Patterns of Logical Argument

We’ve looked at various aspects of turning English sentences into logical statements, and modifying them by negation, converse, and so on. Let’s finish by looking at some questions about standard rules of inference, such as Modus Ponens and the Law of Syllogism.

Four ways to argue

We can start with a question about the basics, from 2002:

Implications in Logic

I don't understand the first four rules to do with implications: Modus Ponens, Modus Tollens, Hypothetical Syllogism, and Disjunctive Syllogism. Can you explain them step by step, please?

These are four kinds of argument (deductive reasoning) that have been discussed since ancient times. (The first two names are Latin, the last two Greek.) Doctor Achilles started by clarifying how he would be typing logical symbols using the keyboard; I’m going to insert in brackets the special characters we use when they are available:

First, just so we're on the same page, here are the symbols I use:

  (P -> Q)     [P→Q]

Means: "If P, then Q." This sentence is true unless P is true and Q is false.

  (P ^ Q)      [P∧Q]

Means: "P and Q." This sentence is true if and only if P and Q are both true.

  (P v Q)      [P∨Q]

Means: "P or Q." This sentence is true if P is true, and it is also true if Q is true. The only way it can be false is if both P and Q are false.

  ~P           [¬P]

Means: "Not P." This is true if P is false.

Now he states and explains each of these “laws”:

Modus Ponens:

The rule for this is:

If you have:

  (P -> Q)

And you also have:

  P

Then you can conclude:

  Q

This follows directly from the definition of

  (P -> Q)

Which is "If P is true, then Q is true."

So, if I know that if P is true, then Q must be true, AND I know that P is true, then I can validly conclude that Q is true also.

More compactly, Modus Ponens (Latin for “method of affirming”) looks like this:

p→q
p  
∴q

It is also called “Implication Elimination”, for obvious reasons; “Affirming the Antecedent”; or the “Law of Detachment”.

An example in English:

If it is raining, the ground is wet. It is raining. Therefore, the ground is wet.

Next,

Modus Tollens:

This is a little trickier. The rule here is:

If you have:

  (P -> Q)

And you have:

  ~Q

Then you can conclude:

  ~P

Here's why:

We know first of all that "If P is true, then Q is true." And we know that Q is false. With Q false, is it possible for P to be true? No!  Because if P were true, then Q would have to be true. So if Q is false, then P has to be false also.

More compactly, Modus Tollens (Latin for “method of denying”) looks like this:

p→q
¬q  
∴¬p

It is also called “Denying the Consequent”, because it involves saying that the conclusion of a conditional statement is false. You may notice that it can be thought of as applying Modus Ponens to the contrapositive of the conditional statement.

An example in English:

If it is raining, the ground is wet. The ground is not wet. Therefore, it is not raining.

Next,

Hypothetical Syllogism:

The rule here is:

If you have:

  (P -> Q)

And you have:

  (Q -> R)

Then you can conclude:

  (P -> R)

Here's why:

We know that "if P is true, then Q is true." And we know that "if Q is true, then R is true." But we don't know ANYTHING about whether any of the letters are actually true or not.

Let's assume (or hypothesize) for a second that P is true. Then, by modus ponens, Q is true. And then by modus ponens again, R is true. So: IF we assume P is true, THEN we conclude R is true. Since we didn't KNOW P was true, we cannot take R home with us, but we can say that "If P was true, then R would be true." This is equivalent to saying "If P, then R" or (P -> R).

More compactly, Hypothetical Syllogism (Greek for “suppositional reasoning”, meaning a syllogism in which each premise contains “if”) looks like this:

p→q
q→r 
∴p→r

An example in English:

If it is raining, the ground is wet. If the ground is wet, cars skid. Therefore, if it is raining, cars skid.

Finally,

Disjunctive Syllogism:

The rule here is:

If you have:

  (P v Q)

And you have:

  ~P

Then you can conclude:

  Q

[This also works if you have (P v Q) and ~Q, you can conclude P.]

Here's why:

We know first of all that "P or Q is true." We also know that P is false. If P or Q is true, and P is false, then Q has no choice but to be true. So we can conclude that Q is true.

More compactly, Disjunctive Syllogism (Greek for “pulling-apart reasoning”, meaning a syllogism in which one premise contains “or”) looks like this:

p∨q
¬p 
∴q

An example in English:

It is raining, or the sprinkler is on. It is not raining. Therefore, the sprinkler is on.

A 1998 question had asked about this last one:

Disjunctive Syllogism

I am enrolled in a Elementary Logic and Set Theory class, and we have an assignment to find out definitions of the words: Modus Ponens, Modus Tollens, and Disjunctive Syllogism. I have already found the first two, but am stuck on the last one. So my question is, what does it mean?

Doctor Mike answered:

The disjunctive syllogism is  (P v Q) and notQ --> P. In words:

  "If one or the other is true,
  and one of them is false,
  the other must be true."

Here is an example:

-  Given: Either Congress meets in Washington D.C. OR pigs can fly.
-  Given: Scientific evidence shows conclusively that pigs cannot fly.
-  Conclusion: Congress meets in Washington D.C.   
   
I hope this helps. 
   
"Thanks for writing."  OR  "Pigs can fly."

What if the form doesn’t fit?

Students can be given the premises of an argument and asked to draw a conclusion, if possible. This requires a little extra knowledge of how forms work, as shown in this question from 2011:

If It Doesn't Follow a Form, Don't Draw a Conclusion

Sometimes with the law of syllogism, the p's and q's don't line up with the formula. What are some examples where there is no conclusion to the laws of syllogism? What about for the laws of detachment?

I've tried books and Internet sites and found nothing to help me with my problem. I can't even find any examples online.

This was the start of an extended conversation that was archived in two parts. I replied, just asking for clarification:

I'm not sure I understand. These laws always DO have a conclusion, and they are always correct.

It sounds like you are asking about something very different: situations in which it may LOOK like these laws apply, but they really don't. The answer there would depend on what YOU think is close enough to the real thing as to confuse you.

Is this an assignment you were given? How is it actually worded?

It turned out that Crystal was referring to an argument with no valid conclusion, not the law itself. She nicely stated and illustrated the laws she was referring to (Hypothetical Syllogism and what I have called Modus Ponens above), showing what kind of problems she was asking about:

I know what the law of syllogism is, and what the law of detachment is. I know how to draw conclusions from syllogisms like this:

  If a quadrilateral is a square, then it contains four right angles.
  If a quadrilateral contains four right angles, then it is a rectangle. 

It would be:

  If a quadrilateral is a square, then it is a rectangle.

And for the law of detachment, when it is in this form, I know how to solve it:

   p->q is a true statement, if p is true then q is true

These are easy because they follow the form.

But she had some problems that either didn’t fit any form, or were hard to identify:

However, there are some examples that I am not sure about. I get really confused and have no idea how to proceed when the statements are mixed up -- when they are not in this form:

  p->q and q->r are true then p->q is a true statement

For example,

  Conditional: If a road is icy,
               then driving conditions are hazardous.
  Statement: Driving conditions are hazardous.

Since this is with the law of detachment, wouldn't you conclude "it is icy outside"? My teacher said that because there are two conclusions, you can't make up another conclusion. I don't get it.

Here's another, which deals with the law of syllogism:

  Conditional 1: If you spend money on it,
                 then it is a business.
  Conditional 2: If you spend money on it,
                 then it is fun.

My teacher said that this has no conclusion, either. I asked about it, but she didn't really explain it to me well. And my math book also doesn't help me at all. Why does this one have no conclusion?

I just don't understand these different forms. I've labeled these examples as p and q and stuff, but no matter how much I try to figure it out, I can't understand why they would have no conclusion. It makes absolutely no sense!

Now I had a really good idea of her confusion, and could answer:

First, the Modus Ponens (Detachment) example:

I suppose what your teacher meant is that the second statement affirms the conclusion of the first statement rather than its condition. 

The law of detachment has the form

  p->q, and p, therefore q

That is, if you know that q happens whenever p happens, and you also know that p did happen, then q must happen.

The example about icy roads and driving conditions is NOT like that; it has the form

  p->q, and q

You can't apply this law; in fact, there is no law that you can apply, so you have no way to make a conclusion.

Look at the details of the example, which illustrates the issue nicely. (Not all examples are even true, much less showing why the logic makes sense.) The conditional statement says that if the road is icy, then driving becomes hazardous. That makes sense. What it does NOT say is that ice is the ONLY thing that can make driving hazardous! There are other reasons to be careful when you drive -- flooding, sun glare, bars having just let out, or whatever. So if someone tells you, "Look out, the driving is hazardous right now," you can't know what the cause is. It might be hazardous for any number of reasons. You CAN'T conclude with any confidence that the road is icy.

Does that make sense?

In fact, attempting to conclude “pq, and q, therefore p” is such a common mistake that it has a name: the Fallacy of the Converse, or the error of Affirming the Consequent. That is, it involves confusing the statement pq with its converse, qp, from which p could be concluded — the arrow goes the wrong way.

Next, the syllogism example:

The law of syllogism says that ...

  if p->q and q->r,

... then you can conclude that ...

  p->r.
   
It's like a pipeline of reasoning: if p is true, then q is true, and therefore so is r. The conclusion of one statement has to be the condition of the next, so the reasoning flows in the right direction.

The example about how you spend money has the form

  p->q, and p->r
   
The pipes don't connect the right way to be able to say that 

  q->r or r->q

This is a less clear example; here's a situation of the same form that may be easier to follow:

  If it rains, I will be wet.
  If it rains, the road will be slippery.

Can I conclude that if I am wet, the road will be slippery? No, maybe I'm wet because I just took a shower.

Can I conclude that if the road is slippery, I am wet? No, maybe there was an oil spill on the road.

So we can't make any definite conclusion; we don't know enough to say more than we have been told.

Summarizing:

The idea here is that if you can't put the argument in some form from which you can make a conclusion, then you can't make a conclusion.

There are other forms you don't know yet, so you can't always be sure that no conclusion can possibly be drawn; but when statements do not conform to any of the forms you know, that is all you can say. YOU can't draw a conclusion, simply because none of the rules you know applies.

Put another way: when given logical statements that do not follow the forms you know, conclude "No conclusion is possible (for me)"!

What if the order doesn’t match the pattern?

The next day, Crystal wrote back in a new thread, about a slightly different issue:

Disorderly Deduction

I know what the law of syllogism is, and honestly I've written to Dr. Math about it a lot and I'm starting to understand it better (thank you!). But here is a problem that I don't get because it doesn't follow the "if p->q and q->r then p->r" law of syllogism form.

  If the sum of the angles of a polygon is 720 degrees, 
  then it has six sides.
   
  If the polygon is a hexagon, 
  then the sum of the angles is 720 degrees.

My teacher said that the conclusion would be 

  If a polygon is a hexagon, 
  then it has 6 sides.

That doesn't follow (to use an analogy introduced to me by a math doctor recently) the "pipes" of the law of syllogism.

I just keep getting more of these weird questions from my teacher, but when I look online for help, the only syllogisms I find are ones I already know how to do.

I responded again:

Hi, Crystal.

Let's translate this into symbols. 

First, what are the simple statements here?

  If the sum of the angles of a polygon is 720 degrees,
  then it has six sides. 

  If the polygon is a hexagon,
  then the sum of the angles is 720 degrees.

I see the following, just taking them in order as they come:

  p = the sum of the angles of a polygon is 720 degrees
  q = it [the polygon] has six sides
  r = the polygon is a hexagon

Using those symbols, we have

  if p then q
  if r then p

Or, in fully symbolic logic terms,

  p->q
  r->p

Clearly we don't have EXACTLY "p->q, q->r," but that's just because we didn't define p, q, and r in the most helpful order.

Since you have absorbed my comparison to plumbing, let's see whether we can arrange these "pipes" so they line up correctly:

   r->p
   p->q

Can you see how this becomes this?

   r->q

The pattern doesn’t care about order:

The key here is that the rules are not all about the particular letters you use, but about relationships. (A plumber doesn't get confused if he pulls out two pipes from his truck in the wrong order!) If you have two conditional statements such that the conclusion of one is the condition of the other, then the law of syllogism says you can connect them together to make one new conditional statement.

Translating our conclusion, r->q, back into words using our definitions, we have

  if r, then q

  If the polygon is a hexagon, then it [the polygon] has six sides.

Or, smoothing out the readability a bit,

  If a polygon is a hexagon, then it has six sides.

Crystal had more questions:

Okay, so this is another question on the law of syllogism!

I understand the hexagon question now (thank you!); but is it because these statements ...

  p->q
  r->p

... are basically biconditionals that you can rewrite them like this?

  r->p
  p->q

I'm just trying to understand it more, and my teacher isn't doing a good job explaining it to me. But I've emailed about this question before and it's getting a lot easier for me to understand!

I answered:

No, nothing here is a biconditional. In fact, for this particular example, everything would have been true if the conditionals were replaced with biconditionals -- which is why this example is not a good example to illustrate the issues. 

The conclusion would still be valid if we replaced everything with different statements -- even total nonsense:

  If a borogove is mimsy,
  then mome raths outgribe. 

  If toves are slithy,
  then a borogove is mimsy.

We can conclude that

  If toves are slithy,
  then mome raths outgribe. 

That's because if toves are slithy, then a borogove is mimsy, and since a borogove is mimsy, mome raths outgribe.

Do you see why?

(If you are a Lewis Carroll fan, you will recognize my use of terms from the nonsense poem Jabberwocky; the author was a logician.)

Crystal again:

So in your nonsense example, you can draw that conclusion because some of the p's and q's "cross out"? and because the statements are assumed true, you can just write the remaining hypothesis and conclusion as an if-then statement?

I just want to make this clear.

I've emailed a lot to Dr. Math and it's helped a LOT!!!

I concluded:

Yes, you can think of it as canceling p's or q's in the sense that it doesn't matter what the letters are, only whether they match up. 

But they have to match up in the right way: one a conclusion; the other a condition.

In the form "p->q, q->r implies p->r," we have

  p->q
     q->r
  -------
  p---->r

In my example, "p->q, r->p" becomes this:

     p->q
  r->p
  -------
  r---->q

Here, the p's connect the two conditional statements.

The order is different, but the relationship is the same.

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