∴ "If exactly one being is F and exactly one being is G and nothing is F-and-G, then exactly two beings are F-or-G." ∴ (((∃x)(Fx∧¬(∃y)(y≠x∧Fy))∧(∃x)(Gx∧¬(∃y)(y≠x∧Gy)))∧¬(∃x)(Fx∧Gx)) ⊃ (∃x)(∃y)(((Fx∨Gx)∧(Fy∨Gy))∧(x≠y)∧¬(∃z)((z≠x∧z≠y)∧(Fz∨Gz))) or ∴ (((∃x)(Fx • ∼(∃y)(∼y=x • Fy)) • (∃x)(Gx • ∼(∃y)(∼y=x • Gy))) • ∼(∃x)(Fx • Gx)) ⊃ (∃x)(∃y)(((Fx ∨ Gx) • (Fy ∨ Gy)) • (∼x=y • ∼(∃z)((∼z=x • ∼z=y) • (Fz ∨ Gz))))
Oh, and where did you get that tautology should not be proven? — Nicholas Ferreira
The first sentence is the english paraphrase of "1+1=2", — Nicholas Ferreira
Why does the conclusion permits "(x or y) or (x & y) to be F & G" if it is said that nothing is simultaneously F and G? — Nicholas Ferreira
sime
This is said in the antecedent, not in the conclusion — Nicholas Ferreira
But why would the conclusion need to explicit something that already has been said in the premise? I mean, in "P⊃Q", for instance, if you analyze only the consequent, you'll see that "Q" permits "¬P", which is denied by the antecedent. I'm not quite sure what kind of analysis are you doing but I think that analyzing only the conclusion without considering what was stated in the premise isn't the right way. — Nicholas Ferreira
Do you think you can prove that 1+1=2?
Of course, but you were saying about the consequent only, not about the entire implication. That is why I said that in "P⊃Q", "Q", alone, permits "¬Q".P⊃Q doesn't permit Q⊃¬P in a consistent logic — sime
Hm, I don't know if I understood. For instance, in the sentence "(¬(∃x)(Fx∧Gx) ∧ Fx) ⊃ (Fx∨Gx)", you would say that the consequent "Fx∨Gx" permits "Fx∧Gx"? (It's just an example for me to understand, I'm not saying this is the case)That case is different to the set-theoretic case, where Fx ∨ Gx permits Fx ∧ Gx and is therefore a weaker statement than the latter. — sime
Well, why couldn't we treat it like so? I mean, I could say that the antecedent is the premise and the consequent is the conclusion, and since the conclusion follows from the premise, I could represent they in a conditional statement. I didn't understand part of your latter paragraph... English isn't my native language and i'm not familliar with a lot of terms you used.Of course, in a sense your antecedent might be said to contain your "conclusion" as a weaker premise, but i think it is a mistake to think of your right-hand side as a conclusion because it must forever remain tied to the antecedent if it isn't to be misinterpreted as allowing F and G to be overlapping sets containing multiple members... assuming of course, that you want to represent the number 2 as a union of pairwise disjoint singleton sets. — sime
The statement of "1+1=2" in Peano arithmetic is:“1”, “+”, “=“, and “2” have specific meanings by convention. So, “1+1=2” is a tautology. It has to be true given the meanings of the terms used. There is nothing to prove. — Noah Te Stroete
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