The Logicians Corner

[drug_mentor]

I realise this thread is a bit of a long-shot, you don't come across too many logic enthusiasts in day to day life, I imagine the odds are even poorer on a drug forum. Nonetheless, I thought I may as well put this out there in the hopes there are some fellow bluelighters who share my enthusiasm for formal logic.

I thought that perhaps a good starting point of discussion would be to compare different approaches to natural deduction, and to discuss the different merits of the various approaches. Perhaps if enough interest is generated we could share sequents and proofs, and/or discuss the meta-theories of different formal systems.

I am familiar with first-order predicate calculus with identity. Personally I use a Gentzen inspired form of proof discovery which was developed by E.J. Lemmon, albeit with some slight modifications. I hope to be able to go into some detail about the features and merits of this approach, but will refrain from doing so unless this thread generates some interest.

Despite being relatively unfamiliar with modal logic and set theory I hope to familiarise myself with them in the future, and I welcome any discussion of them in this thread.

In order to increase the odds of getting some worthwhile responses I am also happy for some discussion of computer logic to take place in this thread, but it would be ideal if such discussion could be related to philosophical logic or other philosophical issues in some way.

 

[polymath]

I had to learn the basic logic and set theory necessary for making mathematical proofs when I was a first-year student in uni, and we also had some Boolean algebra in a digital electronics course. Never really bothered to study anything really abstract, like the multi-valued logics where there are other possible truth values in addition to simple "true" and "false"...

Boolean algebra was kind of fun, because it's so simple that you have variables that can have only two possible values, 0 and 1 (or true and false), and you still can't immediately see how to simplify an expression like (x+z)(x*+y)(z+y) to the shortest equivalent expression possible (x,y,z are Boolean variables, x* means the truth value opposite to x). Then you learn how to use Karnaugh maps to do that.

 

[Dracarys]

I like formal logic. I aslo did predicate calculus, as well as set theory and modal logic in my first year in uni. However, these formal logical systems are a bit primitive. When you actually engage in philosophical issues that require logic, you'll discover that logic is often much more vague or ambiguous. Game theory is a good example of what i mean. To actually proof something in game theory is quite tricky if you want the proof to be as solid as proof in predicate calculus, because you're dealing with all kinds of assumptions. So then it innevitably comes down to proving wich of your assumptions about rational behaviour are the right ones. I'm hoping to find the time in the near future to dig deeper into that sort of stuff.

 

[polymath]

Despite being relatively unfamiliar with modal logic and set theory I hope to familiarise myself with them in the future, and I welcome any discussion of them in this thread.

I looked up the Wikipedia article about modal logic, and I immediately encountered a problem with applying it to statements involving real numbers... In addition to being true or false, statements in modal logic can be described with the modalities "possible" or "necessary".

What if we choose a completely random real number that is larger than 0 and smaller than 1, is it "possible" that the random number we get is exactly 0.5? It's just as possible as any other number, but the probability of getting it is zero, because it's only one number in an infinite set of equally probable numbers. Can something be "possible" but still have zero probability?

E: Looks like it can...

 

[swilow]

I'm interested. I just have no idea about formal logic. Is anyone will to try and give a kinda synopsis of the idea's?

 

[Dracarys]

I'm interested. I just have no idea about formal logic. Is anyone will to try and give a kinda synopsis of the idea's?

Yeah, the most basic idea, wich is supposedly a sort of invention from aristotle (but maybe he heard it from someone else, who knows?) is that you in a sense standardise logical operations. So Aristottle intorduced categorical logic. It's a collection of reasonings of the type "if all A=B and all B=C, then all A=C". So he thought that you could eventually fit all human reasoning in these schemes, simply by varying the terms "all", "some" and "none". This more or less became the dominant principle in formal logic until the beginning of the twentieth century. The works of Frege, russel, tarsky, wittgenstein and many other logicists eventually lead to propositional, predicate and modal logic, as well as set theory. These forms of logic are somewhat more advanced than aristottle's logic.

 

[polymath]

One important thing for a novice to learn is that some words have slightly different meaning in logics than in everyday language. If A and B are logical statements and we say "A or B is true", it's also possible that both A and B are true, logical "or" is not the same as "either or".

Another thing to learn right away is the difference between "A implies B" and "A is equivalent with B". If x is an unknown number, the statement "x is greater than 7" implies that "x is not 2", but you can't use that the other way around, saying that "if x is not 2, then x is greater than seven". OTOH, the statements "x+1=6" and "x=5" are equivalent, they both imply each other.

And one thing that isn't intuitively obvious, is that in logics, it's possible to "prove" ANY statement, even ridiculously false ones, if you start by assuming something that is false. This is the basis of the reductio ad absurdum method of proving that some statement must be false.

 

[Nixiam]

Logic is like an aged wine.

Secured to the pallet.

 

[drug_mentor]

Boolean algebra was kind of fun, because it's so simple that you have variables that can have only two possible values, 0 and 1 (or true and false), and you still can't immediately see how to simplify an expression like (x+z)(x*+y)(z+y) to the shortest equivalent expression possible (x,y,z are Boolean variables, x* means the truth value opposite to x). Then you learn how to use Karnaugh maps to do that.

I am totally unfamiliar with boolean logic. It is pretty late here and I only briefly had a look at it, whilst I am sure it is very useful it seems to mostly apply to maths. Unfortunately I stopped studying maths as soon as it was no longer required in high school, a decision I regret now.

I like formal logic. I aslo did predicate calculus, as well as set theory and modal logic in my first year in uni. However, these formal logical systems are a bit primitive. When you actually engage in philosophical issues that require logic, you'll discover that logic is often much more vague or ambiguous. Game theory is a good example of what i mean. To actually proof something in game theory is quite tricky if you want the proof to be as solid as proof in predicate calculus, because you're dealing with all kinds of assumptions. So then it innevitably comes down to proving wich of your assumptions about rational behaviour are the right ones. I'm hoping to find the time in the near future to dig deeper into that sort of stuff.

This is interesting. I am slightly familiar with game theory and I am aware of many-valued logic, but I really don't know enough about either to attempt to construct a proof.

I am aware classical logic has its limitations but I am fairly certain it has a number of applications in epistemology and the philosophy of language. It is also really useful for formally constructing every day arguments, although admittedly for many every day arguments you can work out their validity more easily without going to the trouble of converting it into the language of the predicate calculus.

I looked up the Wikipedia article about modal logic, and I immediately encountered a problem with applying it to statements involving real numbers... In addition to being true or false, statements in modal logic can be described with the modalities "possible" or "necessary".

I don't think the primary applications of modal logic are mathematical. Aside from allowing a wider scope of everyday arguments to be formally represented I believe it also has applications in model theory.

I'm interested. I just have no idea about formal logic. Is anyone will to try and give a kinda synopsis of the idea's?

When I find the time I would be quite happy to write up the syntax and primitive rules of inference for propositional logic, and provide some examples on how to construct a proof. I may also give a quick treatment of the semantics if I can find an effective way to post truth tables on here.

When you are relatively proficient with propositional logic it isn't difficult to pick up the predicate calculus, which has a considerably wider application. All the rules for propositional logic carry over to predicate logic and you only need to add four quantifier rules and a couple of rules for identity.

OTOH, the statements "x+1=6" and "x=5" are equivalent, they both imply each other.

Is this something you would need boolean logic or perhaps set theory to prove, or could you do it with predicate calculus? It is pretty late here and I have some benzo's in my system but I am having trouble figuring out how one would prove this with predicate logic, although I have rarely used logic for mathematical applications.

And one thing that isn't intuitively obvious, is that in logics, it's possible to "prove" ANY statement, even ridiculously false ones, if you start by assuming something that is false. This is the basis of the reductio ad absurdum method of proving that some statement must be false.

This is true, but since the proof would be resting on assumptions that are not true it is not really problematic.

There are some quirky aspects to classical logic, like the paradoxes of material implication. In classical logic an if....then.... conditional statement is true if the consequent is true and/or the antecedent is false. This is very counter intuitive and does not correspond to the way these sorts of statements are used in the English language. A conditional statement where the antecedent is known to be false is called a vacuous truth.

 

[polymath]

I am totally unfamiliar with boolean logic. It is pretty late here and I only briefly had a look at it, whilst I am sure it is very useful it seems to mostly apply to maths. Unfortunately I stopped studying maths as soon as it was no longer required in high school, a decision I regret now.

If x and y are boolean variables that can be either true or false, we denote the logical "x OR y" with "x+y" and "x AND y" with "xy". The truth value opposite to x is denoted by x* or some alternative denotations. Boolean algebra doesn't need to have anything to do with numbers, it's just a way to play with truth values in a way that looks like calculating with numbers. For example, the statement "x+x*=true" holds for any x because every boolean variable is either true or false. Also, we have "xx*=false", because x and x* can't both be true. Other statements that are true in boolean logic include "x+x=x" and "x(x+y)=x".

Is this something you would need boolean logic or perhaps set theory to prove, or could you do it with predicate calculus? It is pretty late here and I have some benzo's in my system but I am having trouble figuring out how one would prove this with predicate logic, although I have rarely used logic for mathematical applications.

Number systems all have some kind of set of axioms, statements that are assumed to be true. The truth values of all other statements are derived from the axioms. Technically, "6" is just a name we give to the number "5+1" by convention when using Hindu-Arabic numerals, so the statement "x+1=6 is equivalent with x=5" doesn't really give us any new information.

 

[thujone]

boolean algebra is not that complicated, even with a shaky understanding you can reason through building a processor circuit using logic gates and memory modules. it's basically like the hardware version of programming a single function. my preferred approach to all decision-making is divide and conquer -> binary decision model. i would even go so far as to call it my faith. i don't believe there is any problem in the universe that cannot be reduced to a binary decision

 

[swilow]

When I find the time I would be quite happy to write up the syntax and primitive rules of inference for propositional logic, and provide some examples on how to construct a proof. I may also give a quick treatment of the semantics if I can find an effective way to post truth tables on here.

When you are relatively proficient with propositional logic it isn't difficult to pick up the predicate calculus, which has a considerably wider application. All the rules for propositional logic carry over to predicate logic and you only need to add four quantifier rules and a couple of rules for identity.

If you don't mind. I would like that. I feel so uneducated.

Even a link to something. I like logic. I guess I don't know what it is...

 

[Nixiam]

Jesus. Polymath and d_m make me feel I know nothing.

You can learn so much just talking to people.

I'll post something better tonight lol, right now I'm a bit uncoordinated.

 

[herbavore]

boolean algebra is not that complicated, even with a shaky understanding you can reason through building a processor circuit using logic gates and memory modules. it's basically like the hardware version of programming a single function. my preferred approach to all decision-making is divide and conquer -> binary decision model. i would even go so far as to call it my faith. i don't believe there is any problem in the universe that cannot be reduced to a binary decision

This is very interesting and intriguing to me. I seem incapable of this kind of thought. My mind floods with every possibility, every point of view, every overlap. When people describe "black and white thinking" as a problem, I think, "Try living with an eternally grey way of thinking--you may soften your viewpoint!" And often I am frozen into inaction. My eldest son seems to, by nature, have the kind of mind you are describing. At a very early age he would divide any problem confronting him into an A/B equation. Then he would methodically weigh each side. In the end he would make his decision for action and then (and this is another part that amazes me) not look back. He is the person you want around in a crisis. I have tried, and continually failed, to adopt some of the wisdom I have learned by observing him over the years. In the end I decided to apply what I could probably call my faith: we all contribute something necessary to the whole, as messy and lacking as it may be on its own.

 

[tantric]

if you're actually curious, i can speak a bit about buddhist three value logic, where 'unknowable' is a proper state and has information

 

[polymath]

^ Any sufficiently complex logical system allows creation of statements that are "undecidable", i.e. can't be proven to be either true or false. For any logical/mathematical statement, such as "if x is a real number, then x² + 2x + 3 is greater than or equal to 2", there's some minimum number of logical steps that are needed to prove it. An undecidable statement is something that can't be proved with a finite number of logical steps - the proof would have to be infinitely long.

 

[vortech]

Something I love about corners is that low-frequency sound waves (the BASS!) are +9db louder than they are in the center of the room. This is because of a simple-to-calculate +3db increase near each surface, and corners have 3 surfaces. OK that's enough logic for me for one day.

 

[thujone]

This is very interesting and intriguing to me. I seem incapable of this kind of thought. My mind floods with every possibility, every point of view, every overlap. When people describe "black and white thinking" as a problem, I think, "Try living with an eternally grey way of thinking--you may soften your viewpoint!" And often I am frozen into inaction. My eldest son seems to, by nature, have the kind of mind you are describing. At a very early age he would divide any problem confronting him into an A/B equation. Then he would methodically weigh each side. In the end he would make his decision for action and then (and this is another part that amazes me) not look back. He is the person you want around in a crisis. I have tried, and continually failed, to adopt some of the wisdom I have learned by observing him over the years. In the end I decided to apply what I could probably call my faith: we all contribute something necessary to the whole, as messy and lacking as it may be on its own.

i do tend to get along better with people who think differently, there's a better sense of mutual understanding and less conflicts caused by dumb hubris.

 

[polymath]

Something I love about corners is that low-frequency sound waves (the BASS!) are +9db louder than they are in the center of the room. This is because of a simple-to-calculate +3db increase near each surface, and corners have 3 surfaces. OK that's enough logic for me for one day.

Let me challenge this statement logically: Suppose the only source of sound in the room produces sound of extremely low intensity 0.001dB. Is this sound amplified to 9.001 dB in the corners? I don't think so.

 

[Dracarys]

if you're actually curious, i can speak a bit about buddhist three value logic, where 'unknowable' is a proper state and has information

Yeah, and then there are even more forms of three valued logic. Like where something is both true and false at the same time (i don't remember how it's called. something like dialetheism' i believe). It's suggested by a guy named priest as a solution to the liars paradox. His reasoning is that the 'neither true nor false' solution can easily be defeated by 'this statement is either false, or neither true or false'. Only his solution does not have that problem.