MHB Working with decidable predicates

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I am currently reading this article

Here

I was wondering how to calculate the contents of it
Mainly the first equation equaling 1 or 0
 
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This article does not seem to have numbered equations, and the first formula on its own line is not an equation.
 

Thread 'Deductive proof in logic formal systems'

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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