter that tests other 2 is really worth holding on to as form of inference.

Moe did not win the lottery then if Moe did win the lottery then he was never born

I think that reading ends any hope of defending 2. Q would be a cause of not P, and so P cannot imply Q.
[/quote]I think he was doing it to minimize the need for more operators than necessary. The actual minimal expression that encompasses and, or, and not, that covers the spectrum of things like my own function diagram above for all possible binary values. The implication is (not-A or B) is all that is technically needed to cover all of those expressions using two variables and two premises to an argument.

I'm not sure what you meant about the '2' and your example. That happens to be my "(2)" above as relayed by Lemmon's text along with that (1), not Russell.

[Scott Mayers]

Yes. I'm not in doubt of this. The OP was using binary truth values and appeared to confuse the the term, 'contingent' as a value along side of the truth values when the term describes the table's truth values as a whole.

In practice contingent truth means the same as probabilistic truth. It means the same as domain/context-sensitive truth.

It is almost superfluous calling it that since I am having a hard time coming up with non-contingent truths on the spot.

I defined them well. Non-contingencies are either Tautologies or Inconsistencies (via contradiction using binary only)

[Garry G]

Yes. I'm not in doubt of this. The OP was using binary truth values and appeared to confuse the the term, 'contingent' as a value along side of the truth values when the term describes the table's truth values as a whole.

In practice contingent truth means the same as probabilistic truth. It means the same as domain/context-sensitive truth.

It is almost superfluous calling it that since I am having a hard time coming up with non-contingent truths on the spot.

Just to clear up a point. I am using the phrase contingently-true in the sense explained in the OP. Something that is contingent and true. It is true.

Modally: (♢¬p ⋀ ♢p) ⋀ p. In S5 and most modal logics (♢¬p ⋀ ♢p ⋀ p) → p is a theorem.

If the phrasing is an irritant then replace the phrase contingently-true with any term from the set C or treat the set as a series of disjunctions:

C = {extraneous, incidental, accidental, coincidental, fortuitous, haphazard, redundant, superfluous, unneeded, extra, additional, secondary, gratuitous, happenstance, temporary, peripheral, marginal, expendable, dispensable, concomitant, trivial, insignificant, supplementary, random,...}

Add your own suggestions to the set but not equivocal terms like uncertain, possible, unsure, potentially, plausibly, ambiguous, unknown, feasible, thinkable, and so forth. The characteristic of the set is that each term when applied to a state of affairs must assert the state of affairs is true (but contingently true), and when negated also affirm the state of affairs is the case .

Non contingent terms are the negation of C. So if C = coincidental then not C = not coincidental. If p is coincidental it is true. It p is not coincidental it is true.

Examples of non contingent statements:

my gold fish does not speak English
London tower is in London not in Paris
Without gravity planets could not form
my dog is a mammal
In a vacuum light is faster than sound
planet Earth is not flat
I was born in the past.
I am not my own grandfather.
2 + 2 = 4
the word 'contingent' has ten letters.
all men are mortal

...the list is endless.

any problem thinking of examples switch to a term from set C that works for you. The form of the argument does not change.

[Garry G]

The relation of the etymology of "contingent" is 'con-'(with) + 'tangent' (touch), and I believe the past logicians chose it in light of something that is 'touchable' is sometimes and untouchable at other times. So it fits with how you are using it colloquially in the same way. Something that doesn't "have" be be true is one that is either one exclusively but minimally something that has been touched (ie sensed) at least once AND at least untouched at least once. Otherwise they'd be essentially always true or always false.

Thank you for that. Both interesting and informative.

Depending on what your need to design some logic for color though requires more information. All you technically need is one of those wheels, assign the background as 000 in the additive (adding white light). Then you have an '8-bit' code for those colors. I'd need to know more to understand if what you want to use that for? How are you relating this to "material implication"?

the reference to basic colour theory in post 7 was an attempt to illuminate a point about how to interpret Boolean values correctly. It is an analogy to help explain why this system has three values that are true. I then followed up with an appeal to consistency. so the appeal to colour theory does not exist on its own.

This is how the analogy works: RGB primary colours are the complements i.e. 100, 010, 100. The minimum value 000 in colour theory is black and black denies the presence of light. Shine red, green and blue light on a wall and white light is produced. So the primary colours are property of white light. the absent of light i.e. the minimal value black, is their denial. Taking white as an analogy we look to interpret true = 11. the values 01 and 10 become primary value for true analogous to RGB primary colours in colour theory. So we require an interpretation of 10 and 10 that leaves both properties of truth while also remaining complements and denying the minimum value false. and that is how we get three values that are true and one value that is false.

I'm not sure what you meant about the '2' and your example. That happens to be my "(2)" above as relayed by Lemmon's text along with that (1), not Russell.

I was responding to:

(2) If given not-P, then you can conclude the conditional, if P then Q. [-P ⊢ P → Q]

and I think the example:
Moe did not win the lottery then if Moe did win the lottery then he was never born
demonstrates why [-P ⊢ P → Q] is a poor form of reasoning.

[Garry G]

Second, the Modus Ponens comes out not valid as well. If confirmed, that would be terminal.

Just remembered why modus ponens is valid.

The table written as implication is never false. It is reliable but not a theorem because of the appearance of C and N in the truth function.

But Modus poems is not really an implication inference. If we write the same table and remove the false values from the antecedent then the inference modus ponens is valid wherever the antecedent is true i.e. T, N, C and the consequent is T. Most inferences that are never false (reliable) i.e. those with truth functions T, N, C, will now turn out valid if the the antecedent is assumed true.

Which is to say :
p ⋀ (p ⇒ q) ⇒ q
written with a colon in front to indicate it is reliable (bot not theorem) may also be rendered:
p, (p ⇒ q) ⊨ q
The logical consequence symbol indicates wherever the premises are true the consequent is true. That is C ⇒ C or T, N ⇒ N or T, T ⇒ T.

On the tables below false is removed from the consequent and replaced by a placeholder *. We are only interested in what the three true values imply. Analysed that way modus ponens is valid.

I'll be posting less today as I'm off to work. But I think we have resolved an issue.

modus ponens.png (24.78 KiB) Viewed 2430 times

Compare that too paradox 1, which started with no true possibilities. We can block the inference because we cannot assume the antecedent is true.

EXLPOSION.png (42.07 KiB) Viewed 2430 times

[Scott Mayers]

The relation of the etymology of "contingent" is 'con-'(with) + 'tangent' (touch), and I believe the past logicians chose it in light of something that is 'touchable' is sometimes and untouchable at other times. So it fits with how you are using it colloquially in the same way. Something that doesn't "have" be be true is one that is either one exclusively but minimally something that has been touched (ie sensed) at least once AND at least untouched at least once. Otherwise they'd be essentially always true or always false.

Thank you for that. Both interesting and informative.

I'm still unclear of even your OP to be using the four values when what you are thinking is more likely:

"Tautology", rather than 'true'. While they are related, 'true' is an individual assignment of comparative fitness, but with the other terms of 'contingency' you use, those are not comparable. Tautology, contingency, and consistency, are comparisons of whole formulas and relate to the outcomes of all possible values. If you insist on 'contingently true', it is more appropriate as:

"Contingent" unqualified. There is no such particular non-contingent claim and so is unneeded. What is ALL true or ALL false for all values is what 'non-contingent'' means. Thus, IF you are meaning the value of all possibilities, whether true or false for particular values, A contingent 'truth' if you insist on it has to mean "Contingent AND True". This is because a non-contingent truth is just either all true or all false, concepts already exhaustively defined as either a 'tautology' (meaning 'always true'), or Inconsistent (always false).

I don't get the example distinction you gave in the examples for what is non-contingent. If you meant that given you goldfish can't speak any language, the statement is simply not true AT PRESENT but may be in some other place, you MAY mean "Contingent but false" rather than non-contingent. If something is not presently true but still may be, you are meaning to speak of possibilities. The classifications you have are confusing.

When using 'truth' values among other values, they should relate: x is 'true', x is 'not-true', x is 'true and not-true', x is 'neither true nor not-true'. If you are using 'degrees', these then use numbers that propose some percentage of something 'true in x versus not-true in x' and these can have multiple values.

So I need to ask you to tell me if you are opting to describe values of INDIVIDUALS or GROUPS. You seem to mix them together.

I need to determine your definitions and the common principle of collected values they share in kind. Try the genus-specie definitions to both the particular values you use AND to the collection. This may help clear some of this up.

Depending on what your need to design some logic for color though requires more information. All you technically need is one of those wheels, assign the background as 000 in the additive (adding white light). Then you have an '8-bit' code for those colors. I'd need to know more to understand if what you want to use that for? How are you relating this to "material implication"?

the reference to basic colour theory in post 7 was an attempt to illuminate a point about how to interpret Boolean values correctly. It is an analogy to help explain why this system has three values that are true. I then followed up with an appeal to consistency. so the appeal to colour theory does not exist on its own.

This is how the analogy works: RGB primary colours are the complements i.e. 100, 010, 100. The minimum value 000 in colour theory is black and black denies the presence of light. Shine red, green and blue light on a wall and white light is produced. So the primary colours are property of white light. the absent of light i.e. the minimal value black, is their denial. Taking white as an analogy we look to interpret true = 11. the values 01 and 10 become primary value for true analogous to RGB primary colours in colour theory. So we require an interpretation of 10 and 10 that leaves both properties of truth while also remaining complements and denying the minimum value false. and that is how we get three values that are true and one value that is false.

This is a bad example for describing boolean variables without clarity on those definitions above. And note too that while Boolean algebra is extended to multivariables, it is NOT a multivariable algebra.

I'm not sure what you meant about the '2' and your example. That happens to be my "(2)" above as relayed by Lemmon's text along with that (1), not Russell.

I was responding to:

(2) If given not-P, then you can conclude the conditional, if P then Q. [-P ⊢ P → Q]

and I think the example:
Moe did not win the lottery then if Moe did win the lottery then he was never born
demonstrates why [-P ⊢ P → Q] is a poor form of reasoning.

Okay, but I already gave such an example in that original post that is identical in form. The statement IS still 'valid'. It is just not 'sound'. The poverty of the meaning requires an extended logic. Propositional logic does not break into the propositions. Predicate calculus begins this in the first-order logic. These are not actually paradoxes. At least they are not real paradoxes OF THE SYSTEM. The "paradoxes" spoken of implication are only superficial about the soundness of the statements because we lack a means to assure the propositions are actually related. This has to be done outside the system as is the input premise values are assigned with respect to reality.

In fact, even your statement example can be true. "Moe did not win the lottery" is the given FACT going in. The conclusion is dependent upon it in the identical same way as the consequent of an implication is necessarily true if the antecedent is true. The consequent as a conditional only asserts that if, AFTER knowing it is not true that he won the lottery, we then try to assume he did, this would assure that we are not talking about the living Moe [thus is non-alive, unborn or dead]. The conclusion is a consequent of the given as a conditional and so the value of the conclusion is necessary to be true IF the given is true but may be false or true if the given was false.

What may be more clear might be to find a way to indicate the nature of Q to be a 'dummy variable' versus P.

It is similar to the "OR introduction" rule in Propositional Calculus. It says that Given P, then the statement, (P or Q) is true. The Q in this is a dummy variable that may or may not be true. But since we are given P, the statement, 'P or Q' is true by accident. This is not a statement about P and Q but about the RELATIONSHIP of the 'or' operations with respect to what is given.

If you want to (no one stops anyone), you CAN clarify this to avoid the problem by, say, italicizing Q, place brackets around it, or some other indicator. In computer science, this is often treated as a "don't care" value. AND, while it seems it serves no value, these dummy variables actually DO have functional value. For instance, if you are trying to make some component match the timing in parallel to another, you might add a 'dummy' variable (or constant in some places) that makes the timing of some quicker component 'buffer' to the second component's longer time.

[Garry G]

You reply deserves a longer considered response and I will do so they evening. But this stood out.

Moe did not win the lottery then if Moe did win the lottery then he was never born
demonstrates why [-P ⊢ P → Q] is a poor form of reasoning.

Okay, but I already gave such an example in that original post that is identical in form. The statement IS still 'valid'. It is just not 'sound'.

It is very difficult to see how we cannot assume P: "Moe did not win the lottery " is true on some set of possibilities. However the argument allows an interpretation that provides the falsifying condition. So the conclusion P then Q, forces us to assume not P is false. So we assume P is true. And if Moe did win the lottery ticket then he was never born. So we cannot assume the premise is true and we cannot assume it is false. But by itself the premise looks harmless. What is causing the problem is the argument. This counter example demonstrates [-P ⊢ P → Q] forms a contradiction.

The poverty of the meaning requires an extended logic.

True

Propositional logic does not break into the propositions. Predicate calculus begins this in the first-order logic. These are not actually paradoxes.

I just described a paradox above. Extended logics may eliminate the paradox by proving the argument is invalid. Why we hold on to material implication when it goes so horribly wrong? I could understand it if it only lead to oddities and slightly puzzling cases. But not when it leads to contradictions.

[Logik]

Why we hold on to material implication when it goes so horribly wrong?

Because there is no logic that doesn't go wrong.

All models are wrong - some are useful
Worrying Selectively
Since all models are wrong the scientist must be alert to what is importantly wrong. It is inappropriate to be concerned about mice when there are tigers abroad.

Is why we still need the humans to exercise their better judgment than to blindly trust the authority of logic.

But not when it leads to contradictions.

Some contradictions are mice - some contradictions are tigers. Paraconsistent logics allow us to navigate that.

https://en.wikipedia.org/wiki/Paraconsistent_logic
https://en.wikipedia.org/wiki/Dialetheism

[Speakpigeon]

Why we hold on to material implication when it goes so horribly wrong?

Because there is no logic that doesn't go wrong.

Prove the logic of the human mind is wrong.
EB

[Logik]

Prove the logic of the human mind is wrong.
EB

I can't speak about 8 billion people all at once, but your logic is definitely faulty.

Evidence as above. You are asking me to prove a negative.

[Speakpigeon]

Prove the logic of the human mind is wrong.

I can't speak about 8 billion people all at once, but your logic is definitely faulty.
Evidence as above. You are asking me to prove a negative.

Zenologic, it's you making a claim you can't prove.
EB

[Logik]

Zenologic, it's you making a claim you can't prove.
EB

Are you claiming that I can't prove my claims?

Prove it.

[Speakpigeon]

Zenologic, it's you making a claim you can't prove.

Are you claiming that I can't prove my claims?
Prove it.

I'm not claiming you can't. I'm observing you haven't proved your claim despite being asked to.
Proof by empirical fact. Good science.
EB