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

1. Jul 7, 2011

### Tosh5457

I don't know much about this subject, so this is probably a very basic question.

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?

2. Jul 8, 2011

### collinsmark

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" [Broken].)

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" [Broken]).

Last edited by a moderator: May 5, 2017
3. Jul 8, 2011

### micromass

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.

4. Jul 8, 2011

### Tosh5457

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

So, from wikipedia:
Since mathematics only works with deductive reasoning, is logic more general than mathematics?

5. Jul 8, 2011

### micromass

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.

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.

6. Jul 9, 2011

### Hurkyl

Staff Emeritus
Conjecture: All life depends on liquid water to exist.