View source for Bastard's object

{{Stub page|msg=It would be good to have a category of bastards' objects and a stricter definition. This will do for now. [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 11:54, 5 May 2018 (UTC)|grade=D}}



::: {{Caution|{{XXX|{{abbr|not to be confused with}}}}''not to be confused with:'' a [[counter-example]]}}, see [[#VsCounterExample|below]] 
__TOC__
==Definition==

==={{anchor|VsCounterExample}}Vs a [[counter-example]]===
Given a statement;
* {{M|\forall X[\varphi(X)]}}<sup>In words:</sup><ref group="Note" name="FOLinWords1"/> made for some {{M|\varphi}} 
to "prove or disprove it" we must establish either the statement holds, or it does not.

{{Begin Inline Theorem}}Informal discussion:
{{Begin Inline Proof}}A ''[[counter-example]]'' is a demonstration that a statement cannot be true. Suppose for example there is an item, {{M|\mathcal{B} }} in our language for which {{M|\varphi(\mathcal{B})}} is known or easily found to be ''false'' 

This demonstrates a so called "[[proof by counter-example]]" as we have given an instance {{M|\mathcal{B} }} such that we have {{M|\neg[\varphi(\mathcal{B})]}} (the {{M|\neg}} symbol means "not" or [[negation]], to say {{M|\neg[\varphi(\mathcal{B})]}} holds (is true) means that {{M|\varphi(\mathcal{B})}} is ''false'', see ''[[principle of excluded middle]]''), obviously this means that {{M|\varphi(X)}} ''cannot'' be true '''''for all''''' {{M|X}}

Informal examples:
# the claim "all multiplies of 5 are odd" can instantly be shot down by pointing out that "10 is a multiple of 5 and even" 
# all sequels are worse than the originals
{{End Proof}}{{End Theorem}}

If we show that:
* {{M|\exists Y[\neg[\varphi(Y)]]}}<ref group="Note">There exists a "{{M|Y}}" such that we have ( not the following ( {{M|\varphi(Y)}} ) )</ref>
Then such a {{M|Y}} is said to be a "counter example". 

As such we see that counter examples are useful tools when forming proofs, where as a ''bastard's object'' is a concept which shows that something isn't very useful (in terms of logic) or that a concept is informal.
==Notes==
<references group="Note">
<ref name="FOLinWords1">
{{Begin Inline Theorem}}Discussion on the basics of reading [[FOL]] statements:{{Begin Inline Proof}}This statement, {{M|\forall X[\varphi(X)]}}, is a [[first order logic]] statement (''FOL'') expressed in a [[first order language]] whose symbols are that of logic. See [[Reading FOL statements]] for a broader overview. The statement above has 2 parts:
* {{M|\forall X[\ldots]}} means "for all things (in our language) it follows that we have ( {{M|\ldots}} )
* {{M|\varphi(X)}} is a {{link|formula|FOL}} (sometimes overlaps with and is called a ''[[predicate]]'')
Combining these:
* {{M|\forall X[\varphi(X)]}} means "for all X there is, the statement {{M|\varphi(X)}} holds" 
** This statement alone doesn't say what {{M|\varphi}} is, {{M|\varphi}} is said to be a [[free variable]] 
Any free variables left when one has read an entire statement are assumed to be "for all of them that there are", for example:
* {{M|\forall X[\varphi(X)]}} is just short for {{M|\forall \varphi[\forall X[\varphi(X)]]}}; note that this statement does not have {{M|\varphi}} "free" any more.
* {{M|\exists\varphi[\forall X[\varphi(X)]]}} does not have {{M|\varphi}} free either and says "there exists a {{M|\varphi}} such that ( for all {{M|X}} we have ( {{M|\varphi(X)}} ) )
These are clearly different statements
{{End Proof}}{{End Theorem}}{{XXX|Fix formatting}}</ref>
</references>