# How do I formulate "petitio principii" in the formal terminology of logic?

**URL:** <https://boards.straightdope.com/t/how-do-i-formulate-petitio-principii-in-the-formal-terminology-of-logic/354134>\
**Category:** Factual Questions\
**Created:** [April 25, 2006, 1:17pm UTC](https://boards.straightdope.com/t/how-do-i-formulate-petitio-principii-in-the-formal-terminology-of-logic/354134 "2006-04-25T13:17:47Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![Wallenstein](https://avatars.discourse-cdn.com/v4/letter/w/b5a626/32.png) [@Wallenstein](https://boards.straightdope.com/u/Wallenstein)\
**Post date:** [April 25, 2006, 1:17pm UTC](https://boards.straightdope.com/t/how-do-i-formulate-petitio-principii-in-the-formal-terminology-of-logic/354134/1 "2006-04-25T13:17:47Z")

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Is it possible to describe the principles behind “begging the question” - _petitio principii_ - using formal logic terminology (ie. P’s, Q’s etc)?

I only took one semester of logic before heading gleefully off to the lush fields of marxist theory, so am struggling a bit.

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [April 25, 2006, 1:22pm UTC](https://boards.straightdope.com/t/how-do-i-formulate-petitio-principii-in-the-formal-terminology-of-logic/354134/2 "2006-04-25T13:22:41Z")

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I think it falls under the general heading of circular logic. The general form is this:

(P [symbol]®[/symbol] P) [symbol]®[/symbol] P

Any specific instance will have a bit more to the argument, but you’ll see something like that in there.

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**Author:** ![Wallenstein](https://avatars.discourse-cdn.com/v4/letter/w/b5a626/32.png) [@Wallenstein](https://boards.straightdope.com/u/Wallenstein)\
**Post date:** [April 25, 2006, 1:38pm UTC](https://boards.straightdope.com/t/how-do-i-formulate-petitio-principii-in-the-formal-terminology-of-logic/354134/3 "2006-04-25T13:38:34Z")

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Thanks for that 🙂

Can I just check… does _petitio principii_refer only to tautological statements - i.e. _circulus in probando_ claims - or can it also refer to statements whose premise is itself unproven.

An example would be “because capital punishment reduces overall levels of crime, we should execute more people”.

This is the sort of example that Fowler uses, but I wasn’t sure whether it fits the example on Wikipedia:

p implies q  
q implies r  
r implies p  
suppose p  
therefore, q  
therefore, r  
therefore, p.

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [April 25, 2006, 4:14pm UTC](https://boards.straightdope.com/t/how-do-i-formulate-petitio-principii-in-the-formal-terminology-of-logic/354134/4 "2006-04-25T16:14:44Z")

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I’m not sure about all those Latin terms. Informal logic has never been my strong suit.

The argument you posted, in its simplest form, is ((P [symbol]®[/symbol] Q) & (Q [symbol]®[/symbol] R) & (R [symbol]®[/symbol] P) & P) [symbol]®[/symbol] P. That’s equivalent to (P & Q & R) [symbol]®[/symbol] P, which is a tautology. So it can’t be used to prove that P is true.

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**Author:** ![CookingWithGas](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cookingwithgas/32/485_2.png) [@CookingWithGas](https://boards.straightdope.com/u/CookingWithGas)\
**Post date:** [April 25, 2006, 4:19pm UTC](https://boards.straightdope.com/t/how-do-i-formulate-petitio-principii-in-the-formal-terminology-of-logic/354134/5 "2006-04-25T16:19:57Z")

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> [@ultrafilter](#):
>
> I think it falls under the general heading of circular logic. The general form is this:
> 
> (P [symbol]®[/symbol] P) [symbol]®[/symbol] P
> 
> Any specific instance will have a bit more to the argument, but you’ll see something like that in there.

Not sure I follow this form as a demonstration of begging the question. In words, this means

if (If P, then P), then P

but the statement

If P, then P

is a tautology and therefore always true. So the original expression reduces to

If TRUE, then P

Or

P

That is not begging the question, it’s just an assertion that P is true. You do not need to assume that P is true to reduce the expression to a statement that P is true.

Begging the question presumes the truth of what you are trying to prove as a necessary condition of the proof. The example quoted by e-logic is typical.