MHB Writing a statement into symbolic logic

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The discussion focuses on translating the statement "The square of every odd integer is one more than an integral multiple of 4" into symbolic logic. Participants suggest different symbolic representations, with one proposing the formula $$\forall n \in \mathbb Z: n\text{ odd} \to n^2 \bmod 4 = 1$$ and another suggesting $$\forall m\exists n\,(2m+1)(2m+1)=4n+1$$. The conversation emphasizes the importance of defining the logical vocabulary and interpretation for clarity. Overall, the goal is to accurately represent the mathematical statement in symbolic form. The discussion highlights the nuances of translating mathematical expressions into logic.
cbarker1
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Dear Everyone,

I need to translate this following statement into a symbolic logical form of the statement:

The square of every odd integer is one more than an integral multiple of 4.

Thanks,

Cbarker1
 
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Cbarker1 said:
Dear Everyone,

I need to translate this following statement into a symbolic logical form of the statement:

The square of every odd integer is one more than an integral multiple of 4.

Thanks,

Cbarker1

Something like:
$$\forall n \in \mathbb Z: n\text{ odd} \to n^2 \bmod 4 = 1$$
? (Wondering)
 
Cbarker1 said:
The square of every odd integer is one more than an integral multiple of 4.
You can denote the whole statement by a single letter, say, $P$. This is to show that in order to make the problem meaningful, the problem author must specify the signature, or vocabulary: constants, functional symbols and predicate symbols that can be used in the formula. Ideally the author should also specify the interpretation of that signature because this statement is written differently over natural numbers and over reals.

But, guessing the author's intent, the answer is probably
\[
\forall m\exists n\,(2m+1)(2m+1)=4n+1.
\]
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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