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This is the context:

In other words, P says

“This logical sentence does not have a proof shorter than n.”

or

“I do not have a short proof.”

We call such a logical sentence a Parikh sentence. Let us determine if this sentence is true or false. If P were false then a (short) proof of P does exist. But how can there be a proof of a false statement within a consistent system? So the sentence is not false and must be true. As we saw above with Gödel’s incompleteness theorem, just because a statement is true, does not mean it is provable. Now let’s consider the following relatively short proof that a (long) proof of the Parikh sentence exists:

If the Parikh sentence does not have a proof, then in particular it does not have a short proof. Then we can easily check all proofs less than n and see that none of them prove P. Summing up: if the sentence cannot be proved, then we can prove it.

Source: The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us by Noson S. Yanofsky

This is about Parikh’s Theorem. The problem I am having, because I am not a native speaker, is with "in particular" part. Does it mean "specially"? Or does it mean "definitely/certainly"? Or does it mean something else completely? Because I Don't get it how "specially" can apply to this sentence. If this is not a good question for this forum, Tell me to remove it. Thank you in advance.

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3 Answers

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Do you know "a fortiori"?

It's the "a maiore ad minus" argument, concluding from a general to a more special ("particular") case.

As in:

"All cats are mammals. In particular, all black cats are mammals."

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A red shirt is a 'particular' or 'specific* kind of shirt.

So, I could say:

"If I don't wear a shirt, then in particular I don't wear a red shirt"

This is just like the:

"If there is no proof, then in particular there is no short proof"

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The usage of "n particular" in those contexts means that we made some general assumption or observation (such as "no proof at all"), which logically entails a more specific observation (such as "no short proof"), and this more specific case is the one we're interested in.

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