Mathematical Logic.

[rupol2000]

Is mathematical logic formal logic? Is it reducible to language and its grammar?

 

[rupol2000]

We apply certain logic in Mathematics

Which one? What is specific?

 

[rupol2000]

All logic is mathematical

I think that mathematical logic is only part of the general logic, which comes down to the grammar of the language. Logic is a more general science of thought and does not depend on formalism.

In fact, mathematical logic and the grammar of a language are one and the same.

 

[App'z]

The Big Bang is B.S. its an old thought that back ground radiation signaled something. The Universe keeps proving its much bigger than the Physicist have said it is. What happened is stuff gets together and causes gravitation which makes time pass and light to begin to happen. Sure there have been big bangs, thas what happens when stuff gets pissed off. Still only electromagnetic waves travel at the speed of light approximately. all the crap about some really big universe are correct, it is unlimited by time and space. Get ready, you are insignificant ! You know stupid F'rs are wanting to live on the Moon and Mars. Well thats their stupid disire to make their lives misserable just like those floating around in a Space station for 6 months trapped in a 3 wall of space unable to even take a drive on a Saturday afternoon. just floating there useless to most all especially themselves ! Dumb Askholes !

MSN

www.msn.com

 

[itfitzme]

I think that mathematical logic is only part of the general logic, which comes down to the grammar of the language. Logic is a more general science of thought and does not depend on formalism.

In fact, mathematical logic and the grammar of a language are one and the same.

Okay, how about;

The rules of grammar are definitely not logic. Words can be strung together with perfect grammatical structure and be make no sense, both in being illogical as well as in conveying no meaning.

Mathematical logic is founded in measures of physical attributes, such as "These are two apples. Those are two apples. These and those are four apples." Grammatical rules allow for "These are two apples. Those are two apples. These and those are twenty apples." The first is mathematically true and grammatically correct. "2+2=4". The second is grammatically correct but mathematically false, "2+2=20".

Logical fallacies identifies numerous grammatically correct forms of combining words which are logically false.

Post hoc ergo propter hoc is a logical fallacy that identifies a class of statements that may be grammatically correct but are logically incorrect. Statements that convey the meaning that "Since event Y followed event X, event Y must have been caused by event X. ", or "There were no cockroaches in the apartment till you moved in. Clearly your to blame." is false, in isolation of context, because it assumes a causal connection based on simply order of events. While causality requires that the cause precede the event, it is not the fundamental reason for a causality. Causality requires some sort of chain of events, where matter and energy are transferred, connecting the cause and effect.

Logical fallacies sound reasonable because they trigger some familiar logical element or experience that while it may be true, is insufficient to reach the conclusion. And, they tend to be presented in a way that is grammatically correct. Like optical illusions to the eye, they are language illusions. There is a class of logical fallacies, Fallacies of Ambiguity and Grammatical Analogy

 

[itfitzme]

Logic means reasoning. The reasoning may be a legal opinion or mathematical confirmation. We apply certain logic in Mathematics. Basic Mathematical logics are a negation, conjunction, and disjunction. The symbolic form of mathematical logic is, ‘~’ for negation ‘^’ for conjunction and ‘ v ‘ for disjunction. In this article, we will discuss the basic Mathematical logic with the truth table and examples.

Tautology Boolean Algebra Set Theory Conjunction Mathematical Logic (AND, OR & NOT) | Formulas and Examples The basic mathematical logics are conjunction, disjunction, and negation are explained with truth tables and examples. Visit BYJU’S for more information. byjus.com

Gödel's incompleteness theorems -​

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First Incompleteness Theorem: "Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F." (Raatikainen 2015)​

 

[BothWings]

View attachment 599406

Then I'd have to admit something that I'd prefer not to...

I did ace the nuke entrance exam at the ASVAB station when I joined the military if that tells you anything.

But that was many, many years ago.

*****CHUCKLE*****

When I was 19 I also scored very well on an ASVAB test. However I did not join the military.... But they ended up calling me on an almost monthly basis for several years. Several members of my family on both sides have been involved in low level intelligence operations.... With the exception of my half sister who was a lieutenant colonel at the Pentagon. The story begins with my grandfather who worked at the Dayton Ohio Air Base during World war II which was host to some operations more secretive than most other air bases. My grandmother also worked at the library of Congress furnishing military leaders with information by phone and telegram. I had an uncle who designed weapons at the Picatinny arsenal in Delaware, and my father served in the Airborne.... Drafted as a German citizen living in the US!


The United States government keeps tabs on military families and continues to try to recruit their children and children's children In the same capacity until someone gives some reason not to. By explaining the family unit, they are able to keep things much more tight and under wraps if they simply targeted potential recruits on an individual basis.

 

[rupol2000]

The rules of grammar are definitely not logic. Words can be strung together with perfect grammatical structure and be make no sense, both in being illogical as well as in conveying no meaning.

In mathematical formal logic, these are simple grammar substitutions, there is no sense

 

[rupol2000]

Post hoc ergo propter hoc is a logical fallacy that identifies a class of statements that may be grammatically correct but are logically incorrect.

I think that this is just from the general logic and not from the formal one.

 

[rupol2000]

Mathematicians essentially reject languages that, when grammatically correct, allow for a contradiction. Russell's paradox just reveals the contradiction of the grammatically correct expression from the point of view of Cantor's theory "the set of sets that do not contain themselves." Due to the fact that a grammatically correct expression leads to a paradox, he believes that this is a contradictory theory and it is not suitable.

Carl Hewitt considered contradiction to be the norm. But this is not mainstream mathematics.

There is some hypocrisy here, since arithmetic is inconsistent, but it is still used.

 

[itfitzme]

Logic means reasoning. The reasoning may be a legal opinion or mathematical confirmation. We apply certain logic in Mathematics. Basic Mathematical logics are a negation, conjunction, and disjunction. The symbolic form of mathematical logic is, ‘~’ for negation ‘^’ for conjunction and ‘ v ‘ for disjunction. In this article, we will discuss the basic Mathematical logic with the truth table and examples.

Tautology Boolean Algebra Set Theory Conjunction Mathematical Logic (AND, OR & NOT) | Formulas and Examples The basic mathematical logics are conjunction, disjunction, and negation are explained with truth tables and examples. Visit BYJU’S for more information. byjus.com

Challenge:

Please show the one to one connection between the system of basic arithmetic and the fundamental mathematical logic of negation, conjunction and disjunction. That is, how to express basic arithmetic in terms of these three logical functions of ~,^,v.

Seriously, I'm thinking that the connection between the two is exemplified in the detail of coding from the source code that executes in the CPU through the compiled top level computer language that is supported by the mathematical runtime DLLs executing math functions to produce, lets say, the simulation output of the game ARK: Survival Evolved.

I believe that the OP statement is true, I'm just not up to the challenge because showing it is literally writing computer code. I wrote code for a Z80 processor. To get from the Boolean algebra implemented by the CPU requires, not just the set of rule for ~,^,v, but also a set of rules for implementing the algorithm that accessed memory.

Truth tables lead us from ~,^,v to NOT, OR, AND, XOR, XAND. Those, implemented on numerical representations like ones and two compliments then lead to basic arithmetic. Basic arithmetic to algebra, to series and sequences, to sums and limits, to calculus and beyond.

All of these require that the operators act upon mathematical objects. In implementation, those are memory registers, or variables. I see things like linear algebra as being a set of rules for applying more basic mathematics to specific types of objects, matrices. Those objects, both algebraic variables and more complex objects like matrices, are representations of physical objects which have both quantity, the number that the variable represents, and unique physical attributes expressed in units, such as mass, volume, density, etc. There are logical rules for what objects can be combined in formulas of operations. Those logical rules are generally context dependent because they are implemented in the higher level algorithm and are an expression of the problem that is being solved. That is, what are the sequence of operators necessary to display a pack of Raptors taking down a T-Rex? Heck, what are the sequence of operators necessary to display the occlusion off it's skin?

Reasoning, itself, is simply a process of examining a situation and conclusion in terms of all of the body logic that applies. Reasoning is a verb,, the act of applying knowledge and logic. One always hopes for an orderly, efficient, and exhaustive examination but cost of time is certainties moral enemy.

Now, legal opinion, reasoning based on a history of legal opinion, while it may have it's own set of rules, having a set of rules alone does not equate to 'logic". One might ask if logic necessarily requires perfect memory and information. I would suggest yes. The history of legal opinion requires neither. It attempt to employ fundamental logic in it's decisions but is faulted by the reality of uncertainty.

Anyways, to my challenge was find a path from basic~,^,v to "reasoning may be a legal opinion or mathematical confirmation." And, of course, basic science is grounded in mathematical confirmation. That can be expressed in terms of an analysis of data using models that are expressible as computational programs. Admittedly, the further we get from basic science to more applied sciences like economics. Never the less, we continue to see the advancement in AI breaking down the barriers to implementing that knowledge base in reasoning, up to the limits of apparently making the same stupid cognitive errors that humans make. There are many examples, the implementation of which are not to different in implementation from more familiar and fun simulations like ARK: Survival Evolved. They all are grounded in an every more detailed translation of higher order algorithms to basic Boolean implementation of fundamental logical operators. I would say that the sciences, as long as they remain firmly committed to mathematical confirmation, rigor, and constant pursuit of internal consistency, can be demonstratable shown to be rooted in the fundamental mathematical logic of negation, conjunction and disjunction.

But, and it's a big ol' but, The implementation of the root logic requires an ever expanding layering of other exogenous systems of rules that are defined by the context. For example, there are exogenous rules developed from counting and combining physical objects with attributes. Like attributes can be added, even follow some basic rules of arithmetic, like cancellation, but others may not because the object are physically incompatible. As well, proximity in space and time are necessary for causality, and thus context defines what object attributes can be operated upon. Two 2 oz apples and a blender equals 2 oz applesauce but two oz of rocks and a blender doesn't equal rock sauce. There are decisions about what operators can be employed to act upon what object attributes. There is a higher structure to it than the three operators alone.

Just as the elemental particles and physical laws present a system of logic that combines and builds to create all the unique galaxies of the Universe and all the complex living organisms that jostle about Earth, the expression of that isn't all the combinations and permutations in one massive ball of all possible outcomes, but a crescendo of states that interact according to even more emergent properties until we get the unique expression of it all, the Universe as it is Now and at each unique instance of ongoing time.

Coffee time!

 

[itfitzme]

I think that this is just from the general logic and not from the formal one.

Right, which is what grammar implements.

 

[itfitzme]

Mathematicians essentially reject languages that, when grammatically correct, allow for a contradiction. Russell's paradox just reveals the contradiction of the grammatically correct expression from the point of view of Cantor's theory "the set of sets that do not contain themselves." Due to the fact that a grammatically correct expression leads to a paradox, he believes that this is a contradictory theory and it is not suitable.

Carl Hewitt considered contradiction to be the norm. But this is not mainstream mathematics.

There is some hypocrisy here, since arithmetic is inconsistent, but it is still used.

Mathematics is incomplete, not inconsistent. Gödel's incompleteness theorems. I'm still looking for the inconsistency theorem. Alternately, the inherent contradiction theorem. This would be express, basically, as 1=0. An Contratiction Theorem would say that a consistent system of axioms whose theorems can be listed by an effective procedure that inevitably leads to statements that are contradictory, essentially 1=0. At most we have incompleteness or any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system. The second part of which is that it cannot demonstrate it's own consistency. But note that Godel's defines this as any consistent system of axioms is 1) incomplete, that is can't prove every statement it can make and 2) can't prove it's own completeness, not that it is incomplete.

Alternately, from 1) there exists an unprovable formula (in the language of that theory). and 2) there exists a formula such that both it and its negation are unprovable.

 

[rupol2000]

Mathematics is incomplete, not inconsistent. Gödel's incompleteness theorems

Usually no one understands what Gödel wrote about. He had problems with his head.

What he wrote seems to be something like if we decided to do a deduction without premises, or a theory without axiomatics, and Gödel would tell us that this would not work.

 

[itfitzme]

Incompleteness

 

[rupol2000]

The very concept of mathematics is essentially pseudoscientific. It combines several different disciplines, such as geometry, arithmetic, algebra, logic, grammar, but it is not clear what all this is connected to say that this is one and the same general science.

 

[rupol2000]

And the very desire to "axiomatize mathematics" smacks of some kind of idiocy. Axioms make sense only within the framework of a specific theory, and besides, this speculative approach itself was rejected by scientific methodology, this is quasi-ecclesiastical charlatanism.

 

[itfitzme]

A most basic form of logic and reason is the "fault finding" system. It basically takes any given state object and finds a context within which a "fault" can be found. That "fault" then become a reason for rejecting the presented object, be that a car or a conclusion.

As far as systems of logic go, it is about as fundamentally incomplete as possible and consistently presents statements that contextually irrelevant though are grammatically correct and emotionally satisfying because they would be logically correct under a familiar but unrelated context. Axioms are essentially contextual definitions. Godels Incompleteness Theorems are not of axiomaticless systems. By definition, they are axiomatic. They are collections of axioms and the very nature of the proof, as so well described by Derek Muller of Veritasium, is basically, they can't prove everything. They can't prove them true or false. (My favorite proofs were proofs that something can't be proven. )

Now, there are some that have interpreted this to mean that there are things that can't be proven. Godel did not show this as it is a statement as to ultimate provability and Godel's Incompleteness Theorem only deals with any specific set of axiomatic statements, not on all. It is essentially incomplete.

Another misinterpretation is that Godel's Incompleteness Theorem states that every system ultimately leads to a proof that is contradictory. Rather, what he shows is that any system of axioms can be manipulated to present two contradictory statements, neither of which can be proven true or false. This, is an extension of the first part, that there are statements than can't be proven true or false. Not just one, but two related contradictory statements than can't be decided either way.

Basically, he showed mathematicians that whatever proof they are working on could be a lost cause, they can't prove it either way, and they're wasting their career.

 

[rupol2000]

A most basic form of logic and reason is the "fault finding" system. It basically takes any given state object and finds a context within which a "fault" can be found. That "fault" then become a reason for rejecting the presented object, be that a car or a conclusion.

Bullshit. The basis of logic is ordinary deduction. The course of thought from the general to the particular.

 

[itfitzme]

And the very desire to "axiomatize mathematics" smacks of some kind of idiocy. Axioms make sense only within the framework of a specific theory, and besides, this speculative approach itself was rejected by scientific methodology, this is quasi-ecclesiastical charlatanism.

Axioms are necessarily requires, period. There is no "thinking" without axioms.

Like I said, "fault finding" isn't a viable form of logic and reason. Statements can be had that are grammatically correct but lacks any logic.