# What probability that a premise is true is sufficient for the premise to be believed?

Imagine that there is some probability, for any given premise, that the truth of some conclusion depends on the truth of that premise. This means that as the number of unproven premises increases, so does the probability that these premises and that conclusion are not coincidental. How high should the probability be, to be deemed acceptable?

• It depends very much on the context. For a proposition that is life-critical, you'd better be very sure. For things that will cost you a lot of money if you're wrong, somewhat less so. For things that don't matter much, better than 50:50 might be enough. Commented Feb 7 at 0:44

Any probability is fine. For a single premise, the valid conclusion inherits the probability of the premise. For multiple premises, the valid conclusion inherits the statistical concatenation of the probabilities of the premises.

For instance, you and I are playing a dice game. If I roll 4 or higher, I get to roll again. If I succeed twice, you go. If you succeed twice, we tie, otherwise I win the round.

We can input premises:

I roll 4 or higher on roll 1. AND I roll 4 or higher on roll 2. AND [You roll 3 or under on roll 3 OR you roll 3 or under on roll 4].

And argument:

Premises as above. Rules as above. Therefore I win.

The probability of "I win" is the statistical concatenation of the probabilities: .5 AND .5 AND [.5 OR .5] = 0.5 x 0.5 x [1-0.5 x 0.5] = 0.1875

For situations in which input conditions are not exhaustive, for the associated output, one should use a greater-than-or-equal-to for the probability of the conclusion, not an equals sign.

For instance:

• It is raining (with probability 30%)
• If it is raining, the sidewalk is wet
• Therefore the sidewalk is wet (with probability >=30%: someone may have spilled a bucket of water on the sidewalk, for instance).

Which is similar to the exhaustive premises in

• It is raining (with probability 30%)
• If it is raining, the sidewalk is wet
• OR something else has wet the sidewalk (with probability 20%)
• Therefore the sidewalk is wet (with probability 1-(1-0.3)x(1-0.2) = 44%).

Your question is misguided in at least two ways. Firstly, people hold many beliefs without ever making any explicit association between their beliefs and probabilities. Secondly, even where an explicit judgement might be made on the basis of probabilities, there is no standard threshold the attainment of which justifies belief. For example, the standard required to satisfy physicists that the Higgs boson exists was much higher than the standard adopted in criminal law cases.

The mechanic has called... your car is repaired, and your fuel pump is replaced... come get it at your convenience.

Paid and all that... you are on your way out of the auto shop with your keys, when the mechanic says...

"Oh. By the way. I am only pretty sure the fuel pump was the problem,

it may be one of three other things, but they are all very unlikely...

so... the good news is... even if it wasn't the fuel pump, your car probably won't burst into flames on the highway..."

"...probably not."

Are you going to "believe"?

I would be asking for more details personally.

I tried to reserve "belief" for things that don't matter much,

... and use "informed conclusion" for things that matter a lot, or "reserve judgment pending more information" for things that can't be concluded.

What probability that a premise is true is sufficient for the premise to be believed?

Suppose a coin is flipped and hidden from view by a curtain. The coin could be fair 50/50 or it could be 90/10 in favor of heads.

Until the curtain is drawn and the actual state of the coin is observed, you can't be certain of the state of the coin.

Probability does not exist mind independently. Once you and the rest of the philosophical community recognizes this, your answer will become clearer.

There is no probabilistic threshold above which certain premises should be accepted. This isn’t because we don’t know what the probability is. Rather, it is because it doesn’t exist.

There is no objective probability to the statements “fairies exist”, “leprechauns exist”, or heck even “the sun will rise tomorrow.” We assign probabilities as a tool based on how strongly we believe in certain things.

But how strongly we believe in something has no bearing on what’s true. The majority of the world strongly felt that the earth was flat many centuries ago.