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I have four propositions $A, B, C, D$ each of which is either true or false. These statements are completely arbitrary and have no relation to each other. Is it logically sound to make the statement, $P(A=T)=\frac{1}{2}$, that is, the probability of A being true is one half? - I realised my initial question had this as a presupposition so I thought it best to check this too.

My actual question is this:

Given the same $A, B, C, D$ and then add in these relations

$TWO(A, B, C)$ (two of A, B and C are true, one is false)

$A \rightarrow D$

$B \rightarrow D$

Can we conlude $P(D=T)=\frac{2}{3}$ or is there a technical reason why applying probabilities to logic is fallacious in some sense? (I'm new to formal logic and appreciate I may have misused a technical term somewhere)

Thanks in advance, Ben

Ben Crossley
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  • In the first paragraph, you wrote $A,B,C, D$ 'have no relation to each other', while in the specific question, $A\to D$ and $B\to D$ are imposed. – Berci Jun 06 '19 at 14:45
  • These additional relations are 'later' established. I think the best way to explain this would be with sudoku. In the beginning, you may have a few empty cells that bear no relation to each other but as the puzzle evolves you may learn of some relations between cells. Maybe $A1=3 \rightarrow A2=4$, I began with some propositions that had no relation and then added in these relations. I hope that is reasonable. – Ben Crossley Jun 06 '19 at 14:52
  • Do you mean $A, B, C = 0.5,...$ find $P(D)$? If $A, B, C$, and $D$ are all equally likely then you either have your answer ($P(D)=P(A)5$ by definition) or a contradiction (e.g $\frac{2}{3}=\frac{1}{2}$). – R. Burton Jun 06 '19 at 15:34

1 Answers1

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It depends on what the statements are, but most likely not.

You can only do this if you know that $A,B,C$ are all equally likely to occur. This is obviously a major point in probability -- all outcomes must be equally likely. A more subtle note is that they are actually probabilistic. This means they can't be statements whose truthiness or falseness is determined, for instance a statement asserting that the twin prime conjecture is true. Even though this is unproven, it's still either true or untrue; there's no chances involved.

auscrypt
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  • I would have thought that the twin prime conjecture is currently at $P(T)=0.5$ I think that probabilities would only come into it when XOR generalisations come into play. I've just made another question asking about that term too. https://math.stackexchange.com/questions/3253158/name-for-this-family-of-operations – Ben Crossley Jun 06 '19 at 15:13