Paradoxes of Material Implication

[Garry G]

Since EVERY model has counter-examples which violate the "impossibility" criterion,

What counter examples violate modus ponens?

p, (p ⇒ q) ⊨ p

and simplification?

p ⋀ q ⊨ p

Some forms of inference are not seriously up for debate for the lack of counter examples.

Whilst different logics have different applications there is still a core set of intuitions that if we give up on we give up on the capacity to think straight. Paradoxes 6 and 7 do not belong to that set.

You seem to be stretched between two stools. On the hand you admit material implication is flawed, in fact every logic has its limits on your view, but at the same time you are defending its most egregious conclusions and trying to mitigate just how bad they really are. The fact you feel the need to argue for 6 indicates you sense the tap and pipe example could be fatal. Not just a corner where implication meets its limits but a fatal incoherence.

[Logik]

Since EVERY model has counter-examples which violate the "impossibility" criterion,

What counter examples violate modus ponens?

p, (p ⇒ q) ⊨ p

and simplification?

p ⋀ q ⊨ p

Temporal paradoxes.

Like this one:

https://repl.it/repls/PaltryHorribleTelevisions

Code: Select all

def impl(a,b)
  return ( (not a) or b )
end

$toggle = false
def p
  $toggle = (not $toggle)
end

def q
  true
end

impl( ( p and q) , p)
=> false

It's called a Race condition

When the truth-value of P changes after you have established it, and before you've had a chance to evaluate its implications.

[Speakpigeon]

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?

I don't really know, I can't read your mind.
So, you'd have to explain precisely why you think that the fact that "either A or not A" is a tautology and therefore not contingently but necessarily true falsify my claim that "there's nothing else but either contingently true or not contingently true".
EB

[Logik]

"there's nothing else but either contingently true or not contingently true".

Do you have any examples of non-contingent and non-axiomatic truths?

We have been looking for those for a long time.

[Scott Mayers]

There's a lot to read here before I can contribute fairly but I need to note that I understood "material implication" simplest explanation as:

P ⊢ Q → P

Which begs whether the arbitrary meaning (semantic matter) of Q is or is not valid. If given any P, how can it seem to imply the material reality of some antecedent without affirming that Q exists first?

The problem is resolved when we recognize that the conditional is itself true but says nothing of the actual material nature of Q. It may be 'sound' or not but the statement is valid. Similarly, if

Q → P ⊢ P

...is allowed IF P is coincidentally real or not, because the statement again is itself giving a conditional. What would prevent this being true is only if Q and not-P are both true. The 'material' meaning of P here is irrelevant because the 'matter' of the statement is the implication between Q and P, not of their particular meanings.

To be complete, this last statement should be:

Q → P ⊢ P ∨ -Q

[Logik]

To be complete, this last statement should be:

Q → P ⊢ P ∨ -Q

If you provide me with truth-tables for your operators I will construct you a temporal system in which the above is false.

[Scott Mayers]

By the way, is there a more convenient method to printing out diacritic or unusual logical/mathematical expressions here than finding, copying, and pasting? Is there a tag we can be permitted here? [This site blocks all BBCode even though it says it is 'ON']

[Scott Mayers]

To be complete, this last statement should be:

Q → P ⊢ P ∨ -Q

If you provide me with truth-tables for your operators I will construct you a temporal system in which the above is false.

I don't know what you mean when you ask these things other than if you treat the operator symbols as non-constant. The variables are Q and P, the operators of this system are 'constants' semantically understood. You are welcome to illustrate what you mean but if you are treating the operators as variable, that is irrelevant when the context here is already understood by the conventional understanding of the operators here.

An "and" instruction might be written arbitrarily as,

& A, B

rather than

A & B

or you can redefine the symbol uniquely to make "&" mean "or". But then these are not relevant because this just begs the meaning of the symbols of operation as variable, not the semantic/material meanings of them.

[Garry G]

Code: Select all

def impl(a,b); return ( (not a) or b ); end

$toggle = false
def p; $toggle = (not $toggle)
end

def q; true; end

impl( ( p and q) , p)
=> false

It's called a Race condition

When the truth-value of P changes after you have established it.

Ok I don't code so the syntax is strange to me.

I think this says something alone the lines of:

a ⇒ b ⊨ ¬(a) ⋁ b

q ⊨ (p ⋀ q) ⇒ p

I am not sure why we move from the lower case a, b to p, q. but I think you defined implication with the lower case letters. I don't know what the stuff about a toggle means.

but clearly p & q => p is a theorem and so the argument ought to be true, and it is recognised as mistake if a program returns false.

this is my understanding: the program arrives at the wrong answer because the programmer has failed to make two inputs mutually exclusive. And one slower signal just fails to get there in time to cancel the other signal.

I'm assuming that his kind of problem is not just down to bad circuit design and bad programming. If it is just bad design then my answer is the same as the picture of the tank and pipe. The design is fixable. This is not a logical problem and the logical principles are not in any danger. It is just poor plumbing so to speak. So we look for ways to change the hardware, software, programming to keep to the standard dictated by the logical principle (p ⋀ q) ⇒ p. On this view the computer has let us down not the logical principle.

But the real test here is not whether some program fails to obey (p ⋀ q) ⇒ p is true. We might all fail to obey it or not recognise it. The tell is not that some system fails to meet a standard, but that things go wrong when they do meet the standard. If there were examples of race conditions that lead to (p ⋀ q) ⇒ p is true proving hazardous for a program then you may have a point. I don't think you have a case without that kind of counter example.

Go back to paradox 6 and the tank/tap/pipe example. The problem is not that example fails to obey the paradox. the problem is that if we treat the paradox as if it is a true guide as to what to do we fail to empty the tank. You need examples of programs running basic principles like modus ponens or simplification correctly and then going wrong because they made the wrong decision.

[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?

I don't really know, I can't read your mind.
So, you'd have to explain precisely why you think that the fact that "either A or not A" is a tautology and therefore not contingently but necessarily true falsify my claim that "there's nothing else but either contingently true or not contingently true".
EB

My point was that you said "there's nothing else but either contingently true or not contingently true" and then used a tautology to support the statement but a tautology is something that is always true, i.e. necessarily true, and not a contingency. I put the query at the end as I wasn't sure if you were actually pointing-out your contradiction in an attempt to also show another thing that wasn't "either contingently true or not contingently true".

[Logik]

Ok I don't code so the syntax is strange to me.

I think this says something alone the lines of:

a ⇒ b ⊨ ¬(a) ⋁ b

q ⊨ (p ⋀ q) ⇒ p

I am not sure why we move from the lower case a, b to p, q. but I think you defined implication with the lower case letters. I don't know what the stuff about a toggle means.

It doesn't matter - they are just temporary variables. I can rename them to p and q. It will do the same thing.

but clearly p & q => p is a theorem and so the argument ought to be true, and it is recognised as mistake if a program returns false.

Recognized by whom and why?

The problem of "define error" or "define a mistake" boils all the way up to human ethics. "Right" and "wrong".

This is where we will diverge. You treat logic as prescriptive and axioms of logic as laws. I don't. They are arbitrary choices.
They may be useful axioms. Useful to somebody and in some context, but I am free to reject them as and when it suits me if they get in my way.

The key thing to see here is that axioms are the product of arbitrary choice. And in turn - the theorems that result are the consequences of the axiom that you've chosen.

If I choose a different axiom - I will get different theorems.

this is my understanding: the program arrives at the wrong answer because the programmer has failed to make two inputs mutually exclusive. And one slower signal just fails to get there in time to cancel the other signal.

Now look where you have ended up. You are already claiming that you know what the "right" answer is.
If you already know the truth - what do you need logic for?

The most important development in logic in the last 100 years is the Curry-Howard correspondence

It says that a mathematical proof is the same beast as a computer algorithm.

And so in a constructivist paradigm p and q are objects. ANY mathematical objects. That behave in any way you can imagine them behaving.

And this statement is nothing more than a proposition: p & q => p

IF I can find values for p and q such that the above evaluates to "true" then that is proof for your claim.

I'm assuming that his kind of problem is not just down to bad circuit design and bad programming. If it is just bad design then my answer is the same as the picture of the tank and pipe. The design is fixable.

The design is fixable. Logic is not.

Logic is not what you think it is. It's not a source of truth. It's a framework for computation.

This is not a logical problem and the logical principles are not in any danger.

What would you say the "logical principles" are?


Identity? x = x ⇔ True

I can (have) constructed systems where it evaluates to False.

non-contradiction? (p & ~p) ⇔ False

I can (have) constructed logical systems where it evaluates to True.

Axioms.... All axioms are arbitrary.

Aristotelian logic is dead. Church and Turing killed it.

[Garry G]

To be complete, this last statement should be:
Q → P ⊢ P ∨ -Q

If you provide me with truth-tables for your operators I will construct you a temporal system in which the above is false.

I realise I'v made a mistake in a previous post. Several of the formula listed as theorems are only found never false. Not quite a theorem in the system described in the OP. But still not false on any permutation of truth possibilities.

I'd like to see how this table for a modus ponens type inference fairs with race condition problems. I am hoping they would prove more resistant to race conditions. I have no idea really. But there is a four way race between three true values and one false value. So just curious if these tables prove a harder nut to crack.

modus ponens.png

[Logik]

To be complete, this last statement should be:
Q → P ⊢ P ∨ -Q

If you provide me with truth-tables for your operators I will construct you a temporal system in which the above is false.

I realise I'v made a mistake in a previous post. Several of the formula listed as theorems are only found never false. Not quite a theorem in the system described in the OP. But still not false on any permutation of truth possibilities.

I'd like to see how this table for a modus ponens type inference fairs with race condition problems. I am hoping they would prove more resistant to race conditions. I have no idea really.Bbut there is a four way race between three true values and one false value. So just curious if these tables prove a harder nut to crack.
modus ponens.png

OK, but the difference here is that the truth-table is prescriptive. In the sense that those are the results that you EXPECT IF the inputs (p and q) were what they were.

In the exact same sense that we were given times-tables in high school. The result of 7 and 8 when the operation is "multiplication" is 56.
These are just short-hands so that you don't have to do the computation yourself. It's all pre-calculated for you.

And that's all fine. The point is that you are still ONLY talking about DETERMINISTIC logical consequences, not empirical facts.

And so the point I am making is that every time your logical conclusion disagrees with reality

Your logic is wrong. Not reality.

All I am describing (rather verbosely at that) is the garbage-in/garbage-out problem.

[Logik]

We are getting really esoteric here and there is no need.

Logic is a tool for computing consequences.
A tool for computing consequences is incredibly useful to entities (humans!) who anticipate/plan for the future and have clear goals/objectives in mind.

If said human set itself a goal (for reasons we need not discuss or understand) to empty a barrel of water.
And the barrel of water (for reasons we need not discuss or understand) had TWO drainage taps connected in series instead of one.

Then this human would ask itself these questions A PRIORI:
Would opening both taps be sufficient to achieve my goal?

In theory - yes.
In practice - very likely, unless reality has other plans.

IF the human succeeds in emptying the barrel then A POSTERIORI the human can then go on to say: I know how to empty that barrel.
IF the human failed in emptying the barrel then A POSTERIORI the human would go on to say: I didn't factor X into account.

Where X is the thing which prevented you from achieving your objective.

There's even more shenanigans here. If you were to perform the barrel-emptying task 100 times and your procedure worked 99 times. Then you would say "I have done this 99 times - I know how to empty a barrel". And then on the 100th time you will bump into some failure mode you hadn't anticipated before.

[Garry G]

And so the point I am making is that every time your logical conclusion disagrees with reality

Your logic is wrong. Not reality.

All I am describing (rather verbosely at that) is the garbage-in/garbage-out problem.

Well it depends what threshold you set that means a principle disagrees with reality. I'm posting a lot so I'll point you to my previous post on what would count against a standard principle. The problem is not whether a circuit or program fails to repeat the principle, the problem is if things go wrong if they manage to repeat it. That would discount a principle. That would be a logical conclusion disagreeing with reality.

But now you have introduced me to race conditions I am curious whether a system with three true values vs one false value is more resistant and harder to create race conditions for.

I am guessing you will be able to do it, but is it harder? Do these table have a practical utility?