proving the validity of this argument

bonfire09

Either sales or expenses will go up. If sales go up, then the boss will be happy. If expenses go up, then the boss will be unhappy. Therefore, sales and expenses will not both go up.

How would I go about breaking this apart? My attempt goes like this

S- be sales go up
E- expenses go up
B-boss be happy

premise 1. S v E
premise 2. S→B
premise 3. E→~B

∴ ~(SΛ E)

what is a bit confusing is that it has two separate if statements. So I assume their two separate premises? If not, then why?

chiro

Hey bonfire09.

This is too much of a simplistic assumption.

Often expenses do go up as the business grows or needs to increase resources for further demand. It's not a simple thing where both things are mutually exclusive and inverses of each other.

It depends also on the model used for producing the core goods and services.

For example in a factory environment, it is required that the business pay initially for constructing the factory and getting all of the equipment, as well as paying for compulsory employee wages and operation of the factory.

But often in these models, what happens is that after a certain point, producing more actually costs pennies on the dollar in comparison to the cost of producing things before that cutoff so if there is demand for this above the cutoff, this is economical due to the nature of how and what is being produced.

But some things don't work like that. For example if you say run some kind of consultancy business that offers 1 on 1 service for clients, then it means that more people are needed to provide more service if the other employees are unable to meet the needs with respect to either time or other resources.

So in this model, more demand will dramatically increase expenses but it will also increase sales and revenue as well.

You need to factor in the context of business model when analyzing these kinds of things and also realize that every situation depending on the nature of the business, how they produce the good and service, and what that actual good or service is in context.

bonfire09

Sorry but I think I gave too little info. Its an argument and I have to prove if its valid or not using a truth table

alan2

I think that the disjunction of S and E is actually unnecessary. Try a 3 column table with just S, E, and B. You will have 4 rows for the possible combinations of T and F for S and E. You will find one entry in the B column is indeterminate.

Aside from that, I think the question is poorly worded. The word either sounds to me like an exclusive or, which would eliminate the possibility that both S and E are true. But the point is that if S is true and E is true then B is true and B is not true. Therefore, both S and E can not be true.

alan2

Sorry, I should correct myself. You have a 3 row table since the statement of the problem precludes the case S is false and E is false.

bonfire09

well I tried constructing a table like this

S E B SvE S→B E→~B ~(SΛE)
F F F F T T T
T F F T F T T
F T F T T T T
T T F T T T F
F F T F T T T
T F T T T T T
F T T T T F T
T T T T T F F

im guessing its valid according to this table since none of the lines has all true values and a false conclusion.

alan2

It absolutely is valid. I just think that you've overcomplicated it. Rows 1 and 4 are not possible from the statement of the problem. Row 2, S=T, E=F, B=F is impossible, etc. There are only 3 rows which need be considered but I've always been a firm believer in keeping things simple for clarity. Some instructors disagree.