# Is the argument below valid?

If interest rates go down, then I will buy a house. If I buy a house, I will need a loan. Therefore, I will not need a loan if I do not buy a house.

Is this argument valid?

• Proposed title edit: Is P → Q, therefore ~P → ~Q a valid argument? – mic Apr 16 '19 at 14:22
• @MiCl I think there is more than that going on in the question. There are two premises not just one. How does one show that the first premise about interest rates does not provide enough information for a valid argument? – Frank Hubeny Apr 16 '19 at 14:29
• What about: “Is P → Q, Q → R, therefore ~Q → ~R a valid argument?” – mic Apr 16 '19 at 14:33
• @MiCl Yes, "P → Q, Q → R, therefore ~Q → ~R" seems to symbolize the argument. – Frank Hubeny Apr 16 '19 at 14:35
• @MiCl I'm not sure OP knew this is the form of the argument in the post. Formalizing the argument is part of the answer in this case, so I don't think it should be edited into the question. – Eliran Apr 16 '19 at 16:13

Is the argument valid?

No.

"I will not need a loan if I do not buy a house" is the same as "If I do not buy a house, then I will not need a loan".

This is not implied by "If I buy a house, I will need a loan".

Wikipedia describes validity as follows:

In logic, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.

The argument we want to test for validity is the following:

If interest rates go down, then I will buy a house. If I buy a house, I will need a loan. Therefore, I will not need a loan if I do not buy a house.

This can be broken up into propositions with this symbolization key:

• R: "Interest rates go down."
• B: "I will buy a house."
• L: "I will need a loan."

If R then B. If B then L. Therefore, if not B then not L.

We could place the following into a truth table generator. For the truth table generator I am using I would enter the following string:

((R=>B)&&(B=>L))=>(~B=>~L)

This is the result I get: Note the "F" in the third line of the table. This is a line where the premises are true but the conclusion false. Therefore the argument is invalid.

Stanford Truth Table Tool http://web.stanford.edu/class/cs103/tools/truth-table-tool/

Wikipedia contributors. (2019, March 28). Validity (logic). In Wikipedia, The Free Encyclopedia. Retrieved 18:05, April 15, 2019, from https://en.wikipedia.org/w/index.php?title=Validity_(logic)&oldid=889899195

• Do I understand the third line correctly as "in the event that interest rates don't go down (R=false), and therefore I don't buy a home (B=false), I may still need a loan (L=true)"? That makes sense, as the concrete case the OP seems to be missing is that people get loans for many other purposes. – Jon of All Trades Apr 16 '19 at 14:13
• @JonofAllTrades Yes, that would be a way to view the situation. Then the premises "(R=>B)&&(B=>L)" are true, but the conclusion "~B=>~L" is false. That valuation or assignment of truth values to the propositions makes the argument invalid. – Frank Hubeny Apr 16 '19 at 14:24

The last statement suggests that buying a house is the only reason you would need a loan. Not buying a house does not rule out other reasons for needing a loan. Therefore it's logically false.

If it were explicitly stated that you would only ever need a loan when buying a house, it would be logically correct, even though it would be potentially false in reality.

All the upvoted arguments are valid. Here's just another way of phrasing the answer.