Bastard's object

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It would be good to have a category of bastards' objects and a stricter definition. This will do for now. Alec (talk) 11:54, 5 May 2018 (UTC)

Caution:

TODO: Template:Abbr

not to be confused with: a counter-example, see below

Definition

Vs a counter-example

Given a statement;

• [ilmath]\forall X[\varphi(X)][/ilmath]^{In words:} made for some [ilmath]\varphi[/ilmath]

to "prove or disprove it" we must establish either the statement holds, or it does not.

Informal discussion:

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A counter-example is a demonstration that a statement cannot be true. Suppose for example there is an item, [ilmath]\mathcal{B} [/ilmath] in our language for which [ilmath]\varphi(\mathcal{B})[/ilmath] is known or easily found to be false

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

Informal examples:

1. 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"
2. all sequels are worse than the originals

If we show that:

• [ilmath]\exists Y[\neg[\varphi(Y)]][/ilmath]

Then such a [ilmath]Y[/ilmath] 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.