And the next U.K. Prime Minister will be?

[fast]

P1: My hat is black
C: My hat is black

That premise alone leads us to the conclusion. Tie a rope around them and remember the connection. Place one end around P1 and one end around C and remember the link that ties them together.

Now, weaken the argument by saying:

P1: my hat is black
P2: Diesel engines usually last longer than gas engines.
C: my hat is black

The rope of validity remains intact...the argument is valid

Now, crank it up a notch:
P1: my hat is black
P2: Diesel engines usually last longer than gas engines.
P3: it is not the case that Diesel engines usually last longer than gas engines
C: my hat is black

We now have a contradiction, but it doesn’t have the relevancy to conflict with P1 and C. The argument is still valid.

No reason for that. The rope between premise P1 and the conclusion only works for your first two arguments here.

Your last argument says that the conclusion follows from the premises, not just from premise P1 or from P1 and P2. You can't leave any of premises P2 or P3 out.

But, sure, you can define valoud logic all you like as long as you're not trying to convince DBT of your nonsensical logic.

Not that you'd have any chance succeeding if past experience is any guide.
EB

I said the argument is still valid. The argument is valid. Look at the argument, and in it there is a premise that flat out explicitly states the conclusion. The other two premises don’t cause the argument to become invalid. Ironically, it gives us an additional path to conclude validity.

As taught

Look, I think this is all a bit strange. I figure contradictions should just cancel out or something, but that’s not the rules.

As taught

I’m not trying to invent anything here.

 

[steve_bank]

fast

I’m big
I’m not big
Therefore I’m big

Valid

I’m big
I’m not big
Therefore I’m not big
Just as valid

You can not have logical contradictions in the premises.

a = big
!a = not big
(a & !a) is always false logically. Where & is AND in Boolean and you probably call it conjunction.

P1, p2, P3 implies (P1 & P2 & p3)

This is why formal logic and mathematical trumps syllogism.

If you do not grasp that you need a review in logic. Look at the logical fallacy list link.

Sure you can have logical contradictions in the premises.

You cannot say

p1 water is 0 degrees c
p2 water is 100 degrees c
c water is 100 degrees c.

Which is what the first syllogism says. If it were a real problem I would say it is logically indeterminate because the possibility exists for home and store to be the same. Not enough information to make a conclusion.

Assuming in the syllogism home and store are different locations the first is invalid for the same reason the second syllogisms is invalid, sally can not be two places at once.

A pair of contradictory statements are always false.

p1 a
p2 !a
p3 b if a
c b

I'd say invalid because the syllogism defines a contradiction. p3 is impossible because you can not pick a or !a, both are defined simultaneously.

p1 a
p2 if a then turn lights on
p3 if !a then turn lights off

A simple example of how a Boolean equation would be constructed inside an integrated circuit where a represents the state of a switch.

p1 a
p2 a!
p3 if a then turn lights on
p4 if !a then turn lights off

This could not be implemented in logic.

You will have to post an example as to how a valid conclusion can be drawn from a contradiction.

 

[A Toy Windmill]

I started out from the same "premise" and look where I am. The difference between you and me is you ignore what I know, i.e. the "good reason to disallow contradictory premises".

Can you restate the reason for me?

 

[Angra Mainyu]

You will have to post an example as to how a valid conclusion can be drawn from a contradiction.

I already did. In fact, every conclusion can be validly derived from a contradiction. In other words, anything follows from it. But for example:


P1. It is not the case that the Earth is flat.
P2. The Earth is flat.

Then from P2, we obtain:

C1: Either the Earth is flat, or the Moon is made of cheese. [C1 follows from P2 because (P or Q) follows from P.]

Then from C1 and P1 we obtain

C2: The Moon is made of cheese. [C2 follows from C1 and P1 because Q follows from (P or Q) and ¬P.]

So, in two steps, from P1 and P2 we obtain the conclusion that the Moon is made of cheese. This conclusion follows logically from P1 and P2, as only proper logic rules have been used to derive it. Of course, some people claim that those rules are not proper rules. And some of them claim so in an intelligent manner, whereas others - like Speakpigeon's example in this forum - just incur contradiction themselves.

 

[Speakpigeon]

I started out from the same "premise" and look where I am. The difference between you and me is you ignore what I know, i.e. the "good reason to disallow contradictory premises".

Can you restate the reason for me?

It's not published material so I never stated it, beyond the obvious that the principle of explosion, paradoxes of the material implication, and the definition of validity used in mathematical logic all show that mathematical logic is not logic and has nothing to do with logic.

It's toy logic.

Or rather,

Show

sex toy logic...

Show

Very effective to fuck you own mind.

Show

EB

 

[Speakpigeon]

A simple example of how a Boolean equation would be constructed inside an integrated circuit where a represents the state of a switch.

p1 a
p2 a!
p3 if a then turn lights on
p4 if !a then turn lights off

This could not be implemented in logic.

Yes it can and no problem. The result, though, is that the system doesn't do anything. The antecedent of both P3 and P4 are not true so the system doesn't turn the lights on and doesn't turn the lights off.

Where would be the problem? The system doesn't do anything and leaves the lights as they are, either on or off as the case may be.

You could even have two separate inputs giving the state of the switch, one saying "light turned on true" and the other saying "light turned off true" irrespective of whether the light is on or off. No problem whatsoever. And then the bit of code you suggest above will behave just as I explained here.


___________________________

You clearly take Boolean logic to be logic itself. But no, Boolean logic is not logic. It's a simple calculus. A formal method.

It was initially meant as a model of logic, i.e. the logic of human reasoning, but it doesn't work.

Some of it is obviously correct: conjunction, disjunction and negation.

But the implication is junk.

So, it is a formal method, a calculus, but neither logic itself not a correct model of logic.

Everything you claim here is based on this faulty model.

You ain't going nowhere.
EB

 

[Speakpigeon]

Look, I think this is all a bit strange. I figure contradictions should just cancel out or something, but that’s not the rules.

As taught

What would be the justification that these rules should be followed?

There isn't any. So, why are you following them?

All there is are mathematicians talking as if they were the world experts on logic when their expertise is limited to a toy logic.

Mathematical rules refer in really to the toy logic they've invented for themselves, like a child may talk of "mummy" while manhandling a puppet. Mathematicians being the children that they are will speak of "logic" and you fall into the trap thinking they are really talking about logic when in effect they are only really talking about their beloved toy logic.
EB

 

[fast]

You cannot say

I used to say that. Once it got drilled into me that people can say (say, that is) just what it is they’re saying, I struggled to even speak when I recognized how often I said it and tried to refrain. But, that’s an aside.

P1: the water that’s in my pot on my stove is now 72 degrees F.

It’s a premise. It a basis that is apart of an argument that I can utilize to support a conclusion.

P2: it’s not the case that the water that’s in my pot on my stove is now 72 degrees F.

That’s a premise too, and though it conflicts with the previous one, it stands on its own—for you to use or not use at your leisure.

Pick a conclusion. Like posing a question where you can ask one, posit a conclusion.

Here’s one:

C1: some apples are perceived as red

There are two premises. Pick one or the other or both. Do what you can with what you have to work with and try to arrive at the conclusion. Can’t happen.

Try this one:

C2: the water that’s in my pot on my stove is not now 72 degrees F.

P1 isn’t helpful. P2 will work. They conflict, but who cares? Use them in whatever combination that blazes a trail to whatever conclusion is sought to reach.

 

[Speakpigeon]

Is there a reason to deliberately include logical contradictions in the premises?

Good question.

No, there is no good reason to do it deliberately.

However, in real life, people may come to contradict themselves. In particular, it is possible that the technical specification of a system come to include a contradiction. That may be a simple error but it may also be that different people wrote different parts of the document and that nobody spotted the contradiction. I've seen blatant mistakes discovered only late in the development process, so this sort of thing is possible.

Of course, mathematicians had no reason to delve on it.

Except that mathematical logic started essentially with Boole's Algebra, which relied on truth tables to provide definitions, in particular to model the logical implication, what was later called by Russell the "material implication".

This was a gross approximation of the implication but mathematicians were unable to find any better so they settled for it.

Russell discovered paradoxes coming with material implication but thought them harmless. These paradoxes include the case of contradictory premises, and that's where the problem comes from. Because it is an obviously idiotic claim of validity, very early many people have pointed out it was absurd and stupid. To no avail.

We even have now Big Brother mathematicians busy redacting the history of mathematics and less than skilfully explain that, no, there aren't any real paradoxes of the material implication!

Thus, we have this vast literature on the subject and many crackpots, including some mathematicians, who try to "prove" mathematical logic false or try to push alternative systems, each more convoluted and unjustified than the other, and this even though the formula itself is thought mostly harmless.

Claiming valid all implications with a contradictory antecedent is an extraordinary claim and extraordinary claims require extraordinary evidence.

Evidence which is not available because mathematical logic is no empirical science.
EB

 

[Speakpigeon]

Is there a reason to deliberately include logical contradictions in the premises?

Blatant contradictions where one premise is just the denial of another? Perhaps there is no good reason. Perhaps all such arguments are stupid. Perhaps we should be so eager to discard them that we will declare them illegal from the outset. It wouldn't bring the house down to do so.

I come at this from the other direction: until I see a good reason to disallow contradictory premises, they should be permitted. Arguments with contradictory premises may be stupid, but then, there may just be lots of stupid arguments. I know that if you enumerate all the valid statements of propositional logic, most of them will be utterly worthless. But they're still permitted because my distaste isn't compelling enough to disallow them.

Claiming valid all implications with a contradictory antecedent is an extraordinary claim and extraordinary claims require extraordinary evidence.
EB

 

[A Toy Windmill]

You're just on ignore now. Bye.

 

[Speakpigeon]

You're just on ignore now. Bye.

Thanks, that will save me time.

You're the typical specialist and unfortunately you're typical of the average academic. You haven't thought about any of the fundamental issues. Well, to be fair, it's not your pay grade and most people don't do it anyway and if most people did it none of us would be here to talk about it because humanity wouldn't have had the necessary time left to do the basic chores. So, thanks to all the non-thinkers.

Who do we remember in the history of humanity? Well, many "doers", but many of them not for the good reasons, people like Hitler and Stalin. The scientists we remember where all first and foremost thinkers: Galileo, Newton, Einstein and many others. And each time, what they come to say inevitably shows people before just didn't understand, and they were initially criticised when not threatened to be burnt at the stake. It's life. People don't like contradictions because they really don't like when you tell them something that shows there's a job left to be done.

You have absolutely nothing to tell me. I already have identified the crucial points and I already understand the main issues. I now know that mathematicians just don't understand logic at all. They don't even understand what logic is. Philosophers dito, computer scientists dito. It's definitely pathetic but it's life and maybe we have more serious problems ahead of us. But if we do, probably it's not a good thing that we should be under the impression that mathematical logic is all good when in fact it's just irrelevant.

I'm not the only one to say this. It's not even new. Mathematicians themselves have disagreed and keep disagreeing about it and about what previous mathematicians have done, which is why you have different "methods of logic" to being with, something they are not even prepared to admit that it is a symptom. Everything is honky dory. No paradoxes. No contradiction between different theories. No disagreement with Aristotle. No difference between mathematical logic and the logic of most human beings. LOL. Everything is under control! Excellent.

You are a caricature of the self-sufficient academic. You're not prepared to engage or even acknowledge the problem and everything is fine. We understand things and you don't because we are the academics. And I'm a "crackpot". LOL. I certainly don't know many crackpots who can articulate their ideas like I do and you certainly can do any better.

Christ, it's not even saving me time after all!
EB

EDIT
And once you ignore me, you won't have anyone to talk to about logic!

 

[Angra Mainyu]

You're just on ignore now. Bye.

Thanks, that will save me time.

You're the typical specialist and unfortunately you're typical of the average academic. You haven't thought about any of the fundamental issues. Well, to be fair, it's not your pay grade and most people don't do it anyway and if most people did it none of us would be here to talk about it because humanity wouldn't have had the necessary time left to do the basic chores. So, thanks to all the non-thinkers.

Who do we remember in the history of humanity? Well, many "doers", but many of them not for the good reasons, people like Hitler and Stalin. The scientists we remember where all first and foremost thinkers: Galileo, Newton, Einstein and many others. And each time, what they come to say inevitably shows people before just didn't understand, and they were initially criticised when not threatened to be burnt at the stake. It's life. People don't like contradictions because they really don't like when you tell them something that shows there's a job left to be done.

You have absolutely nothing to tell me. I already have identified the crucial points and I already understand the main issues. I now know that mathematicians just don't understand logic at all. They don't even understand what logic is. Philosophers dito, computer scientists dito. It's definitely pathetic but it's life and maybe we have more serious problems ahead of us. But if we do, probably it's not a good thing that we should be under the impression that mathematical logic is all good when in fact it's just irrelevant.

I'm not the only one to say this. It's not even new. Mathematicians themselves have disagreed and keep disagreeing about it and about what previous mathematicians have done, which is why you have different "methods of logic" to being with, something they are not even prepared to admit that it is a symptom. Everything is honky dory. No paradoxes. No contradiction between different theories. No disagreement with Aristotle. No difference between mathematical logic and the logic of most human beings. LOL. Everything is under control! Excellent.

You are a caricature of the self-sufficient academic. You're not prepared to engage or even acknowledge the problem and everything is fine. We understand things and you don't because we are the academics. And I'm a "crackpot". LOL. I certainly don't know many crackpots who can articulate their ideas like I do and you certainly can do any better.

Christ, it's not even saving me time after all!
EB

EDIT
And once you ignore me, you won't have anyone to talk to about logic!

It is not acceptable to make false and disparaging accusations that you ought to know are false. To any interested readers, I would suggest that you take a look at this post, or really pretty much any of the exchanges between Speakpigeon and me, or Speakpigeon and Bomb#20, or Speakpigeon and A Toy Windmill in the Logic and Epistemology forum. Speakpigeon does not understand logic, but that is not a big problem. Speakpigeon's unjust attacks on other people based on falsehoods that Speakpigeon ought not to believe are.

 

[steve_bank]

You will have to post an example as to how a valid conclusion can be drawn from a contradiction.

I already did. In fact, every conclusion can be validly derived from a contradiction. In other words, anything follows from it. But for example:


P1. It is not the case that the Earth is flat.
P2. The Earth is flat.

Then from P2, we obtain:

C1: Either the Earth is flat, or the Moon is made of cheese. [C1 follows from P2 because (P or Q) follows from P.]

Then from C1 and P1 we obtain

C2: The Moon is made of cheese. [C2 follows from C1 and P1 because Q follows from (P or Q) and ¬P.]

So, in two steps, from P1 and P2 we obtain the conclusion that the Moon is made of cheese. This conclusion follows logically from P1 and P2, as only proper logic rules have been used to derive it. Of course, some people claim that those rules are not proper rules. And some of them claim so in an intelligent manner, whereas others - like Speakpigeon's example in this forum - just incur contradiction themselves.

The fact that a contradiction can be ort of an anayisi in a more lengthy argument. But in a conclusion you can select one side or the other in a contradiction.

p1 a
p2 !a
p3...

P1 and p2 are declarations not subject to conditions. I'd accept the possibility further premises in an argument could resolve the contradiction.

I would not accept as valid an argument based soley on a contradiction.

In the car syllogism Sally can't be in two places at once regales of the rest of the argument.

a
!a
conclusion a

Invalid.

 

[fast]

If missy is in her car, her car is in South Carolina
If missy is in her car, her car is in Georgia
If missy is in her car, her car is in Alabama
If missy is in her car, her car is in North Carolina
If missy is in her car, her car is in Florida
Missy is in her car
Therefore, her car is in Florida

Here the contradiction is clear, a car can not be in two places at once.

Yet your argument here is not valid but not for the reason you give!

It is not valid only because it doesn't include the premises necessary to say that Georgia is not Alabama etc.

Because, from a logical validity point of view, it is possible that Georgia is Alabama unless a premise specifies otherwise.

Thus, there's no premises in your argument precluding the car being in all places at once because they are the same place.

Still, yes, it is no valid. But not for the reason you give.

Even your claim that "a car can not be in two places at once" could not be left implicit. There is nothing in logic that says something cannot be in two places at once.

Still a long way to go, Steve.
EB

I don’t see why that’s necessary. They are words with referents that can be fact checked. And by “Georgia”, it should be obvious I mean the state. In fact, if what I meant isn’t the interpretelation, it’s not apart of my argument.

 

[steve_bank]

P1. It is not the case that the Earth is flat.
P2. The Earth is flat.

Then from P2, we obtain:

C1: Either the Earth is flat, or the Moon is made of cheese. [C1 follows from P2 because (P or Q) follows from P.]

Then from C1 and P1 we obtain

C2: The Moon is made of cheese. [C2 follows from C1 and P1 because Q follows from (P or Q) and ¬P.]

You can not arbitrarily ignore one half of a contradiction p1 p2. In the conclusion the moon is made of chees is incorrect.

 

[fast]

You will have to post an example as to how a valid conclusion can be drawn from a contradiction.

I already did. In fact, every conclusion can be validly derived from a contradiction. In other words, anything follows from it. But for example:


P1. It is not the case that the Earth is flat.
P2. The Earth is flat.

Then from P2, we obtain:

C1: Either the Earth is flat, or the Moon is made of cheese. [C1 follows from P2 because (P or Q) follows from P.]

Then from C1 and P1 we obtain

C2: The Moon is made of cheese. [C2 follows from C1 and P1 because Q follows from (P or Q) and ¬P.]

So, in two steps, from P1 and P2 we obtain the conclusion that the Moon is made of cheese. This conclusion follows logically from P1 and P2, as only proper logic rules have been used to derive it. Of course, some people claim that those rules are not proper rules. And some of them claim so in an intelligent manner, whereas others - like Speakpigeon's example in this forum - just incur contradiction themselves.

The fact that a contradiction can be ort of an anayisi in a more lengthy argument. But in a conclusion you can select one side or the other in a contradiction.

p1 a
p2 !a
p3...

P1 and p2 are declarations not subject to conditions. I'd accept the possibility further premises in an argument could resolve the contradiction.

I would not accept as valid an argument based soley on a contradiction.

In the car syllogism Sally can't be in two places at once regales of the rest of the argument.

a
!a
conclusion a

Invalid.

About the car example:

Because a contradiction guarantees validity, I opted to exclude that path for validity by including contraries—where only the falsity of a premise guaranteed unsoundness but not invalidity.

P1: I’m in South Carolina
P2: I’m in Florida
C: I’m in South Carolina

That’s unsound yet valid

It’s unsound because not all premises are true
It’s valid —but not because of a contradiction —there is no contradiction

 

[steve_bank]

P1. It is not the case that the Earth is flat.
P2. The Earth is flat.

Then from P2, we obtain:

C1: Either the Earth is flat, or the Moon is made of cheese. [C1 follows from P2 because (P or Q) follows from P.]

Then from C1 and P1 we obtain

C2: The Moon is made of cheese. [C2 follows from C1 and P1 because Q follows from (P or Q) and ¬P.]

You can not arbitrarily ignore one half of a contradiction p1 p2. In the conclusion the moon is made of chees is incorrect. There is no support in the premises to say the moon is cheese. Conclusion as written does not follow from p1.

p1 and p2 are asserted as fact in the argument. The devil is in the details. In your mind it may be p1 OR p2, but that is not how it is written. Try expressing it in formal logic.

P1. Some think it is not the case that the Earth is flat.
P2. Some think the Earth is flat.
C The Earth may be made of cheese or it may be flat. .

Reasoning I would accept.

I'd say your reasoning applied to real situations would get you in trouble. Clever sophistry and footwork.

 

[Angra Mainyu]

You will have to post an example as to how a valid conclusion can be drawn from a contradiction.

I already did. In fact, every conclusion can be validly derived from a contradiction. In other words, anything follows from it. But for example:


P1. It is not the case that the Earth is flat.
P2. The Earth is flat.

Then from P2, we obtain:

C1: Either the Earth is flat, or the Moon is made of cheese. [C1 follows from P2 because (P or Q) follows from P.]

Then from C1 and P1 we obtain

C2: The Moon is made of cheese. [C2 follows from C1 and P1 because Q follows from (P or Q) and ¬P.]

So, in two steps, from P1 and P2 we obtain the conclusion that the Moon is made of cheese. This conclusion follows logically from P1 and P2, as only proper logic rules have been used to derive it. Of course, some people claim that those rules are not proper rules. And some of them claim so in an intelligent manner, whereas others - like Speakpigeon's example in this forum - just incur contradiction themselves.

The fact that a contradiction can be ort of an anayisi in a more lengthy argument. But in a conclusion you can select one side or the other in a contradiction.

p1 a
p2 !a
p3...

P1 and p2 are declarations not subject to conditions. I'd accept the possibility further premises in an argument could resolve the contradiction.

I would not accept as valid an argument based soley on a contradiction.

In the car syllogism Sally can't be in two places at once regales of the rest of the argument.

a
!a
conclusion a

Invalid.

There is nothing to resolve. Contradictions are not the sort of things that are resolved. My point is that my argument above (i.e., the one you quote) is one in which the conclusion follows logically from P1 and P2, as only proper logic rules have been used to derive it.

 

[Angra Mainyu]

P1. It is not the case that the Earth is flat.
P2. The Earth is flat.

Then from P2, we obtain:

C1: Either the Earth is flat, or the Moon is made of cheese. [C1 follows from P2 because (P or Q) follows from P.]

Then from C1 and P1 we obtain

C2: The Moon is made of cheese. [C2 follows from C1 and P1 because Q follows from (P or Q) and ¬P.]

You can not arbitrarily ignore one half of a contradiction p1 p2. In the conclusion the moon is made of chees is incorrect. There is no support in the premises to say the moon is cheese. Conclusion as written does not follow from p1.

p1 and p2 are asserted as fact in the argument. The devil is in the details. In your mind it may be p1 OR p2, but that is not how it is written. Try expressing it in formal logic.

P1. Some think it is not the case that the Earth is flat.
P2. Some think the Earth is flat.
C The Earth may be made of cheese or it may be flat. .

Reasoning I would accept.

I'd say your reasoning applied to real situations would get you in trouble. Clever sophistry and footwork.

The conclusion follows from P1 and P2, as explained. It does not follow from P1 alone. There is no sophistry in my argument. In formal logic, it is very easy.

P1: P
P2: ¬P.
C1: P or Q. [this follows from P]
C2: Q. [this follows from C1 and P2].