Identifying if the statement is valid or invalid

[sonadoramante]

Homework Statement

Either John or Bill is telling the truth.
Either Sam or Bill is lying.
Therefore, either John is telling the truth or Sam is lying.

Relevant Equations

Let P stand for John is telling the truth.
Let Q stand for Bill is telling the truth.
Let R stand for Sam is lying.

P v Q
R v Q
-----------
Therefore P v R

Let P stand for John is telling the truth.
Let Q stand for Bill is telling the truth.
Let R stand for Sam is lying.

P v Q
R v Q
-----------
Therefore P v R

P Q R P v Q R v Q P v R

F F F F F F
F F T F T T
F T F T T F
F T T T T T
T F F T F T
T F T T T T
T T F T T T
T T T T T T

Therefore the argument is invalid.

 

[Kyouran]

Either Sam or Bill is lying. Shouldn't this be R v ~Q ?

 

[hutchphd]

This is just a suggestion but why not let J,B, and S stand for each guy respectively telling the truth. Sure would make life easier! Less chance to screw it up...half the battle for me...too late now.

 

[sonadoramante]

Thank you!

J= John is telling the truth.
B= Bill is telling the truth.
S= Sam is telling the truth.

J v B
~S v ~B
------------
Therefore, J v ~S

J B S J v B ~S v ~B J v ~S
F F F F T T
F F T F T F
F T F T T T
F T T T F F
T F F T T T
T F T T T T
T T F T T T
T T T T F T

Hence, this is a valid argument. Thanks once again for the feedback.

 

[Mark44]

Therefore the argument is invalid.

Hence, this is a valid argument.

Surely one of these is correct.

There's a wrinkle here that you might have missed. Here is part of the info given in this problem, using notation suggested by @hutchphd.

Either John or Bill is telling the truth.

Let J stand for John is telling the truth.
Let B stand for Bill is telling the truth.

J v B

I don't believe the last statement, J v B, is correct. When people say, "Either <this> or <that>" what they normally mean is the disjunctive OR rather than the conjunctive OR.
In other words, with "Either John or Bill is telling the truth." what they mean is that
1) John is telling the truth and Bill is lying,
or
2) John is lying and Bill is telling the truth.
J v B means "John is telling the truth and Bill isn't OR Bill is telling the truth and John isn't OR both are telling the truth.
The Either part disqualifies both from being truthful, and is saying that exactly one of them is telling the truth.

 

[PeroK]

Surely one of these is correct.

There's a wrinkle here that you might have missed. Here is part of the info given in this problem, using notation suggested by @hutchphd.

I don't believe the last statement, J v B, is correct. When people say, "Either <this> or <that>" what they normally mean is the disjunctive OR rather than the conjunctive OR.
In other words, with "Either John or Bill is telling the truth." what they mean is that
1) John is telling the truth and Bill is lying,
or
2) John is lying and Bill is telling the truth.
J v B means "John is telling the truth and Bill isn't OR Bill is telling the truth and John isn't OR both are telling the truth.
The Either part disqualifies both from being truthful, and is saying that exactly one of them is telling the truth.

Are you sure? I would assume that A or B means A or B or both. Either, but not both, would be an exclusive or: XOR.

 

[PeroK]

Homework Statement:: Either John or Bill is telling the truth.
Either Sam or Bill is lying.
Therefore, either John is telling the truth or Sam is lying.

Another approach is to work backwards. I'll use a capital letter for telling the truth and lower case for lying.

The first interpretation is that the conclusion is that we have one of these three cases $Js, JS, js$. That means either John is telling the truth, sam is lying or both. That means that the only case not possible is $jS$ (john lying and Sam telling the truth).

We need to check that the first two statements exclude the case $jS$. Therefore, we need to show that if john is lying, then Sam cannot be telling the truth. I.e. if john is lying then so is sam.

Assume john is lying. Then (by first hypothesis) Bill is telling the truth. And if Bill is telling the truth, then (by the second proposition) sam is lying. Hence, the first two propositions exclude the case jS; and the conclusion is valid.

You could try this approach with the "exclusive or" interpretation.

 

[Mark44]

Are you sure? I would assume that A or B means A or B or both. Either, but not both, would be an exclusive or: XOR.

That's exactly what I'm saying. The OP translated "Either John or Bill is telling the truth." as if it were "John or Bill is telling the truth." The latter interpretation would be OR; the former would be XOR.

 

[PeroK]

That's exactly what I'm saying. The OP translated "Either John or Bill is telling the truth." as if it were "John or Bill is telling the truth." The latter interpretation would be OR; the former would be XOR.

I'm really confused now.

 

[Mark44]

What I'm saying is that if the problem were stated as "Either John or Bill is telling the truth." then I would agree with the OP's conclusion of J v B.
But the presence of "Either" here suggests to me that it's one or the other telling the truth, but not both.

 

[PeroK]

What I'm saying is that if the problem were stated as "Either John or Bill is telling the truth." then I would agree with the OP's conclusion of J v B.
But the presence of "Either" here suggests to me that it's one or the other telling the truth, but not both.

Ah, so "either ... or" could be taken to be XOR. Okay, but sooner or later we need concrete terminology rather than woolly English.

 

[sonadoramante]

Are you sure? I would assume that A or B means A or B or both. Either, but not both, would be an exclusive or: XOR.

You're right. It is inclusive (OR)

 

[sysprog]

When people say, "Either <this> or <that>" what they normally mean is the disjunctive OR rather than the conjunctive OR.

They're called 'exclusive disjunction', and 'disjunction', respectively (as of course you know).