# What Paradoxical Statements really are

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**What Paradoxical Statements really are**

Gebhard Greiter, 2011

The most simple example of a paradoxical statement seems to be

**S: This statement S is lying.**

If we assume

**S**is lying, then

**S**is not lying; but if we assume

**S**is not lying, then

**S**is lying.

So, what is wrong here?

The problem here lies in what we think a statement is: Under a formal point of view, a state-

ment may be

**, may be**

*true***, or may be**

*false***. The statement**

*empty***Alice is pregnant**

for example cannot be assigned a truth value (

**or**

*true***) as long as we do not know of**

*false*which person Alice exactly we are speaking. So, at least now, it is an

**statement.**

*empty*We see: Depending on the context

**c**in which a statement

**X**occurs, the statement's truth

value v(

**c**,

**X**) may be

**,**

*true***, or**

*false***. The context in this sense is at least a point in**

*undefined*time, but may also contain definitions of the objects the statement refers to.

Formal logic usually ignores this fact, and so mathematicians see

**S**as a problem. I do not:

*are statements*

**Paradoxical statements****X**for which there is no context

**c**

such that v(

**c**,

**X**) is different from

**empty**.

Now we prove that the statement

**S**given above is paradoxical in the sense of this definition.

If not, a context

**c**would exist, such that v(

**c**,

**S**) is

**or**

*true***and is solution the equation**

*false*v(

**c**,

**S**) = v( v(

**c**,

**S**) ==

**)**

*false*In other words: One element

**x**of {

**,**

*true***} would be solution of the equation**

*false***x**= v(

**x**== false )

This equation however is satisfied neither by

**nor by**

*true***(read == as "is identical to").**

*false*1

What we have learned is: The fact that paradoxical statements exist is telling us that to have

only two truth values (

**,**

*true***) is not enough; we also need a value**

*false***.**

*undefined*We also see that mathematical logic should be generalized to

no longer ignore the fact that mapping statements to boolean values is not enough:

Reality is mapping statements

*context-sensitive*to {

**,**

*true***,**

*false***}.**

*undefined*The fact that mathematical logic as you find it in text books is ignoring context (especially

time) and knows only two boolean values, may be a consequence of the fact that at least

mathematical statements can only be

**or**

*true***and are so independent of the point in**

*false*time we look at them:

Mathematical laws are valid in every context.

Other statements however do not have this nice feature - maybe not even statements that

describe physical laws: Researchers believing in String Theory are no longer sure that our

universe is the only one, and they tell us that, should other universes exist, the physical laws

valid there may differ from the physical laws valid in our universe.

Author's homepage: http://greiterweb.de/spw/

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