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

[Speakpigeon]

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.

Interesting point.

You're correct that this is what we do in informal arguments, i.e. whenever we have a conversation about the real world rather than about a ... formal argument. We assume the usual interpretation of the vocabulary used.

However, in this case, the question is essentially whether an argument is sound, not whether it is valid.

Arguments in ordinary conversations are nearly always very simple, broadly what Aristotle described in details. Thus, validity is usually not an issue, at least not with most people. The arguments used are very nearly always valid. Disagreements usually are about the truth of the premises.

Arguments are not even usually made entirely explicit so that we use enthymemes rather than proper syllogisms.

But we're discussing validity here, not soundness. And the key is to come back to the distinction between validity and soundness.

Validity is assessed on a 100% formal basis, meaning that you can't assume that the same word used several times in the argument always refers necessarily to the same thing, except for the logical vocabulary such as "and", "not", "is" etc.

In the argument here, there is no formal basis to assume that Georgia and Alabama for example don't possibly overlap, to some extent or even completely, and if they did, the car could be both in Georgia and Alabama.

But it goes in fact much deeper than that. Logic doesn't say anything about the properties of space. In particular, logic doesn't say that A included in B and B included in C implies A included in C. The spatial notion of "inclusion" isn't a logical concept. Thus, if you want to make up an argument about space, you need to specify whatever properties of space you want to take into account and you can only do that through some more premises.

Where the usual interpretation of the terms used comes in is when you assess soundness. By definition, you assess soundness on the basis of your assessment of the truth of the premises and you can only do that by accepting some interpretation of the words used. You can't assess the soundness of an argument about the Moon without deciding first what the word "Moon" will be taken as meaning in this instance. And then, whether you accept a premise saying for example that the Moon is a satellite of Earth is up to you but it will inevitably depends on what interpretation you accept for the word "Moon". Note also that assessing soundness requires that you accept an interpretation without the support of any good logical reason. In other words, soundness is assessed on the basis of what you believe about the world. You can always try to analyse concepts to get down to fundamental entities, but your argument will have to mention such entities, using some words, and it will be up to you to decide whether what the premises say about them is true or not.

Thus, validity is necessary to soundness, but it is not sufficient. You can't completely reduce soundness to validity. Physics in particular may be understood as the theoretical analysis of the concepts used in our arguments, to get down to arguments only using fundamental entities such as quarks and what not, but even physics has to stop the analysis at some point and decide of the truth of the premises, and this not on logical ground but on empirical ground. Thus, application of logic is not ... logical. At least not even 50% logical, which explains why we have never ending debates whatever the issue. How do we get to agree that God exists on logical ground only?
EB

 

[Speakpigeon]

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

Indeed.

However, here, you shamelessly contradict yourself since you are now willing to assess validity precisely by ignoring the usual interpretation of South Carolina and Florida, something you said "wasn't necessary". Here:

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.

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.

To make the argument not valid, you would need to add a premise P3 saying that if you are in Florida, then you're not in South Carolina.
EB

 

[Speakpigeon]

as only proper logic rules have been used

Proper logic rules? Who decides what are proper logic rules?

Ah, yes, I see, mathematicians do ...

No, not for me, and obviously not for most people, here and anywhere. Never.
EB

 

[Angra Mainyu]

as only proper logic rules have been used

Proper logic rules? Who decides what are proper logic rules?

Ah, yes, I see, mathematicians do ...

No, not for me, and obviously not for most people, here and anywhere. Never.
EB

No, no one "decides" what they are. Rather, one assesses what they are. Not everyone is equally good at figuring that out, though. But the rules that were used are that from (P or Q) and ¬P, Q follows, and from P, (P or Q follows). If part of your brain is malfunctioning and you fail to see that those things do follow, that is a problem for you and for others you accuse for no good reason, but such is life.

Interestingly, nearly everyone who studies logic - logicians, philosophers, mathematicians - realize that those rules are proper.

 

[fast]

Indeed.

However, here, you shamelessly contradict yourself since you are now willing to assess validity precisely by ignoring the usual interpretation of South Carolina and Florida, something you said "wasn't necessary". Here:

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.

To make the argument not valid, you would need to add a premise P3 saying that if you are in Florida, then you're not in South Carolina.
EB

I don’t think so. I could be wrong.

 

[Speakpigeon]

No, no one "decides" what they are. Rather, one assesses what they are.

So mathematicians waste their time assessing rules without deciding which are correct. I gets better an better.

Not everyone is equally good at figuring that out, though.

So who is going to decide who is good at it?

Ah, yes, I see, nobody ever does. Instead, mathematicians just assess. Assess is the new ultimate reality.

If part of your brain is malfunctioning

Sure, if, but how do you prove my brain is malfunctioning? Do you just "assess"?

Interestingly, nearly everyone who studies logic - logicians, philosophers, mathematicians - realize that those rules are proper.

Proper? What do that even mean in this context?

Rules are conventions decided by people. Proper just means these dudes agree among themselves. Who cares?

Again, why would these rules be correct? Rules are never correct. They are conventions. Those who agree with the conventions will judge them "proper". Beating your children every now and then could be a proper rule. So, you may choose to care about it but other people have no good reason to do that. A good reason would be that a method of logic would be the correct model of human logic.

Interestingly, most people I have tested with contradictory premises disagree with mathematicians. Obviously, they fail to be biased.

And most logicians before Boole didn't accept the principle of explosion, so even your claim here is junk.
EB

 

[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.

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].

I am being serious and blunt. In the environments I worked in if you put forth the kind of reasoning you used to declare the moon is made of chees on a real problem you would be labeled incompetent. You do not seem to understand logic.

 

[fast]

There’s an old saying about getting out of a situation the same way you got into it, but life’s not always that easy. The difficulties of doing things one way is not always equivalent to doing things inversely. Jumping off a diving board and into the water is one thing, but jumping out the water and onto the diving board is quite another.

Taking the premises of an argument and propeling our way to a conclusion seems substantively more difficult than safely standing upon a conclusion and gazing out in observance of premises.

If I start with the conclusion “I have salt OR I have pepper,” I can leisurely browse at the available premises:

P1: I have salt
C: I have salt or I have pepper

From that vantage point, I can also start with the conclusion “The Earth is flat OR the moon is made of cheese.”
P1: the earth is flat
C: the earth is flat or the moon is made of cheese

It’s from this perspective where things seem easy—like falling into a hole

But, stand it on its end with no gun powder to power the cannon and I find complexity building.

From a premise of “I have salt” to rolling out a conclusion with another wording seems like something is required. If I program a computer to display only the premises given, then never shall it display something not programmed.

I can go from
C: I have salt or I have pepper
And know that one of two different possible premises are lurking

But flip it and all hell breaks loose
P: I have salt
C: I have salt or Elvis is playing the guitar on the moon made of purple cheese

Who saw that coming?

 

[A Toy Windmill]

It is non squitter. The conclusion does not follow from the premises. The syllogism is not valid, conclusion does not follow from premise.

P1-P3 are irrelevant to C. C does not follow from P4 P5.

You are wrong on the irrelevance of P3. The conclusion follows from P3, P4 and P5 and can be formalized:

P3) JH != JC
P4) PM = BJ || PM = JH
P5) PM = JC
C) PM = BJ

From P4 and P5, we have (by substituting PM for JC in the left disjunct)

C1) PM = BJ || JC = JH

From P3, we rule out JC = JH in P4, leaving

C) PM = BJ

 

[Angra Mainyu]

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].

I am being serious and blunt. In the environments I worked in if you put forth the kind of reasoning you used to declare the moon is made of chees on a real problem you would be labeled incompetent. You do not seem to understand logic.

I am being serious and blunt. In the environments I work in, if you imply that I ever suggested that the kind of reasoning I put forth would provide any good reason to even suspect that the Moon is made of cheese, you would be corrected immediately because you do not even understand the arguments you are replying to. But my logic is correct.

 

[A Toy Windmill]

There’s an old saying about getting out of a situation the same way you got into it, but life’s not always that easy. The difficulties of doing things one way is not always equivalent to doing things inversely. Jumping off a diving board and into the water is one thing, but jumping out the water and onto the diving board is quite another.

Taking the premises of an argument and propeling our way to a conclusion seems substantively more difficult than safely standing upon a conclusion and gazing out in observance of premises.

If I start with the conclusion “I have salt OR I have pepper,” I can leisurely browse at the available premises:

P1: I have salt
C: I have salt or I have pepper

From that vantage point, I can also start with the conclusion “The Earth is flat OR the moon is made of cheese.”
P1: the earth is flat
C: the earth is flat or the moon is made of cheese

It’s from this perspective where things seem easy—like falling into a hole

But, stand it on its end with no gun powder to power the cannon and I find complexity building.

From a premise of “I have salt” to rolling out a conclusion with another wording seems like something is required. If I program a computer to display only the premises given, then never shall it display something not programmed.

I can go from
C: I have salt or I have pepper
And know that one of two different possible premises are lurking

But flip it and all hell breaks loose
P: I have salt
C: I have salt or Elvis is playing the guitar on the moon made of purple cheese

Who saw that coming?

There's another way to flip this around. Suppose you start at some conclusions and you want a computer to come up with premises from which to derive them. Mathematicians like to do this when they axiomatize a mathematical field down to simple principles. But scientists do it too. For instance, Newton might start with Kepler's laws for planetary motion, and wish to derive the simplest principle that can produce those laws.

So we're doing something like:

C: Planets move according to Kepler's laws.

and now we go looking for premises. And behold! I've found two:

P1: Planets move according to Kepler's laws.
P2: Elvis is playing the guitar on the moon made of purple cheese

Again. WTF?

 

[Angra Mainyu]

No, no one "decides" what they are. Rather, one assesses what they are.

So mathematicians waste their time assessing rules without deciding which are correct. I gets better an better.

You grossly misrepresent what I said.
People assess whether rules are correct. They do not decide it. Either they are correct, or not.

So who is going to decide who is good at it?

Ah, yes, I see, nobody ever does. Instead, mathematicians just assess. Assess is the new ultimate reality.

No, everyone assesses questions they think about. Some people think about logic rules. Most don't. Some people are good at logic. Others aren't.

Sure, if, but how do you prove my brain is malfunctioning? Do you just "assess"?

I do not prove it - proofs are for mathematics -, but I have shown it conclusively to any reader being rational and paying attention - for example, by showing that some of your beliefs on these matter contradict others.
Of course, you will never realize that. But similarly, if someone claims the Moon Landing was a hoax, and insists on a conspiracy theory after they have been shown the relevant evidence, I reckon their brain is malfunctioning, as they are making improper epistemic probabilistic assessments. Others will insist that Jesus walked on water. And nearly all of those will insist that I'm the one in error. Well, such is life. They're still wrong. As are you.

Proper? What do that even mean in this context?

I'm using the word "proper" in the usual sense of the word, in English.

Rules are conventions decided by people. Proper just means these dudes agree among themselves. Who cares?

Some rules are conventions decided by people. But usually they decide them for some reason. Other rules are not conventions, but the rules of properly functioning English-speaking humans, which are the same for at least pretty much all present-day languages (I'm not sure whether there are exceptions in humans having language; it seems improbable).

And before you misrepresent my position again (I can see your error coming), no, I did not claim that there is not a single proper human logic for all colloquial languages. I think there probably is, and if there isn't, at least there is one for each language (and it is common to many if not all of them). In formal languages, one can restrict logic in different ways (as one does in mathematics), though one uses our intuitive logic in the metalanguage (my reply that we do not know whether humans have a natural capacity, etc., is explained in that thread, in which I showed why that was so on the basis on your own parameters for assessing what is part of human nature, what is an
"inherent capacity", etc.). In other words, you bungled that thread too.

Now, you talked in other threads about mathematicians using their intuitive logic, rather than what you call "mathematical logic". Well, of course, mathematicians do not use logic formalizations most of the times, in most of the proofs - other than those working on mathematical logic -, but then, what you defined as "mathematical logic" is not tied to any formal system, and it is in fact in line with the logic intuitions that most mathematicians have.

Well, guess what?
Mathematicians are generally way above average at doing logic intuitively - no formalization, no nothing. In part, that is just talent - some of us were well above average at math in primary and high school, before we learned anything about formal logic - and in part, training - most of which, again, happens without formalization of logic.

You said it yourself: mathematicians usually use their own intuition to prove things, rather than formalize it. Try to do that yourself. You can't, because you're not nearly as good as intuitive logic (not "intuitionistic", but "intuitive") as any mathematician is. Your participation in these threads (e.g., your failure to even realize you incurred contradictions, even after that has been repeatedly shown) provide conclusive evidence of this. The fact that you will never see your errors is no more of a problem than the fact that nearly supporters of ML conspiracy theories, YEC, will never realize that their ideology/religion has been debunked, repeatedly. It provides very weak evidence against the assessment that you are not good at logic - which is not to say you're below average; you very probably aren't, but you're definitely not good enough for these exchanges. You are also 'in the grip of an theory' so to speak, which is bad as always, but it's not the only problem: your failure to see your own contradictions when they are explained to you is not accounted by your irrational commitments to your beliefs about mathematicians.

Interestingly, most people I have tested with contradictory premises disagree with mathematicians. Obviously, they fail to be biased.

No, they fail at logic, in those particularly unusual cases. People's logic intuitions tend to work well enough in most contexts of their daily lives. But the most distant one is from those contexts, human intuitions are generally less reliable. But some people are just better than average. Just as some people are (much) better at running, punching, or at picking up what others are feeling than the average, some are (much) better at logic. And they can get better (anyone can, even though to different degrees) by training, like - say - learning math (whether or not they study mathematical logic).

And most logicians before Boole didn't accept the principle of explosion, so even your claim here is junk.

There is such things as progress. There is a reason people come to accept something: someone provides good reasons for it. It's not as if Aristotle actually had the ideas you believe he had, or that he explicitly rejected arguments for the principle of explosion.

 

[Speakpigeon]

But flip it and all hell breaks loose
P: I have salt
C: I have salt or Elvis is playing the guitar on the moon made of purple cheese

Who saw that coming?

2 + 3 = 5;
Therefore, 2 + 3 = 5 or it is not true that 2 + 3 = 5.

x = 10 or x = 112;
Therefore, x = 13 or x = 112 or x = 11 or x = 114 or x = 15 or x = 117 or x = 17 or x = 118;

x is even;
Therefore, x is even or x is not even.

The premise of this argument is true;
Therefore, the premise of this argument is true or the conclusion of this argument doesn't follow.

The accused is guilty;
Therefore, the accused is guilty or the accused has been framed by the police.

All these are valid but it seems you feel somewhat doubtful about that kind of implication... Maybe it is because in ordinary conversations we sometime use "or" to introduce a contradictory supposition.
EB

 

[steve_bank]

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].

I am being serious and blunt. In the environments I worked in if you put forth the kind of reasoning you used to declare the moon is made of chees on a real problem you would be labeled incompetent. You do not seem to understand logic.

I am being serious and blunt. In the environments I work in, if you imply that I ever suggested that the kind of reasoning I put forth would provide any good reason to even suspect that the Moon is made of cheese, you would be corrected immediately because you do not even understand the arguments you are replying to. But my logic is correct.

You are apparently using a form of a boo trapping argument.
Again the wording of your syllogism. Therefore the moon IS made of cheese. There is no logical connection to the premises.

The second conclusion is bootstrapped. You draw the first conclusion then you bootstrap the second by selecting one of two contradictory statements p1 p2.

In court bootstrapping is forbidden, it is seen as sophistry to draw a conclusion without any evidence circumstantial or otherwise. Your syllogism works around the lack of evidence in the premise regarding the second conclusion.

If this were a Perry Mason episode and I the judge I would say 'Counselor reword your argument or move on'.

 

[steve_bank]

fast

Taking the premises of an argument and propeling our way to a conclusion seems substantively more difficult than safely standing upon a conclusion and gazing out in observance of premises.

That is the crux of applying logic. I look at it as top down vs bottom up. Induction vs deduction.

I want to build a bridge. I can start with a known form of bridge design and work backwards to define how it gets built.

I can look at the conditions of the site and the requirements and work towards selecting a design.

In reality it is rarely one or the other. One goes back and forth with some trial and error between conclusion and premises. Start with a design and work backwards finding it will not work. Start with the original derived premises and work toward a different design. It can take multiple cycles.

There are rules of logic but there is no mechanistic set of rules on how to apply logic to achieve a goal.

There is electrical circuit theory but there are no rules on how to put together a circuit to do a task.

Staring out we study theory and simple applications. Then from experience and observation we learn how to apply theory. Two engineers may not come up with the same circuit solution to a problem, while both may work fine.

It is experience and theory distilled to reasoning is how I see it. A function of the brain.

 

[fast]

I’m paraphrasing and adding, but I was asked why I would follow the rules of logic even when questions linger about justification. It’s an established convention that leaves no one confused when everyone is on the same page. I figure I’m doing well just to know some basics. I’m not out to expose what might be genuine flaws. If I’m going to vote on whether or not I think an argument is valid, I think I’m doing better service to those I speak with to align my vote with what accords with accepted convention.

Sure, I might argue a point here or there (and get my feet wet, so to speak), but as I come to terms with how things are taught, I adjust my positions accordingly.

His logic is taught is my benchmark. I can consider what others are saying, but I think I’m doing good to keep an eye on what extent one deviates from convention.

 

[Speakpigeon]

If I’m going to vote on whether or not I think an argument is valid, I think I’m doing better service to those I speak with to align my vote with what accords with accepted convention.

???

What "convention"?!

Mathematicians don't even agree among themselves on any one "convention", so there is no convention, even less an "accepted convention".

And even if they did, it would remain a convention among mathematicians not necessarily accepted by non-mathematicians.

Why don't you speak French? You know, it's a convention. Along French people, sure, but at least it is really a convention.

In other words, you choose to accept the convention as if you couldn't decide for yourself whether an argument is valid or not.

I wonder how you can go through life without harm since you are clearly not very well versed in the details of this convention. How did you survived until now without it?
EB

 

[fast]

If I’m going to vote on whether or not I think an argument is valid, I think I’m doing better service to those I speak with to align my vote with what accords with accepted convention.

???

What "convention"?!

Mathematicians don't even agree among themselves on any one "convention", so there is no convention, even less an "accepted convention".

And even if they did, it would remain a convention among mathematicians not necessarily accepted by non-mathematicians.

Why don't you speak French? You know, it's a convention. Along French people, sure, but at least it is really a convention.

In other words, you choose to accept the convention as if you couldn't decide for yourself whether an argument is valid or not.

I wonder how you can go through life without harm since you are clearly not very well versed in the details of this convention. How did you survived until now without it?
EB

I tried damn it!

True story. Funny story too. I took an English class, a Spanish class, and a French class all in the same quarter (they later switched to the semester system). One day in Spanish class, the teacher would read something in Spanish and a student would stand, translate to Spanish, and speak it aloud.

When it came around to my turn, I did my part, except afterwards (right after speaking), the teacher laughs and says “that came out 1/3 Spanish, 1/3 English, and (in an inquisitive tone said) 1/3 French. The class smiled and I explained I was taking all three. They were rolling that day!

 

[Angra Mainyu]

You are apparently using a form of a boo trapping argument.
Again the wording of your syllogism. Therefore the moon IS made of cheese. There is no logical connection to the premises.

Yes, there is. The connection is that applying the logic rules I used in the argument, the conclusion follows from the premises.

The second conclusion is bootstrapped. You draw the first conclusion then you bootstrap the second by selecting one of two contradictory statements p1 p2.

In court bootstrapping is forbidden, it is seen as sophistry to draw a conclusion without any evidence circumstantial or otherwise. Your syllogism works around the lack of evidence in the premise regarding the second conclusion.

I don't know what you mean by "bootstrapped", but no serious court would forbid the deduction of (P or Q) from P or the deduction of Q from (P or Q) and ¬P. Of course, in a serious court, any argument with contradictory premises intended to provide support for the conclusion would be rejected, and with good reason. However, as I have already said (and it should be clear from my posts), I am not remotely suggesting that such argument would provide evidence for the conclusion. For that matter, the following argument is also valid (i.e., the conclusion follows from the premises).


P1: Either the Moon is made of cheese, or the Earth is flat and the Moon is made of cheese.
C: The Moon is made of cheese.

If this were a Perry Mason episode and I the judge I would say 'Counselor reword your argument or move on'.

It would be difficult to deal with a judge who does not understand what I'm saying, despite repeated clarifications. But fortunately, I'm not in that situation.

 

[Angra Mainyu]

I’m paraphrasing and adding, but I was asked why I would follow the rules of logic even when questions linger about justification. It’s an established convention that leaves no one confused when everyone is on the same page. I figure I’m doing well just to know some basics. I’m not out to expose what might be genuine flaws. If I’m going to vote on whether or not I think an argument is valid, I think I’m doing better service to those I speak with to align my vote with what accords with accepted convention.

Sure, I might argue a point here or there (and get my feet wet, so to speak), but as I come to terms with how things are taught, I adjust my positions accordingly.

His logic is taught is my benchmark. I can consider what others are saying, but I think I’m doing good to keep an eye on what extent one deviates from convention.

Here's another reason to support the rules in question: they are truth-preserving. Whenever the premises are true, the conclusion is guaranteed to be true.

For example, consider the following rule:

From (P or Q) and ¬P, you deduce Q.

Now, if it turns out that (P or Q) is true and ¬P is true, then P is false, right? But (P or Q) is true, so Q is true. Hence, the rule above preserves truth.

Imagine now that you reason by combining two rules, each of which preserves truth. Well, guess what? The use of the two also preserves truth, because assuming your premises are true, the conclusion after the first step (applying one of the rules that preserve truth) will be true, and then the same happens when you use the second rule. Similarly, if you combine several rules that preserve truth, the result preserves truth.

There are of course different systems that are truth-preserving. But there is one that is the strongest, in the sense that any conclusions that follow from a premise by means of a truth-preserving method also follows from this, strongest method. It's classical logic (there are arguments against is usage in mathematics, for example based on the idea that not all mathematical statements are either true or false; that could lead to a much more interesting discussion if people were interested).