The list

Logic Level 1

You find a forgotten list on an ancient piece of paper while cleaning your professor's office. It has the following written down:

Exactly 1 statement on this list is false.
Exactly 2 statements on this list are false.
Exactly 3 statements on this list are false.
Exactly 4 statements on this list are false.

How many statements on the list are in fact false?

0 1 2 3 4

8 solutions

Denton Young
Jul 6, 2017

Relevant wiki: Truth-Tellers and Liars

Clearly there can only be one true statement, since all the statements say different things. Since there are four statements, that means that $(4 - 1) = 3$ statements are false.

Moderator note:

Note this answer works even if there are $n$ statements:

Exactly 1 statement on this list is false.

Exactly 2 statements on this list are false.

...

Exactly $n-1$ statements on this list are false.

Exactly $n$ statements on this list are false.

It can't be the case all the statements are false (otherwise the last statement would be true, forming a contradiction). Therefore there must be at least 1 true statement. Since all statements are mutually exclusive, there must be exactly 1 true statement, implying the true statement is "Exactly $n-1$ statements on this list are false."

These commenters (and the answer) are all wrong for 2 reasons: 1. They all assume there is at least one true statement. That is not given and, therefore, all of the statements could be false and none true; if there are only 4 statements on the paper. however, ... 2. They also all assume there are only 4 statements on the paper. There could be dozens. Since statement 4 is stated as factual, it is true.

Sam Johnson - 3 years, 10 months ago

If every statement was false, then the statement "Exactly 4 statements on this list are false." would be true and thus a contradiction.

Jason Dyer Staff - 3 years, 10 months ago

Not necessarily. If there were six more statements, 4 does not have to be true

Cendra Lynn - 3 years, 10 months ago

If there were six more statements, the question would have said so. "It has the following written down" means that what is shown are the only things written down.

Denton Young - 3 years, 10 months ago

so.....which one is the TRUE statement?

Mirthon Manon - 3 years, 10 months ago

Statement 3.

Denton Young - 3 years, 10 months ago

And what will happen if all the statements are true?

Aditya Prasad - 3 years, 10 months ago

Where does it say one statement is true? All four could be false, making 0 the certain correct answer

Cendra Lynn - 3 years, 10 months ago

The question was how many statements are FALSE. I guarantee that's more than 0. In fact, it's 3. There are only 4 statements, and they can't all be false (otherwise statement 4 would be true, a contradiction.)

Denton Young - 3 years, 10 months ago

Peter Macgregor
Jul 17, 2017

Since every pair of statements say contradictory things there are either NO true statements or exactly ONE true statement.

First suppose there are NO true statements. Then statement 4 is false, which means that at least one of the statements is true, which contradicts the premise that there are NO true statements.

So there must be exactly ONE true statement, and $\boxed{three}$ false statements.

We can go a little further and say that statement number three is true, and the other three statements are false.

How do we know if there are any statements it might be just a list of butterflies or a shopping list

Brett Lanham - 3 years, 10 months ago

The post did not contain any info about the situation when I looked at it so any of the answers could be true or false

Brett Lanham - 3 years, 10 months ago

Are you saying that we need to clarify that each of these statements must be true or false only?

Pi Han Goh - 3 years, 10 months ago

Munem Shahriar
Jul 17, 2017

He just asked for how many false statements. There are four statements and all statements can't be false among the four statements. Only one statement is true and the rest are false. Therefore $4 - 1 = \boxed{3}$

Are you sure that $4 - 3 = 3$ ? Check your post again (haha)!

Toby M - 3 years, 10 months ago

It's a typo, Sorry about it.

Munem Shahriar - 3 years, 10 months ago

Why aren't all statements true? Therefore, all are possibly false. Isn't all about perspective since we don't see the actual statement ore content to evaluate scientifically, logically or emotionally - which in all cases, is subjective?

Richard Taylor - 3 years, 10 months ago

In any possible world, all statements cannot be true together. Exactly 1 and Exactly 2 are incompatible for example. And not all statements could be False, since that'd make the last statement true making a contradiction.

Isn't all about perspective since we don't see the actual statement ore content to evaluate scientifically, logically or emotionally - which in all cases, is subjective?

The formal interpretation of the problem is this: If there is a consistent model in which all the statements are assigned a truth value, then how many of the statements would have been assigned True .

The formal interpretation above is a concrete perspective. However, you made a point that philosophers including Saul Kripke has thought about before:

Kripke proposes a solution in the following manner. If a statement's truth value is ultimately tied up in some evaluable fact about the world, that statement is "grounded". If not, that statement is "ungrounded". Ungrounded statements do not have a truth value. Liar statements and liar-like statements are ungrounded, and therefore have no truth value.

Wikipedia

Agnishom Chattopadhyay - 3 years, 10 months ago

Jeremy Ho
Jul 19, 2017

Letting $P(n)$ be the statement 'Exactly n statements on this list are false', we know $P(0)\lor P(1)\lor P(2)\lor P(3)\lor P(4)$ . We also know $(\forall i\neq j)\,\,\,P(i)\land P(j) = False$ .

Therefore there can be no more than two true statements, so $P(0)=False$ . Therefore at least one of the statements must be true.

$\implies P(3)$

Venkatachalam J
Jul 16, 2017

In the given list of 4 statements, clearly If one statement is true then the remaining 3 statements are false.

Therefor there are exactly 3 false statements in the list.

Mohammad Khaza
Jul 7, 2017

i think there can only be one true statement, as all the statements say different things. Since there are four statements, that means (4-1)=3 statements could be false

I think when the fourth statement says exactly four statement on this list is false meaning it is itself is false including other three and contradicts itself. When third statement says exactly three meaning other three statement is false, same goes for second and first statement but 3rd statement gives maximum possible wrong statement.

Anu Ai - 3 years, 11 months ago

Joshua Powles
Jul 20, 2017

For $n=4$ of these statements, statement $n-1=3$ must be true as the others lead to contradiction. Statement $n$ would require itself to be false as well as true, and statements below $n-1$ would require more than one of the statements to be true at the same time.

Timmy Kostolansky
Jul 17, 2017

Only one of the statements can be true because all statements contradict one another. Thus, the only true statement is the third one and there are 3 false statements .

Yep, as with other single solution T/F problems I look for the statement that fits the premise . If that makes sense.

Tristan Graham - 3 years, 10 months ago

The list

8 solutions

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