Paradoxes of Material Implication

[Logik]

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 if things go wrong if they manage to repeat it. That would discount a principle. That would be a logical conclusion disagreeing with reality.

You are welcome to run my algorithm a million times. It will produce the result you DON'T expect every time.

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.

No. Race conditions are synchronization problems. You are working with stale or incorrect information which is no longer consistent with the state of reality. Or as your grandmother says "Assumption is the mother of all fuckups". Trust but verify etc.

The error in all logic is that it's done on a 2-dimensional, static piece of paper. Whereas the universe is 4 (or more?) dimensional and rather dynamic.

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

Sure. They are as useful as me saying "I have no idea how to parse these things without pen and paper, they are so unintuitive".

Now you have given me a table. Now I don't have to pull out my pen and paper and just consult your table. They save time.

[surreptitious57]

The error in all logic is that its done on a 2 dimensional static piece of paper
Whereas the universe is 4 ( or more ? ) dimensional and rather dynamic

It is not an error if it is tested empirically in order to see how true it really is
It should therefore be a part of the process rather the entirety of the process
The map may be a representation of the territory but they are not actually the same

[Logik]

It is not an error if it is tested empirically in order to see how true it really is
It should therefore be a part of the process rather the entirety of the process
The map may be a representation of the territory but they are not actually the same

You can empirically verify the model: OpenTap1 ∧ OpenTap2 ⇔ BarrelWillEmpty
You can verify it every day for10 years in a row. It works! By any statistical measure that is a significant result.
If your goal is to empty the water the water of THAT particular barrel - open the two taps and that's that. Job done.

And then! A terrible winter. It snows for the 1st time in 100 years! It's -7 degrees Celsius outside.

The water freezes in the barrel and your model no longer works.

[Scott Mayers]

I'm confused at what this thread is about other than to state what material implication is. I still didn't get a direct response from my simplest post and so kind of feel that Gary may be just a sock-puppet. [Something that tends to occur if 'seemingly' new people treat you as irrelevant to respond to without apparent notice.]

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.]

It seems odd to assign the PARTICULAR assignment of values as 'contingent' without multi-valued systems that permit a truth value that is both true and false simultaneously of a single input type, as the OP mentions. If you begin with multi-valued logic, you also then have to include 'cycle' operators.

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".

And the second one is:

(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.

[Logik]

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.

The implication concept is trivial to comprehend. Because it's so strict it's black-and-white.

Validity (logic) 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.

Unless you are flexible on the meaning of 'impossibility' then deduction mandates 100% certainty of conclusions.
By contraposition. Deduction mandates 0% probability of error in conclusions when the premises are true.

If there is ANY probability>0 of A being true and B being false, it's not valid deduction.

It's an ideal. Because in this universe 100% certainty/0% probability of error is.... idealism.

[surreptitious57]

The water freezes in the barrel and your model no longer works

But new knowledge however is acquired [ which is always useful ] which is that when water is frozen it cannot flow
The mistake is to assume that the experiment will produce the same results even when the initial conditions change
When the water is in liquid form and the taps can turn and there are no blockages then the barrel will always empty
But when any of these conditions do not apply then the barrel will not always empty

[Arising_uk]

...
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. ...

I'm way above my pay grade here and have been trying to follow the things you have been pointing out that have been ocurring in CompS and Logic recently but does the above really say this? From what I can understand of what you say the CSC says that a certain kind of computing, functional programming, is the same as a certain type of mathematical logic, intitionistic logic. Is the the same as the claim that all mathematical proofs can be reduced to algorithms?

[Logik]

Is the the same as the claim that all mathematical proofs can be reduced to algorithms?

Yes. Although... language.

Not "reduced to". Transposed/translated to.

Proofs and algorithms are mathematically isomorphic objects.

[Garry G]

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.

I don't think this resolves the dissatisfaction with arguments of this form.

The tables in the OP invalidate P ⇒ (Q ⇒ P) when P is C and Q is N. Therefore on this view: when P is a contingently-true and Q is non-contingently true the argument is falsified. Let me try for a counter example that hopefully illuminates why this analysis makes more sense.

P: our dog Aristotle likes to watch x factor
Q: Aristotle is a mammal

'If our dog Aristotle likes to watch x factor then if he is a mammal he likes to watch x factor.'

Clearly the material nature of Q does make a difference to how the argument sounds and if Q is material then the argument is false. It is true standard tables tell us this argument must be ok for the reasons you articulated. The question is whether we admit the paradox and the resist the urge to read Q as material to the argument. If the reason for insisting it is immaterial is based on truth tables there are operators for a truth table system in the OP that say otherwise. The question is which ones do we trust and which are the more conservative. The system in the OP is weaker than standard logic. It is definitely more conservative.

As an interesting aside: I followed the four valued tables as a prescription to contrive the above example. That is I looked for an example where P is a non contingent proposition and Q contingently-true because that is what the table for the argument said to look for to invalidate the inference. It was then easy to concoct an argument that places this argument form in its worse light.

[Logik]

From what I can understand of what you say the CSC says that a certain kind of computing, functional programming, is the same as a certain type of mathematical logic, intitionistic logic.

Further to this point. In as much as they are isomorphic and any mathematicians should leave it at that, the order in which you say things seems to matter to humans (because it implies the order of causality or something).

Intuitionistic logic e.g intuition can be formalized on a Turing machine.

While it says nothing about the world or how it relates to mathematics - it is simply an argument for (self) expression.
Any phenomenon/process that you experience - you can express (model? describe?) as an algorithm, on a computer.

What's different between a paper and a computer? Time. Computers can represent dynamics/change/calculus. Paper can't.
Because algorithms are step-by-step instructions on how to get the "correct" (desired?) answer given what you know.
Naturally - you already have the "right" answer. You know exactly what you want the algorithm to conclude. You just have to join the dots to the premises.

When formulated that way, it should be a DUH!

[Scott Mayers]

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.

The implication concept is trivial to comprehend. Because it's so strict it's black-and-white.

If it was 'trivial' people wouldn't make interpretation mistakes about what is or is not a 'material implication'.

Validity (logic) 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.

Unless you are flexible on the meaning of 'impossibility' then deduction mandates 100% certainty of conclusions.
By contraposition. Deduction mandates 0% probability of error in conclusions when the premises are true.

If there is ANY probability>0 of A being true and B being false, it's not valid deduction.

It's an ideal. Because in this universe 100% certainty/0% probability of error is.... idealism.

Are you ADDING definitions? I did not particularly say that quote in my post you responded to and it appears that you are responding to it as though I did. I only wrote definitions of terms specifically useful for whole 'well-formed-formula' types. "Validity" is already understood. The concepts that are in issue with me are about collections of all the possible binary truth values of an expression.

A B |.....f0.....f1.....f2.....f3....f4.....f5.....f6.....f7.....f8.....f9....f10...f11...f12...f13...f14...f15
0 0 |..... 1..... 1..... 1..... 1..... 1..... 1..... 1..... 1..... 0..... 0..... 0..... 0..... 0..... 0..... 0..... 0.....
0 1 |..... 1..... 1..... 1..... 1..... 0..... 0..... 0..... 0..... 1..... 1..... 1..... 1..... 0..... 0..... 0..... 0.....
1 0 |..... 1..... 1..... 0..... 0..... 1..... 1..... 0..... 0..... 1..... 1..... 0..... 0..... 1..... 1..... 0..... 0.....
1 1 |..... 1..... 0..... 1..... 0..... 1..... 0..... 1..... 0..... 1..... 0..... 1 .... 0..... 1..... 0..... 1..... 0.....

Each of the above uses only a binary value system, with all possible functions in two variables. If for any new formula that you find that maps to some expected function, it is "Tautologous" if all values are mapped the same expected function as all ones; if "Inconsistent" no mapping occurs anywhere and is a zero for all values; If for all other assignments which are at least one match and one mismatch, these are 'contingent'.

Example, for 'A v B', this is f8 above and is a 'contingency' because for when the input assignments are A = 0 and B = 0, the function value for this is 0 AND all other assignments for this function is 1.

f1 is Tautologous, and f15 is Inconsistent. The rest are Contingent. All but f15 are Consistent.
These are not necessary to define this way but are the ones that the past logicians had convened to agree to define these as for the metaproofs.
When using these to judge comparisons of some formula expression of any type to these main functions regardless of operation choices, ALL values must map as Tautologies for each of these completely for the system to be 'complete'.

[By the way, I recommend saving my html on the chart in a word processor if you want to use it here again. The code for the blank spaces is "<color=#E1EBF2> tag, replacing anything in between to the background color here.]

[Logik]

Are you ADDING definitions? I did not particularly say that quote in my post you responded to and it appears that you are responding to it as though I did. I only wrote definitions of terms specifically useful for whole 'well-formed-formula' types. "Validity" is already understood. The concepts that are in issue with me are about collections of all the possible binary truth values of an expression.

The "quote" is the definition of "logical validity" found here: https://en.wikipedia.org/wiki/Validity_(logic)

The "validity" criterion is explicit about the "impossibility" and says nothing about boolean logic whatsoever. Constraining the definition to two-valued logics is your own addition.

Because the definition is not explicit I am applying it to the broadest possible set of logics possible: ALL logics. Including multi-valued/probability logics.

And so whatever it means for A to be "true" and B to be "false" in a probabilistic logic the "validity" criterion is still pretty much black and white.

IF there is ANY possibility (non-zero probability) that A is true while B is false then it follows that A ⇒ B is NOT valid.

Deduction (and therefore material implication) mandates 100% certainty. In this universe that's a luxury.

[Scott Mayers]

Are you ADDING definitions? I did not particularly say that quote in my post you responded to and it appears that you are responding to it as though I did. I only wrote definitions of terms specifically useful for whole 'well-formed-formula' types. "Validity" is already understood. The concepts that are in issue with me are about collections of all the possible binary truth values of an expression.

The "quote" is the definition of "logical validity" found here: https://en.wikipedia.org/wiki/Validity_(logic)

The "validity" criterion is explicit about the "impossibility" and says nothing about boolean logic whatsoever. Constraining the definition to two-valued logics is your own addition.

Because the definition is not explicit I am applying it to the broadest possible set of logics possible: ALL logics. Including multi-valued/probability logics.

And so whatever it means for A to be "true" and B to be "false" in a probabilistic logic the "validity" criterion is still pretty much black and white.

IF there is ANY possibility (non-zero probability) that A is true while B is false then it follows that A ⇒ B is NOT valid.

Deduction (and therefore material implication) mandates 100% certainty. In this universe that's a luxury.

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.

Note that for the binary valued system for two variables here, you can replace A with P, and B with -P, to see how this demonstrates when these are or are not consistent, contingent, or a tautology. If done so, f15 is 'P and not-P', an Inconsistency, and f1 is 'not(P and not-P) == (P or -P) is a Consistent Tautology of all assignment truth values.

Have you ever actually used multi-valued logic, by the way? Even with three terms, this gets complex fast using only two variables. You require a "cycle" function, along with the traditional operators. All that is necessary is a maximum three-valued logic though, along with the binary ones to make up for all the other multivalued systems. You just have the third value represent "add a value" function (the inconsistent contradiction of f15 for binary would be this for the binary and triggers the trinary value system.) This is the 'dimensioning' factor I mentioned before with you.

That is, if you discover any "inconsistency" in a system, most treat this as a 'stop' because for binary it is a contradiction. But the 'stop' IS a credible function of someone using it. If you find a system "incomplete" for some task, it triggers this function or similar one in a higher valued system. That is the ONLY way you overcome the incompleteness from a logical system: add a dimension. If you then get a result that maps to f15, you can have this 'trigger' the next logic up with another value added.

[Garry G]

I'm confused at what this thread is about other than to state what material implication is.

Like all threads they take on a life of their own and people start chasing all sort of rabbits.

The thread was started to discuss the paradoxes of material implication, for those who think the paradoxes are a real problem the system in the OP is offered as a solution. I'd also like see a considered reaction to a system with three truth values and one false value. Those who really don't think the paradoxes are a problem will no doubt be underwhelmed. For those that don't think the paradoxes are a problem then the purpose is to give reasons why they are a problem.

I still didn't get a direct response from my simplest post and so kind of feel that Gary may be just a sock-puppet. [Something that tends to occur if 'seemingly' new people treat you as irrelevant to respond to without apparent notice.

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

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.

It seems odd to assign the PARTICULAR assignment of values as 'contingent' without multi-valued systems that permit a truth value that is both true and false simultaneously of a single input type, as the OP mentions. If you begin with multi-valued logic, you also then have to include 'cycle' operators.

The motivation for the system and thinking behind it was expanded in a post to Logik at post seven on the first page. It is the post with the figure for the additive and subtractive models of basic colour theory.

The value F = T & F, but this is is not a Dunn Belnap style system. The primary motivation build on a correct analysis of extensions of standard logic. That is Cartesian product systems. When that analysis is done the set {11, 10, 01, 00} which is the first extension has to be interpreted with three truth values if 11 is interpreted as true. I'll not regurgitate the arguments of post seven, it is all there.

I don't code so the reference to cycle operators passes me by right now.

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.

[Logik]

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.