Why Is There Exists X Such That X + Y = Z for All Y and Z False?

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The statement "there exists X such that for all Y and Z, X + Y = Z" is proven false by demonstrating that no single value of X can satisfy the equation for all possible Y and Z. If X is set to -1, for example, choosing Y as 2 and Z as 3 results in -1 + 2 = 1, which does not equal Z. Additionally, for any X other than -1, setting Z to X + 1 and Y to X + 2 leads to a contradiction, as it requires X to equal -1 to hold true. Thus, the existence of such an X is impossible. The discussion clarifies the logical reasoning behind the falsehood of the statement.
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Hey guys,

do you now how to prove that the statement "there exists X such that for all Y and Z X+Y=Z" is FALSE. Seems easy but I can't figure out any counter example to the statement.

Thanks
 
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Do you understand WHY it is false? "There exists X such that for all Y and Z, X+ Y= Z" means that we can find a specific X such X+ Y= Z for any Y and Z- in other words, X is chosen FIRST. That is there must exist a specific number X so that X+ Y= Z for any Y and Z.

X+Y= Z is the same as X= Z- Y. Okay, suppose X= -1. Choose Y= 2 and Z= 3. X+ Y= -1+ 2= 1 which is NOT equal to Z= 3. The statement is false for X= -1. Now suppose X is any number other than -1. Let Z= X+1 and Y= X+ 2. Then X+ Y= X+ X+ 2= 2X+ 2. That is equal to Z=X+ 1 only if 2X+2= X+ 1 or X= -1 which is false.
 

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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