Intro to Logic: Valid Deductive Argument Forms

by Luke Muehlhauser on September 28, 2009 in Intro to Logic

logic

Welcome to my course Intro to Logic (index). Here, we learn the basic skills of good thinking and their benefits in real life.

I wrote earlier about the deductive argument; an argument which claims that if its premises are true, then its conclusion must be true. I also wrote about fallacies; ways that arguments can go wrong. Today I’d like to write about four ways that deductive arguments can go right.

Modus Ponens

One form of valid deductive argument is called Modus Ponens. It looks like this:

  1. If P, then Q.
  2. P.
  3. Therefore, Q.

This is clearly valid. Here’s an example:

  1. If pointless suffering exists, then an all-good, all-powerful, all-knowing God does not exist.
  2. Pointless suffering exists.
  3. Therefore, an all-good, all-powerful, all-knowing God does not exist.

Keep in mind that though this argument is logically valid, it might not be sound. One or both of its premises might be false. For example, the theist might deny (2) and argue that all suffering which exists does have a purpose, though often we are not capable of understanding that purpose.

Modus Tollens

A Modus Tollens argument looks like this:

  1. If P, then Q.
  2. Not-Q.
  3. Therefore, Not-P.

An example:

  1. If God exists, then objective moral facts exist.
  2. Objective moral facts do not exist.
  3. Therefore, God does not exist.

Once again, though the form is valid the premises may be highly debatable.

Dilemma

A valid dilemma argument works as follows:

  1. Either P or Q.
  2. If P, then R.
  3. If Q, then R.
  4. Therefore, R.

A famous example is the Euthyphro dilemma, applied to monotheism:

  1. Either what is moral is moral because it is commanded by God, or God commanded it because it is moral.
  2. If what is moral is moral because it is commanded by God, then classical theism is false (because God’s morality would thus be arbitrary, which classical theism denies).
  3. If God commanded what is moral because it is moral, then classical theism is false (because God would thus be subject to an external moral system, but according to classical theism God is not subject to anything).
  4. Therefore, classical theism is false.

This is a valid argument, but of course it might not be sound. The atheist, for example, would deny premise (1).

Simplification

Simplification is so obviously valid it is barely worth mentioning. It works like this:

  1. P and Q.
  2. Therefore, P.

An example:

  1. Socrates is a man and Socrates is a mortal.
  2. Therefore, Socrates is a man.

Brilliant! You’re a master logician now!

There are other valid forms of argument, for example the reductio ad absurdum and various forms of syllogism. We will discuss them later.

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{ 13 comments… read them below or add one }

Josh September 28, 2009 at 10:04 am

“This is a valid argument, but of course it might not be sound. The atheist, for example, would deny premise (1).”

I don’t think that’s entirely the point. I believe the euthyphro dilemma can be phrased as a proof by contradiction, i.e.

1)P (god exists)
2)P implies Q (morality by god)
or
3)P implies ~Q (morality by something else)
4)Q implies ~P
or
5)~Q implies ~P
6)Hence, P implies ~P, so it must be false.

Something like that.

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lukeprog September 28, 2009 at 4:10 pm

Josh,

Yes, there are many ways of representing the euthyphro dilemma.

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IntelligentDasein September 28, 2009 at 7:18 pm

Luke if you need any help with logic, give me a shout out. :-)

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Dace September 28, 2009 at 8:30 pm

It’s interesting – I’d always heard and taken the Euthyphro dilemma to show that morality didn’t depend on God, not that classical theism must be false. Since most Christians take God to be at least subject to the laws of logic, they wouldn’t agree with Luke’s 3. One then wonders whether anyone is a classical theist at all.

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lukeprog September 28, 2009 at 10:04 pm

IntelligentDasein,

I certainly do need help with logic, always. What kind of help did you have in mind?

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IntelligentDasein September 29, 2009 at 5:13 am

Suggestions for entries you should make and what examples should be included. I think you are doing a great job, but I would have included a basic lesson on how to symbolize thoughts with the 4 major symbols (and, or, if-then, not, and equivalence) before I did this entry. Then you could have shown the forms in symbols also.

lukeprog: IntelligentDasein,I certainly do need help with logic, always. What kind of help did you have in mind?

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lukeprog September 29, 2009 at 7:09 am

IntelligentDasein,

I will be talking about translation into formal languages a bit later. I know I’m doing letter substitution already, but that’s fairly intuitive without having to know formal translation rules.

But yes, please send me any suggestions you have, in particular on the order of how you would address subjects. Right now I’m planning to cover syllogisms and some more fallacies and set theory, and then start talking about formal logic, but I may be missing some things I should address before talking about formal logic. After covering a couple formal logic languages, I’ll probably discuss inductive and abductive reasoning a bit more, and then move into probabilistic logic (ala ET Jaynes). But yeah, I could definitely use some help making sure I put things in the right order so I don’t keep backtracking or confusing people.

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IntelligentDasein September 29, 2009 at 11:11 am

Definitely Luke.
I got your back.

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lukeprog October 1, 2009 at 8:36 pm

So… do you have any suggestions?

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Jason December 2, 2009 at 11:14 am

Luke,
I have no formal training in Logic, just trying to learn it on my own, so forgive me if this a stupid question. I’m having some trouble agreeing with the validity of the Modus Tollens form. It seems to assume that the relationship from p to q is the same as from q to p. It seems to me that one could accept the idea that objective moral facts depend on the existence of God, but not necessarily agree that the existence of God requires the existence of objective moral facts.

By using the premises from the first example I think I can illustrate my line of thinking better:

1. If pointless suffering exists, then an all-good, all-powerful, all-knowing God does not exist.
2. An all-good, all-powerful, all-knowing God does not exist.
3. Therefore Pointless suffering does not exist.

I can agree with both 1 and 2, but 3 is obviously false.

My default stance is that I’m just missing something; can you fill in the gaps for me?

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drj December 2, 2009 at 12:42 pm

That would not be a valid modus tollens.. you have inserted a negation into the consequent.

A modus tollens:

If P, then Q
~Q
~P

What you have, is something weird like this:

If P, then ~Q
~Q
~P

Or an equivalent, but with the negation hidden:

If P, then Q
Q
~P

Obviously, not valid.

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lukeprog December 2, 2009 at 6:31 pm

Jason,

Your argument is of the form:

1. If P, then Q.
2. Q.
3. Therefore, Not-P.

But this is not Modus Tollens form.

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Jason December 3, 2009 at 12:38 am

…and the light bulb over my head turns on. Thanks guys.

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