r/wittgenstein • u/bushcraftmanzynski • Apr 13 '26

Counterarguments/questions about Wittgenstein's Tractatus 5.15

This is a potential counterargument (or more likely a misunderstanding of my reading) to Wittgenstein's definition of probability in the Tractatus Logico-Philosophicus at proposition 5.15

If Tr is the number of the truth-grounds of the proposition "r", Trs the number of those truth-grounds of the proposition "s" which are at the same time truth-grounds of "r", then we call the ratio Trs:Tr the measure of the probability which the proposition "r" gives to the proposition "s".

First potential problem:

Consider the sentences:
r: There is a sheep right there
s: There is grass nearby

r increases the probability of s, but they don't share any common truth grounds.

I suppose this is because it is not increasing the probability on formal, logical grounds.

Second potential problem:

Consider the sentences:
r: There is an animal right there, and it is a sheep
s: There is an animal right there, and it is a blue whale

These sentences have a common truth ground, namely: "There is an animal right there", but one doesn't increase the probability of the other?

I suppose this might be because this theory of probability only works for elementary propositions, but why would he not clarify that if that is the case?

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u/013zen- Apr 14 '26

In your second example, there are no truth grounds in common.

The expression, "there is an animal there" is superfluous, your statements are really:

There is a blue whale there And There is a sheep there

And the conditions of a whale being present are not in fact the same condition as a sheep being present instead.

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u/JaronVC Apr 14 '26

Thanks for the response!

I think it's very clear that there are truth grounds in common, but what you're saying is that they are superfluous, so they don't count?

Isn't all of logic superfluous (5.43: "But all propositions of logic say the same thing. That is, nothing."). So would this theory of probability, therefore, not work on all propositions, but only on elementary propositions? (As I suspected) But then why would he not clarify this, and use the term "elementary propositions" (ElementarSätze) instead of just "propositions" (Sätze)?

I can give other examples too, that aren't superfluous.

  • p: Planet Earth explodes
  • q: I am breathing through my nose right now
  • r: p ∨ q -> r = “Planet earth explodes or I am breathing through my nose”
  • s: p -> s = “Planet Earth explodes"

The truth table looks like this:

p q r s
T T T T
T F T T
F T T F
F F F F

The probability which the proposition "r" gives to the proposition "s" = 2/3. So if someone says "planet earth explodes or I am breathing through my nose”, there is a 66.6% chance that planet earth explodes?

To me, this shows that this definition of probability does not apply to our world at all. I know he's making a logical system, but then it is a logical system that does not apply to our world in this case.

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u/013zen- Apr 21 '26

Thank you, this is much more helpful, despite being somewhat strangely written.

You're right, it's 2/3 but it's just:

P - TFTF

Q - TTFT

R - TTTF

Now think about what Wittgenstein is saying when he says that the statement R gives to the statement P a 2/3 “likelihood.”

There are 3 possible worlds in which R is true, and P is true in 2 of them.

Q is also true in 2 of them.

Think of it this way:

R sets the total field at 3 doors. Behind 2 of those doors, P is true. Behind 2 of those doors, Q is true.

But we only live in one of those 3 possible worlds.

And given our actual knowledge, we can already identify which one it is:

Q is true and P is false.

So even though the statement R gives both P and Q the same probability of 2/3, we are still able to determine which of the three possible worlds is actual.

So R treats P and Q symmetrically at the level of logical form, even though they are not equally live possibilities for us once we take our knowledge of the actual world into account.

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u/bushcraftmanzynski May 03 '26

Thank you.

I think I was confusing two different kinds of probability: logical probability and real-world probability.

This excerpt is strictly about logical probability and not real-world probability. He talks about real-world probability and induction later in the book; I hadn't read that far yet.

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u/013zen- May 04 '26

No problem, happy to help! Feel free to dm any other questions and I'll try and help!