Paradoxes of Material Implication

[Garry G]

Standard propositional logic proves a small class of non intuitive theorems. These are sometimes called paradoxes of material implication. Here is a list of better known paradoxes.

1. (p & ¬p) => q
2. p => (q => p)
3. ¬p => (p => q)
4. ¬(p => q) => p
5. (p => q) V (q => r)
6. [(p & q) => r] => [(p => r) V (q => r)]
7. [(p => q) & (r => s)] => [(p => s) V (r => q)]

There is an extension of standard two valued logic free of paradox - at least it disproves 1-7 and proves a set of obvious standard theorems. If someone is able to come up with some nasty unwanted results - then back to the drawing board.

Here's a sketch: the four values are : {T, N, C, F}.

T = True
N = Not contingently true (think: not accidental and not peripheral and not temporary etcetera).
C = contingently true (think: accidental or peripheral or redundant etcetera).
F = False.

A contingent truth is "our cat Aristotle is on the mat". A non contingent truth is "our goldfish Plato does not speak Latin".

The system is due to a large set of adjectives that employed in declarative sentences affirm a state of affairs is the case and when denied also affirm the state of affairs. Which is to say that if a states of affairs is accidentally the case then it is true, but if it is not accidental it is still true, if it is temporary it is true and if not temporary still true, if peripheral then true if not peripheral then also true, if casual then true if not casual then also true and so on. We take the grammatical distinction and draw from it two logical values as properties of truth.

As complements N and C cannot entail each other. They do however sum to True (T) i.e. N + C = T. Thus the system has three values that are true and one value that is false. BUT only T is the designated value. N and C are not designated but they are true and so they cannot entail false.

The system is the Cartesian product of two x two Boolean values {1, 0} x {1, 0} = (11, 10, 01, 00}. So {T, N, C, F} = {11, 10, 01, 00}. The tables for negation, disjunction and conjunction behave as a product system. However the implication tables restricts what N and C imply. To distinguish it from material implication and Lewis' strict implication this alternative is labelled semantic implication.

There is also an example truth table showing how how paradox 4 is disproved. e.g. if it is not the case p implies q when p is false and q is contingently true, then it is false to infer p.

An example of why we might not want 4 in any reasonable system of logic, try this example:let 'p => q' be the "if ...then..." assertion 'if the law is just then innocent people don't go to jail', and therefore paradox 4 '¬(p => q) => p' reads:
' if the law is just then innocent people don't go to jail...is not the case.... then the law is just'.

negation connectives 2vlx 2.png (47.1 KiB) Viewed 2970 times

[Speakpigeon]

I believe you came to the right place here. We happen to have the most famous logician in the multiverse so I'm sure he will compute your claim and show your error. Because, he only ever finds error in all things, so don't you foolishly expect to be vindicated. Brace yourself instead for some ad hominem.
I'll try myself to have a look but I don't use computers to compute anything. I use my own mind and this can only take as long as it takes.
Still, I can say right now that I would agree that the "logic" of material implication is just worse than terrible. It's just... disgusting.
And, of course, someone should do something about it! The sooner the better.
On the face of it, I feel your idea probably wrong. First, logic has nothing to do with "contingence" or "non-contingence". So, if that's essential to your perspective, I would expect this to be wrong. Then again, maybe the term "contingence" may be replaced by something else I don't know what.
Also, there's nothing else but either contingently true or not contingently true, because, whatever A, there's nothing but either A or not A. So, here again, it doesn't look too good "on the face of it".
Further, logic has little to do with "semantic" either, at least outside the semantic of the logical constants, such as "true", "and" etc. So, uh-uh.
The general idea of complicating the calculus sounds a good one. I always took material logic to be just too bloody simple to be anywhere near like the real stuff. But, there's complication and complication.
That being said, it's indeed too complicated to give an answer straight away, if I can even get to understand your complicated system...
So, no doubt Logik will beat me to it.
We also appear to have a few experts at the moment, so they should be able to give at least an opinion but more likely tell you where you went wrong.
You're aware that many people, very bright people, have searched for more than two thousand years, right? And Russell wasn't an idiot either. So, don't expect too much.
You should also know in this day and age you're far from being the only one to keep searching. I hope for you you're really, really bright.
EB

[Garry G]

Thanks for the response.

I feel like I may unwittingly be in another longstanding conversation..

Ah well - anyway - I wouldn't get sidelined by the semantics. I gave the sketch for those who wanted to know more. We may agree or disagree over what counts as a truth value but the real point is whether a system with three True values leads to useful results.

If the semantics is an irritant treat the tables as just another four valued system. You don't need to know what the values mean to check what is proved and not proved. The only novelty is the the implication table - semantic implication - as I called it The merit of this table - if there is one - will be its appeal to those who think the paradoxes listed are a real problem. Logicians who don't think 1-7 are a problem will be underwhelmed.

Unless I've mangled my own tables the implication table definitely falsifies 1 - 7. So the next question is whether there are other paradoxes it fails to falsify. Of this I am not certain. I collated all the paradoxes I could find. There is one more not listed which is equivalent to 4. ¬(p -> q) -> (p & ¬q). So if anyone can find other examples to test that would be helpful. After that the question is whether these tables falsify any standard theorems that can't be done without. There is one I know it fails but not I think disastrous.

The system is weaker than standard propositional logic so we don't have to worry about the system introducing new inconsistencies.

[Logik]

We may agree or disagree over what counts as a truth value but the real point is whether a system with three True values leads to useful results.

In the most abstract way possible - human communication/agreement/disagreement is a three-valued system.

Your view.
My view.
Reality.

Getting two (or more) entities to agree on a 3rd value is called consensus. https://en.wikipedia.org/wiki/Consensus ... r_science)

Philosophy forums aren't the place to look for consensus.

And the dominant purview of all formal logic is one that is greatly disconnected from instrumentalism/constructivism paradigms from which I view the world. e.g you probably don't use logic like I do

In summary: our conceptions of what logic is, what it does and what it's used for are so far disconnected that unless you have some clear success criteria (goals/intentions) in mind, this conversation is unlikely to be fruitful.

[Garry G]

In the OP there are tables for an extension of two valued logic. On face value these tables look like another four valued logic. Only the implication table is novel - I think. I am reasonably sure I've seen all the important four valued logics but I wrong.

I was hoping someone might try and experiment with the tables to test them out and see if they find the results plausible. But maybe that is a naive hope.

Failing that throw some standard theorems in the ring and I'll provide the tables.

If someone wants to draw a defensive line around standard two valued logic and admit no change - then maybe this is not the thread for them unless they are willing to defend 1 to 7. If everyone thinks there is nothing to defend then a short conversation it is.

This is the purpose of this thread. But it could go in other directions.

[Logik]

In the OP there are tables for an extension of two valued logic. On face value these tables look like another four valued logic. Only the implication table is novel - I think. I am reasonably sure I've seen all the important four valued logics but I wrong.

Have a look at N-valued (many-valued) logics. Once you have a generally applicable concept whether you are dealing with boolean, or 4-value logic is neither here nor there. You can translate from one to the other.

https://en.wikipedia.org/wiki/Many-valued_logic

I was hoping someone might try and experiment with the tables to test them out and see if they find the results plausible. But maybe that is a naive hope.

Why not just test them out in your own, favourite programming language?

If someone wants to draw a defensive line around standard two valued logic and admit no change - then maybe this is not the thread for them unless they are willing to defend 1 to 7. If everyone thinks there is nothing to defend then a short conversation it is.

This is the purpose of this thread. But it could go in other directions.

I can argue for and against, but I see that as a false dichotomy either way. I am comfortable with both. I use both. Each has its pros and cons.

Hence why philosophical arguments without clear success/failure criteria is difficult. Horses for courses.

[Garry G]

Have a look at N-valued (many-valued) logics. Once you have a generally applicable concept whether you are dealing with boolean, or 4-value logic is neither here nor there. You can translate from one to the other.

Thank you for the link. It is not unfamiliar. I am not sure what you have in mind when you talk about a generally applicable concept. I'll sketch some more of what I have in mind.

The four valued table is intended to be a conservative extension of two valued logic. As mentioned in the OP the NOT, AND, OR tables are the Cartesian product of 2 x two Boolean values, viz., {11, 01, 10, 00}. We may continue through the extensions N=8, N=16 and so on, but extended systems prove the same set of theorems. They are just Boolean logic in the extended 2-tuple format, or N-value format as we progress through the series. But there is no need to go beyond N = 4.

And there is no point to even bother with the N = 4 extension without the interpretation that forces a revision of the implication table. This is the only table that breaks with the standard algebra of a Cartesian product system; this is due to how the values are interpreted.

Given 10 + 01 = 11 and 11 interpreted as true the values 10 and 01 also preserve truth. That is the basic insight and I'm hoping it is obvious once pointed out. It should not need sustained argument in support.

But I'll go a couple of extra yards.

First I try to illuminate my point (pun intended).

In the real world there is a very natural N = 8 Boolean logic that makes a useful analogy that is hopefully illuminating. White = 111, Black = 000. The order of red, green blue does not really matter, but let blue = 100, green = 010 and red = 001. The secondary colours are yellow = 011, magenta = 101 and cyan = 110. Basic colour theory has a subtractive model (product) and additive model. If you are already aware of this all the better. You likely already see where this is going and I do not need to labour the point.

colour theory.png (37.2 KiB) Viewed 2935 times

White light is produced by adding red light, green light, blue light. If you shine red light, blue light and green light on wall you can produce white light. Black is a shadow and the complement of red, green, blue.

As maximum values Truth is comparable to White. In the N = 4 system where 11 = true, the values 01 and 10 are compared to the primary colours and as primary values they produce the maximum value. Question along the lines of what does it mean for a primary value to produce truth, will be redirected back to basic colour theory.

Just to be clear about why I am bringing up basic colour theory. This is not an argument for why 01 and 10 are each version of true. It does however illuminate how I intend them to be understood, If you get basic colour theory then you at least understand what I mean when I say 01 and 10 are primary values. It also gives a real world example in which the understanding of a maximum value I am pursuing makes sense.

I now appeal to consistency.

If for instance 01 were interpreted as false then: {10, 01} = {True, False}. On this interpretation it is consistent to interpret the maximum value 11 along the lines of "evaluable" or "possibly decidable" but it is inconsistent to interpret 11 as true because 11 literally means "true or false". In standard two valued logic 1 + 0 = 1 and the 0 is annihilated. once we may also say : true or false = true, because false is annihilated.

The vanishing trick not happen to false if associated in an extended system. With the value 10 or 01 from the set of N = 4, n s for example F = 01, then False no longer disappears. And so if we interpret 11 as true, which seems perfectly reasonable, then both appearances of the Boolean 1 in 11 represent some version of truth i.e. 11 = TT.

On the other hand If 10 = T and 01 = F then 11 = TF.

If we want to say 11 = T, that is inconsistent because then T = TF. If we want say 01 is equivocal and use the sign ? for equivocation then 11 = T?

I have now set out my basic point, attempted to illuminate it by way of comparison to colour theory and support it by demonstrating it is the only consistent approach.

A corollary that follows from keeping a consistent interpretation is that standard material implication does not translate into N-valued extensions where N > 2, and when the maximum value is interpreted as true. Hence the revised implication table in the OP.

Further extensions N > 4 prove the same theorems as N = 4. Thus the tables in the OP are sufficiently general.

Why not just test them out in your own, favourite programming language?

Simple answer is I don't code, but I guess I could learn. It seemed quicker to hit a forum. But I'm getting the sense a couple of years learning to code may be quicker.

But also - I'd like to test some of my arguments, the semantics I have come to to try and explain why I get three truth values and so forth.

[PeteOlcott]

It is all explained right here:
https://en.wikipedia.org/wiki/Paradoxes ... mplication
It is not really a paradox at all. Its merely that material implication is defined in a counter-intuitive way.

[Garry G]

It is all explained right here:
https://en.wikipedia.org/wiki/Paradoxes ... mplication
It is not really a paradox at all. Its merely that material implication is defined in a counter-intuitive way.

Maybe explosion - paradox 1 listed in the OP - is defendable, and maybe the first couple of paradoxes are just puzzling. If the list stopped at 1-3 defending material implication is probably sustainable. However the inferences listed become increasingly egregious. Paradox 4 is counter intuitive to the point something is clearly starting to go very wrong. Paradoxes 6 and 7 are untenable.

6. [(p & q) => r] => [(p => r) V (q => r)]

"if tap one is open and tap two is open then the tank will empty then if tap one is open then the tank will empty
or tap two is open then the tank will empty"

safety taps.png (25.17 KiB) Viewed 2918 times

if 6 is correct we should be able to open just tap one to empty the tank, or just tap two to empty the tank, it should not be necessary to open both.

7. [(p => q) & (r => s)] => [(p => s) V (r => q)]

if Moe is in New York visiting the Statue of Liberty then he is in the US and if Moe is in London visiting the Tower of London then he is in England then if Moe is in New York visiting the Statue of Liberty then he is in England or if Moe is in London visiting the Tower of London then he is in the US.

If 7 is correct then somewhere in Google maps we should be able to find the Statue of Liberty in England, or the Tower of London in the US.

I wonder exactly how counter intuitive the conclusions material implication lead to would have to become for its defenders to give up on it? I quit somewhere around paradox 4.

[Logik]

6. [(p & q) => r] => [(p => r) V (q => r)]

"if tap one is open and tap two is open then the tank will empty then if tap one is open then the tank will empty
or tap two is open then the tank will empty"

safety taps.png

if 6 is correct we should be able to open just tap one to empty the tank, or just tap two to empty the tank, it should not be necessary to open both.

Logic is just a modeling tool - LEGO for your mind. It's meant to formalize our intuitions about the world, but the formalization occurs a posteriori of our experiences and is never quite complete.

What is interesting to me is how you have chosen to formalize the system of taps: [(p & q) => r] => [(p => r) V (q => r)] (for the record, I can't even parse that into English without pen and paper, it's that counter-intuitive to me).

I simply saw a Boolean AND: OpenTap1 ∧ OpenTap2 ⇔ WillEmpty

Your formalization seems to go against the KISS principle.

So you have the taps before you. Intuitively you know and understand how they are supposed to work. If both taps are open - the tank will empty.
This is empirically and intuitively true. Except when it isn't.

If the pipes are blocked -> Will NOT empty
If the tank has no water -> Will NOT empty
If the taps are above the water line -> Will NOT empty
If the tank is full and hermetically sealed at the top -> Will NOT empty

And finally (given the picture you have provided us with) - the tank will actually empty only half way. Because of the positioning of Tap 1.

All conceptual models have edge/corner cases. The more "universal" the model - the more edge/corner cases it has. Because we have no universal truths "Material implication" suffers from that faux-universality immensely.

[Speakpigeon]

It is all explained right here:
https://en.wikipedia.org/wiki/Paradoxes ... mplication
It is not really a paradox at all. Its merely that material implication is defined in a counter-intuitive way.

Well, yes, but what's a paradox?
I would say myself that it's a statement that first of all is seriously baffling to us. Second, that it is baffling because we remain uncertain as to why it is baffling. The Liar paradox is a good example. Different people will have different resolutions to offer and most people won't feel good about any of them. Including people who will claim the thing is not a paradox to begin with without being able to explain why exactly. I tested this many times. It's almost fascinating to watch.
So, to me, paradoxes are in the eyes of the beholder. What's paradoxical to you may not be to me. And there are good reasons for that.
So, going back to the so-called paradoxes of the material implication, we need to remember they've been called that by the very people who invented the "logic" of material implication. To them, these were theorems that validly followed from the truth table definition of the material implication. The fact that they called them "paradoxes" is empirical evidence that they themselves had this impression that they were nonetheless false. False but proven true by first principles. Hence, their paradoxical value to these people.
To me, these are just false and obviously false, hence not paradoxical at all. However, they definitely remain paradoxical to the supporters of "material logic", as this thing should be called, at least as long as they'll insist on using this material logic to prove anything.
EB

[Garry G]

So you have the taps before you. You know how they are supposed to work

We have the taps before us. We have the formula before us. The formula is telling us the taps do something they don't. It is because we know what to do that we know there is something wrong with 7.

All models have edge/corner cases. The more "universal" the model - the more edge/corner cases it has. "Material implication" suffers from that immensely.

If by corners and edge cases you mean all models have a counterexample that eventually shows it is a false model, then we are agreed about material implication.

If there is standard propositional logic A which proves paradoxes 1 to 7, and a competing logic B that proved everything provable in A with the exception of 1 to 7, is B preferable or just another logic with little to choose between A and B?

If your answer amounts to "meh" and you don't find B preferable we are pulling in different directions.

The logic put forward in the OP is not the simple case of A versus B. It proves basic standard theorems and logical consequences we would wish to find, viz.,

p ⊨ ¬¬p
⊨ ¬(p ⋀ ¬p)
⊨ (p ⋁ ¬p)
⊨ (p ⇒ p)

The system does the simple stuff right.

The following are never false. Not quite a theorem because their truth function contains an N or C or both. Failing to be a theorem does not mean they are unreliable. They are indeed never false. So the system throws up an in-between class of formulae not false and not theorems. I'll put a colon in front of these instead of the logical consequence symbol.

: (p ⋀ q) ⇒ p
: (p ⋀ q) ⇒ q
: p ⇒ (p ⋁ q)
: q ⇒ (p ⋁ q)
: (p ⋀ (p ⇒ q)) ⇒ q
: (¬q ⋀ (p ⇒ q)) ⇒ ¬p
: [p ⇒ (q ⇒ r)] ⇒ [(p ⋀ q) ⇒ r]
: (p ⇒ q) ⇒ (¬p ⋁ ¬q)

However some standard theorems not usually deemed problematic are falsified on at least one permutation of truth possibilities. The next three are in this class. Neither a theorem or never false they are disproved.

material implication in one direction: ⊭(¬p ⋁ ¬q) ⇒ (p ⇒ q)
transposition both directions: ⊭(p ⋁ q) ⇒ (¬p ⇒ ¬q), (¬p ⇒ ¬q) ⇒(p ⋁ q)
exportation in one direction: ⊭[(p ⋀ q) ⇒ r] ⇒ [p ⇒ (q ⇒ r)]

I don't think this system has any egregious edges. If there is criticism it is that it is too restrictive.

I'd also add I think the disproved inferences are themselves suspicious. Maybe not at the level of poor reasoning seen in paradoxes 6 and 7 but still suspicious. Exportation for example:

[(p ⋀ q) ⇒ r] ⇒ [p ⇒ (q ⇒ r)]
p: 'Moe is waiting hopefully in the restaurant with a rose in his lapel'
q: 'the blind date is going to be enjoyable''
r: "Chloe is on her way to the restaurant and will be there soon'

Yes difficult to parse. The inference is falsified on the permutation of truth possibilities such that : p is non contingent (N), q is false (F) and r is contingently true (C).

So if it was never going to be the case Moe missed this date (p is N), the date is now going to be a disaster (q is F) , Chloe has gone for a night out with the girls instead unaware they are about to walk into the same restaurant where Moe is waiting (r is C).

under those set of conditions the antecedent inference [(p ⋀ q) ⇒ r] is contingently true, the consequent inference [p ⇒ (q ⇒ r)] is false.

[Arising_uk]

Also, there's nothing else but either contingently true or not contingently true, because, whatever A, there's nothing but either A or not A. ...

Er!? But 'either A or not A' is a tautology so not contingently but necessarily true so 'there's nothing else but either contingently true or not contingently true' is untrue, or did I miss your point?

[Logik]

So you have the taps before you. You know how they are supposed to work

We have the taps before us. We have the formula before us. The formula is telling us the taps do something they don't. It is because we know what to do that we know there is something wrong with 7.

Yes. Logic is a model of reality. Not reality itself. All models are wrong - some are useful.

The nuance here is that some model-error is tolerable. Some model-error is catastrophic. Which brings us onto the notions of precision and completeness.

All models have edge/corner cases. The more "universal" the model - the more edge/corner cases it has. "Material implication" suffers from that immensely.

If by corners and edge cases you mean all models have a counterexample that eventually shows it is a false model, then we are agreed about material implication.

My view is far more controversial than that. The validity criterion in logic is clear and incredibly strict:

In logic, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false

Since EVERY model has counter-examples which violate the "impossibility" criterion, then it seems to be it's impossible to satisfy "validity" - I go as far as to say valid deduction is impossible in this universe. Because you are dealing with imperfect knowledge. What people call "valid deduction" is really induction being mistaken for deduction.

And when experience (empirical evidence) contradicts your "sound premise" - scientists call that falsification.

If there is standard propositional logic A which proves paradoxes 1 to 7, and a competing logic B that proved everything provable in A with the exception of 1 to 7, is B preferable or just another logic with little to choose between A and B?

If your answer amounts to "meh" and you don't find B preferable we are pulling in different directions.

My answer is not "meh". My answer is "it depends". What are you using the logics for?
Because I see them as "just tools" I'd have to evaluate the pros and cons of A and B.
If they intersect in utility - I'd keep both and use them in different contexts.
If the one is subset of the other - I'd throw away the subset and keep the superset.

Ultimately though, it sounds to me as if you are looking to propose/impose "standards for logic". And so I find myself using this XKCD for the 2nd time today:

standards.png (49.78 KiB) Viewed 2866 times

While you sure as hell get to choose a "standard" for yourself, and you even get to propose a standard for others. I don't have to adopt your standard if it doesn't fit my use-case.

the system does the simple stuff right.

And this is the crux of the issue. Logic is a modeling tool. First order logic is a very simple tool.

Think: LEGO 2-4 year old.

Reality is very, VERY complex. So complex that we often find ourselves reaching for N-th order logics in physics. Computational complexity theory deals with this notion.

The human tendency/error is to try to universalize and simplify everything: to attempt to use 1st order logic to speak about a reality that is difficult to comprehend even in high-order logics is to try and stick round pegs in square holes.


Formal systems exist entirely in the abstract realm. That we USE them to reason about reality is neither here nor there. They may be suitable for the job at hand. Or not.

This is a position that is broadly labelled as Model-dependent realism

[Garry G]

If the pipes are blocked -> Will NOT empty
If the tank has no water -> Will NOT empty
If the taps are above the water line -> Will NOT empty
If the tank is full and hermetically sealed at the top -> Will NOT empty

If the pipes are blocked the antecedent is false and the inference is proved true. False premises lead to any conclusion. If however we assume the premises are true - that is we assume P & Q are true and R is true and everything is in working order. what counts is the position of tap one and tap two in the same pipe. The rest is immaterial.

And finally (given the picture you have provided us with) - the tank will actually empty only half way. Because of the positioning of Tap 1

I'm also guilty of typos. Both typos and the poor drawings are fixable. Arguments 6 and 7 however are not fixable.