# How can I make a valid conclusion about a set of statements which are in tandem?

The problem is as follows:

If there are no government subsidies for agriculture, then there are government controls on agriculture. If there are government controls on agriculture, then there is no agricultural depression. There is depression or agricultural overproduction. It is a fact that there is no overproduction. Indicate the true alternative.

The alternatives given in my book are as follows:

$$\begin{array}{ll} 1.&\textrm{There are government controls on agriculture.}\\ 2.&\textrm{There is no economic depression.}\\ 3.&\textrm{There are no government subsidies for agriculture.}\\ 4.&\textrm{There are government subsidies for agriculture.}\\ \end{array}$$

I'm confused as to how exactly I should proceed with this sort of riddle. It looks kind of convoluted. I think it requires the use of logic operators but I can't recall exactly how to use them. Does any logic simplification exist which should be used? Can someone help me with a step by step approach?

If there are no government subsidies for agriculture, then there are government controls on agriculture. If there are government controls on agriculture, then there is no agricultural depression. There is depression or agricultural overproduction. It is a fact that there is no overproduction. Indicate the true alternative.

A step by step approach would start from the back:

There is depression or agricultural overproduction. It is a fact that there is no overproduction.

Therefore:

It is also a fact that there is depression.

If there is depression and we know

If there are government controls on agriculture, then there is no agricultural depression

We also know

There is no government control

Therefore from

If there are no government subsidies for agriculture, then there are government controls on agriculture

We know

There must be government subsidies

Giving the correct option

4

For a logical simplification we can use logic gates, but its the same approach:

If there are no government subsidies for agriculture, then there are government controls on agriculture. If there are government controls on agriculture, then there is no agricultural depression. There is depression or agricultural overproduction. It is a fact that there is no overproduction. Indicate the true alternative.

Use the follow to define events:

Let A = Government subsidies
Let B = Government control
Let C = Agricultural depression
Let D = Agricultural overproduction

From the question we know that

NOT A = B
B = NOT C
(C OR D)
NOT D

So following the chain:

NOT D = C = NOT B = A

A third and final way to solve would be to consider each of the options given:

1: If there are government controls, there are no government subsidies. There is therefore no depression. Which means there must be overproduction. This counters the true statement so is FALSE

2: If there is no economic depression, there must be overproduction. This immediately counters the true statement so is also FALSE

3: If there are no subsidies, there is government control. There is therefore no depression, which means there is overproduction, which counters the true statement so this is therefore also FALSE

4: If there are subsidies, there is no government control. There is therefore a depression, which means there is no overproduction, which is TRUE

Hope this helps!

• The logic gates you have used can be exchanged with ¬ for not and ^ for or? and v for and?. How would it be using this notation?. The equal sign here is equivalent to then?. Commented Mar 18, 2020 at 0:57
• @ChrisSteinbeckBell yeah I have simplified the gates massively, but they could be exchanged for official symbols for each and IF ... THEN statements Commented Mar 18, 2020 at 7:25
• @BeastlyGerbil - I believe that your inclusion of NOT (C AND D) is incorrect. I believe that in propositional logic, an "or" statement is always assumed to be an "inclusive or" unless explicit words to the contrary (such as "but not both") are included in the statement. Commented Mar 18, 2020 at 9:46
• @LannyStrack I agree, and the only reason I Included it is because (I think?) that it is impossible to solve without this assumption, not sure if it should say ‘either’ in the question Commented Mar 18, 2020 at 9:48
• @BeastlyGerbil - No, it is not impossible to solve without that assumption. Even with an inclusive or, the combination of (A or B) and (not B) implies that A must be true. Commented Mar 18, 2020 at 9:52

@ChrisSteinbeckBell - Because you haven't replied to the previous answer yet, I am not sure if you are fully comprehending the kind of logical reasoning needed to solve a problem like this.

What follows is not intended to be a complete answer to your question, but just a helpful tip.

When studying logic, it is extremely important to understand the following:

If you know that A implies B, and you also know that B is not true, then you know that A MUST be false.

Logical statements:
1. If there are no clouds in the sky, then it is not raining.
2. It is raining.

From these two statements taken together, we can conclude that the first part of statement 1 must be false. So, there are not no clouds in the sky. In other words, there are (at least some) clouds in the sky.

You can process these statements like this:
1. If (there are no clouds in the sky), then (it is not raining).
2. (It is raining).
Therefore, (there are clouds in the sky).

1. If (A), then (B).
2. Not (B).
Therefore, not (A).

1. A ⇒ B
2. ¬B
∴ ¬A

Stated in a simple way, you can say:

If A implies B, and B is false, then A must also be false.

((A ⇒ B) AND (¬B)) ⇒ ¬A

You should review this concept as long as it takes until you are confident that you fully understand it.

What follows is the whole analysis of the problem. Note that this is redundant with what was already explained in the other answer.

First, apply parentheses to all the individual statements:

If (there are no government subsidies for agriculture), then (there are government controls on agriculture). If (there are government controls on agriculture), then (there is no agricultural depression). [(There is depression) or (there is agricultural overproduction)]. It is a fact that (there is no overproduction).

Let us assign A, B, C, and D as follows: (for any statement including NO, we will remove it, and apply a NOT operator to it)

A: there are government subsidies for agriculture
B: there are government controls on agriculture
C: there is agricultural depression
D: there is agricultural overproduction

Now using this notation, we can rewrite the above sentences as:

If (not A), then (B). If (B), then (not C). [(C) or (D)]. (not D).

Partly converting to symbols, we get:

notA ⇒ B. B ⇒ notC. C or D. not D

where the ⇒ symbol means "implies" or "if...then"

Using simple rules of logic, we can process these statements from last to first:

Note: I use the ⊢ symbol to mean "yields". For example, A ⊢ B means "A yields B", or "from A, we know B"

1. (not D) AND (C or D) ⊢ C
2. C AND (B ⇒ notC) ⊢ notB
3. (notB) AND (notA ⇒ B) ⊢ A

So from the above, we know that A and C are true, and B and D are false.

• Thanks for this input. It was more clearer. But some aspect of it remains unanswered. Let's say among Demorgan's law, absorption or whatsoever. Would any of these be applied to the riddle I mentioned?. Commented Mar 20, 2020 at 0:45
• As I mentioned at the start of my post, I was not providing a complete answer to your question. I didn't think that was necessary because the previously submitted answer is complete. I will expand my post to include a full answer, but please note that the first person who answered has already done that as well. As for your question about the need for applying other rules such as DeMorgan's laws or absorption, NO, it is not necessary to apply any such rules. In fact, the only rules you need to use are the most simple rules of logic, simpler than DeMorgan's laws or absorption. Commented Mar 20, 2020 at 2:03
• The example you mentioned can it work in the other direction?. Let's say if not A then B. Can we conclude not B?. Is your example a syllogism?. Let's say if A then B, and B then C, thus A then C. If not C then not A?. Am I getting the right picture here?. Commented Mar 20, 2020 at 2:10
• I am not sure I understand your first question. Are you asking if we can conclude notB from only (notA ⇒ B)? If that is your question, then the answer is no. You cannot make that conclusion. However, from the two statements (notA ⇒ B) and (notB), you can conclude that A must be true. I guess you could call my example a syllogism, yes, but I think of it as more a basic law of logic. For your final question involving A, B, and C, yes, your understanding is correct. Commented Mar 20, 2020 at 2:40
• By the way, I revised my answer to include a more detailed explanation of the full problem. However, please note that the first person who posted an answer had already done that, so if/when you accept a best answer, it should be that of the other person, not mine. Commented Mar 20, 2020 at 2:42