# How could I translate logic reasoning to the language of mathematics?

• Lingusitics
• Tosh5457
In summary, deductive reasoning is the only type of reasoning that is accepted in mathematics. Statements such as "We can find no nontrivial integer solutions to x^n+y^n=z^n" do not imply that there are no such solutions. It must be proven by deductive reasoning that there aren't any solutions.
Tosh5457

If I want to understand/explain something, I use inductive or deductive reasoning, which are parts of logic. For example, take this inductive reasoning:

Every life form we know of depends on liquid water to exist.
All life depends on liquid water to exist.

How could I translate it to mathematics? And is everything translatable to the language of mathematics?

There is a field of philosophy called symbolic logic, [Edit": although maybe a better term is formal logic] that might fit your needs pretty well. Here's a wiki article about one form:
http://en.wikipedia.org/wiki/First-order_logic" .)

This type of philosophy is the basis of Boolean logic, used in digital [computer] circuits. It can be used to build binary adders; combinational and sequential logic circuits/gates; together those can be used in part to build arithmetic logic units (ALUs). These, together with more logic circuits are what makes computers, cell phones, WiFi routers, Bluetooth devices, Bluray players, etc. You name it.

Although symbolic logic is technically philosophy, mathematical reasoning is itself a philosophy, when you think about it. Both are good at deductive reasoning. Not so with inductive.

Be careful with inductive reasoning. Perhaps every swan you've ever seen is white. You might conclude that all swans are white. That's fine and dandy until you visit the land down under and stumble upon a Black Swan (http://en.wikipedia.org/wiki/Black_Swan" ).

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Mathematics only works with deductive reasoning. Inductive reasoning is not allowed. So your reasoning is not valid in mathematics.

And not all sentences can be translated in mathematics. Mathematics only considers very special sentences: the well-formed formula's.

micromass said:
And not all sentences can be translated in mathematics. Mathematics only considers very special sentences: the well-formed formula's.

What do you mean? Can't every deductive reasoning be translated to mathematics?

So, from wikipedia:
Logic is often divided into two parts, inductive reasoning and deductive reasoning.

Since mathematics only works with deductive reasoning, is logic more general than mathematics?

Tosh5457 said:
What do you mean? Can't every deductive reasoning be translated to mathematics?

Yes, but not in the way you want it. The deductive statement

Cats are always black, so my cat is black

can be translated as $P\Rightarrow Q$. But P and Q have no special meaning here. And a statement about cats can not be translated into mathematics. So only the form of the reasoning can be translated into mathematics, not the meaning of the statements.

Since mathematics only works with deductive reasoning, is logic more general than mathematics?

Mathematics deals with mathematical logic, and this only allows deductive reasoning. The logic you're talking about is "philosophical logic" and could allow inductive statements, but this is not mathematics. It's true that philosophical logic is more general than mathematical logic.

Statements such as

"We can find no nontrivial integer solutions to $x^n+y^n=z^n$"​

does not imply that there are no such solutions. It must be proven by deductive reasoning that there aren't any solutions. And this is the only reasoning accepted.

Tosh5457 said:
Every life form we know of depends on liquid water to exist.
All life depends on liquid water to exist.

How could I translate it to mathematics?
Conjecture: All life depends on liquid water to exist.

## What is logic reasoning?

Logic reasoning is a method of using rational thinking to reach conclusions based on given information or premises. It involves using logical rules and principles to determine the validity of an argument or statement.

## Why is it important to translate logic reasoning to mathematics?

Translating logic reasoning to the language of mathematics allows for a more precise and formal representation of logical arguments. It also allows for the use of mathematical tools and techniques to analyze and evaluate these arguments, leading to more accurate and sound conclusions.

## What are the key components of translating logic reasoning to mathematics?

The key components of translating logic reasoning to mathematics are identifying the premises and conclusions of an argument, defining and representing logical operators and relationships using mathematical symbols, and using axioms and rules of inference to construct a mathematical proof.

## What are some challenges in translating logic reasoning to mathematics?

Some challenges in translating logic reasoning to mathematics include abstract concepts that may not have a direct mathematical representation, the need for precise and formal definitions, and the potential for ambiguity or inconsistencies in translating between the two languages.

## How can I improve my ability to translate logic reasoning to mathematics?

Practice and familiarization with both logic and mathematical concepts are key to improving one's ability to translate between the two. It is also helpful to break down arguments into smaller, more manageable parts and to carefully define and represent each component using mathematical notation.

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