Traditional logic and its usefulness in the past

[PainterGuy]

Hi,

I was reading this article, https://writings.stephenwolfram.com/2015/11/george-boole-a-200-year-view/, and the following excerpt is taken from it.

When George Boole came onto the scene, the disciplines of logic and mathematics had developed quite separately for more than 2000 years. And George Boole’s great achievement was to show how to bring them together, through the concept of what’s now called Boolean algebra. And in doing so he effectively created the field of mathematical logic, and set the stage for the long series of developments that led for example to universal computation.

When George Boole invented Boolean algebra, his basic goal was to find a set of mathematical axioms that could reproduce the classical results of logic. His starting point was ordinary algebra, with variables like x and y, and operations like addition and multiplication.

George Boole came up with his Boolean logic around 1847.

I think that it was Claude Shannon who showed that any numerical operation can be carried out using the basic building blocks of Boolean algebra; as this is also mentioned in this video around 17:00 /watch?v=IgF3OX8nT0w (put youtube.com in front).

Contrary to widespread belief, Boole never intended to criticize or disagree with the main principles of Aristotle's logic. Rather he intended to systematise it, to provide it with a foundation, and to extend its range of applicability.

Source: https://en.wikipedia.org/wiki/George_Boole#Symbolic_logic

My Question:
I understand that these days Boolean logic is the basis for digital computation and its use is widespread in terms of applicability and usefulness.

What was logic really used for, for example, before George Boole formalized it, or even after its formalization by Boole but before adoption as a tool for digital computation? I'm sure it must have had some useful applications before it became the basis for digital computation but mostly, as a layman, I come across examples as the one shown below which is just interesting but IMHO not very useful in terms of real world application. Could you please help me it? Thanks!

All men are mortals
All Socrates are men
All Socrates are mortals
Source: https://en.wikipedia.org/wiki/Term_logic


Helpful links and notes:
https://en.wikipedia.org/wiki/Logic
Aristotelian logic is also called "term logic", traditional logic, and syllogistic logic.

 

[MathematicalPhysicist]

There's a series of books called "Handbook of the History of Logic" by Dov Gabbay.

You might find it interesting to you.

 

[Mark44]

What was logic really used for, for example, before George Boole formalized it, or even after its formalization by Boole but before adoption as a tool for digital computation? I'm sure it must have had some useful applications before it became the basis for digital computation but mostly, as a layman, I come across examples as the one shown below which is just interesting but IMHO not very useful in terms of real world application. Could you please help me it? Thanks!

Logic was used as the basis for deductions, such as this simple one.
According to Pythagoras, all right triangles have sides that satisfy the equation $c^2 = a^2 + b^2$, with c being the length of the hypotenuse, and a and b being the legs of the right triangle.
There is a right triangle whose legs are 5 and 12 units.
Therefore, the hypotenuse of this triangle is 13 units in length.

In addition, when I studied geometry back in the early 60s, we proved a great number of theorems using logical arguments, such as this one: the opposite angles made by a transversal that cuts two parallel lines are equal in measure.

Logic is also the basis of legal arguments, in which a set of given facts leads to an inescapable conclusion.

 

[PeroK]

Logic is also the basis of legal arguments, in which a set of given facts leads to an inescapable conclusion.

Ha ha! If that were the case, there would be no need for the prosecutor and the defence!

 

[Mark44]

Ha ha! If that were the case, there would be no need for the prosecutor and the defence!

Of course, the prosecutor will lay out a logical argument that is intended to convince the jury of an inescapable conclusion. The defense will also attempt to use logic to contradict the prosecutor's argument, and if that doesn't work, might see if an emotional argument will do the trick.

 

[TeethWhitener]

Anyone doing any kind of scholarly work would never be taken seriously if their arguments were not logically rigorous. This was as true 2000 years ago as it is today (maybe more so). Some scholars (Euclid, Newton, Spinoza, for example) chose to make the logical steps of their arguments quite clear, and some (Descartes, Plato, etc) wrote in a more literary style, obscuring the logic of their arguments somewhat. But make no mistake: if these thinkers hadn’t grounded their arguments in basic logic, we would not hold nearly the same regard for them that we do today.

 

[PeroK]

I come across examples as the one shown below which is just interesting but IMHO not very useful in terms of real world application. Could you please help me it? Thanks!

All men are mortals
Socrates is a man
Socrates is mortal

The first point is that a basic lack of logical thought is at the root of a lot of political disputes. For example, someone politically minded might dispute that logic with the counterexample: "my wife is mortal, but she's not a man and certainly not called Socrates". At this point his political supporters laugh and mock your crazy assertion.

There is, in fact, a long list of logical fallacies that you will find everywhere if you look closely. There are dozens of videos on this. E.g.

 

[PeroK]

Of course, the prosecutor will lay out a logical argument that is intended to convince the jury of an inescapable conclusion. The defense will also attempt to use logic to contradict the prosecutor's argument, and if that doesn't work, might see if an emotional argument will do the trick.

With "real" logic that doesn't work. Both legal arguments are explicitly based on a priori conclusion.

There was an infamous case in the UK where the defence lawyer, by luck, found the key evidence that showed their client's innocence. The prosecution had written on it "do not show to the defence"!

 

[TeethWhitener]

With "real" logic that doesn't work. Both legal arguments are explicitly based on a priori conclusion.

For OP’s sake, I feel it’s important to point out that it’s perfectly possible for two parties to logically reach different conclusions if they start from different premises. That’s in fact what a lot of scholarly (and legal) argumentation is based on: trying to show that one or more premises in a logical argument is untenable.

 

[Stephen Tashi]

Hi,

I was reading this article, https://writings.stephenwolfram.com/2015/11/george-boole-a-200-year-view/, and the following excerpt is taken from it.



George Boole came up with his Boolean logic around 1847.

What Boole invented was not boolean logic or boolean algebra in the modern sense of those terms.

https://www.math.uwaterloo.ca/~snburris/htdocs/MYWORKS/PREPRINTS/aboole.pdf

 

[jedishrfu]

Of course, the prosecutor will lay out a logical argument that is intended to convince the jury of an inescapable conclusion. The defense will also attempt to use logic to contradict the prosecutor's argument, and if that doesn't work, might see if an emotional argument will do the trick.

Sadly, this only partially true. Each side tries to frame their arguments with their plausible but flawed ie cherry picked logic relying on juror preconcieved biases, prejudices and tv dramas.

In one case, the defense counsel convinced the jury that the victim had held his cellphone in a menacing manner making it look like a gun to contend that the defendant shot him in self defense.

In another case, the prosecution withheld evidence, a bandana found a hundred yards from the crime scene. The defendant, the husband of the victim was sentenced to many years in jail because of the biased belief that lack of evidence means the husband must have done it even though the prosecutors case was weak and there was other evidence that the husband did not do it.

The innocence project found that evidence and after a big court fight got it tested with the latest DNA science and discovered the real killer as it had his DNA and the DNA of the victim.

Even before a trial begins the defense and prosecutors survey the jury pool looking for jurors that are hard to sway and eliminate them hoping the ones selected will come in with an open mind.

Both sides will also try to frame their case with statements like “you've had kids, they get in trouble but they turn out okay” if the defendant is young spoken repeatedly during vior dire.

There’s logic, psychology, and science at play as well as poker and chess strategies too.

 

[PainterGuy]

Thank you for the help, everyone!

In my humble opinion, presently, when it comes to science, logic is more of a common sense approach. Is science really defined by logic? For example, one might have argued before the relativity that if Newton's laws worked in everyday situations, therefore, logically speaking, they would also work the same way everywhere and at any speed. This is not true therefore the logic was false. The better version of logic would be that one can never be sure. Logically speaking, even there are going to be cases when the relativity might not produce valid results in certain situations.

I don't know what kind of special role logic as a separate discipline played in the past.


Helpful links:
/watch?v=K4ChzesrWKI (put youtube.com in front)
/watch?v=DRx-3jvC918 (put youtube.com in front)

 

[PeroK]

For example, one might have argued before the relativity that if Newton's laws worked in everyday situations, therefore, logically speaking, they would also work the same way everywhere and at any speed.

That is precisely illogical. I think that's what's called a faulty generalisation. That might have been an assumption, but it was not a logical conclusion.

 

[TeethWhitener]

Seconded @PeroK . A logical assertion would have been: if Newton’s laws work all the time, then they’ll work at high relative velocities. They don’t work at high relative velocities; therefore (by modus tollens) they don’t work all the time.

 

[PainterGuy]

So, the point is that science guided itself. The so-called logic is just a way to codify our learning experience to guide us in future. In this case, never take anything with certainty, or there are limits of every theory. I still don't see why they had logic as a separate discipline. In case of science, it should just have been a byproduct of scientific endeavors. Sorry!

 

[PeroK]

So, the point is that science guided itself. The so-called logic is just a way to codify our learning experience to guide us in future. In this case, never take anything with certainty, or there are limits of every theory. I still don't see why they had logic as a separate discipline. In case of science, it should just have been a byproduct of scientific endeavors. Sorry!

Science can't work by logic alone. It needs inspiration, insights, and experimental expertise. Outside of the mathematical logic of the models, logic itself perhaps doesn't play a great part. In fact, early science was (IMO) held back by religious and philosophical thinking. There were too many sacred principles and too much reliance was placed on pure thought as a source of scientific ideas. IMO, it wasn't until the 19th Century that science really broke free. Mathematics grew enormously in areas, such as Riemannian manifolds, that had no grounding in the pure philosphical ideas of previous times. And, of course, there was Darwin's Theory of Evolution. Physics went further in the 20th Century with SR, GR and QM etc.

Logic itself doesn't achieve so much, because it must build on whatever founding assumptions you make. Although, its become fundamental to the foundations of mathematics, of course.

But, false logic can lead you astray. In some ways, the importance of logic is to avoid systematic errors in thinking. That's it's importance to science, IMO. It's the basis for systematic thought and analysis of ideas and data.

 

[fresh_42]

Logic itself doesn't achieve so much

Good that there aren't logicians around.

It is getting interesting if we consider constructive mathematics or ternary logics. I wonder if we could handle uncertainty by one of those. It would be a fundamental break with our current paradigms, but who knows.

 

[TeethWhitener]

Good that there aren't logicians around.

It is getting interesting if we consider constructive mathematics or ternary logics. I wonder if we could handle uncertainty by one of those. It would be a fundamental break with our current paradigms, but who knows.

That’s actually one big driver for the development of paraconsistent logics—they can accommodate contradictions without allowing the principle of explosion that occurs in classical logic.
https://en.m.wikipedia.org/wiki/Paraconsistent_logic

 

[MathematicalPhysicist]

That’s actually one big driver for the development of paraconsistent logics—they can accommodate contradictions without allowing the principle of explosion that occurs in classical logic.
https://en.m.wikipedia.org/wiki/Paraconsistent_logic

seems like another fad...

 

[TeethWhitener]

seems like another fad...

Well, given that we have no way of showing that most interesting axiomatic systems are consistent from within themselves, the (epistemic) possibility that, e.g., ZF contains a contradiction is ever-present. There are two ways to approach this possibility: 1) replace the axioms with different axioms and cross your fingers and hope and pray that this system doesn’t contain a contradiction (and rinse and repeat as necessary), or 2) develop a formal system that can accommodate contradictions. Personally, I think both ways are worth pursuing.

 

[MathematicalPhysicist]

Well, given that we have no way of showing that most interesting axiomatic systems are consistent from within themselves, the (epistemic) possibility that, e.g., ZF contains a contradiction is ever-present. There are two ways to approach this possibility: 1) replace the axioms with different axioms and cross your fingers and hope and pray that this system doesn’t contain a contradiction (and rinse and repeat as necessary), or 2) develop a formal system that can accommodate contradictions. Personally, I think both ways are worth pursuing.

I read one book on Paraconsistent Logics and it didn't convince me we should abandon classical logic.
But for the sake of curiosity it's good to keep studying this type of logic; there's also Paracomplete Logics.
But besides statements like the liar I don't see any statements used in everyday language that are true contradictions.
Well, until someone will find an inconsistency in ZF; why worry?
(-:

 

[PainterGuy]

Thanks a lot, everyone, for the help and your time!

Logic itself doesn't achieve so much, because it must build on whatever founding assumptions you make. Although, it's become fundamental to the foundations of mathematics, of course.

But, false logic can lead you astray. In some ways, the importance of logic is to avoid systematic errors in thinking. That's it's importance to science, IMO. It's the basis for systematic thought and analysis of ideas and data.

It was really helpful.