Formal Logic 101 – Part 2: Argument Structure and Validity, and Intro to Inference Rules

Formal logic is primarily concerned with the verbal (or written) expression of reasoning. Similar to the math teacher who always demanded you show your work (even if you had the right answer) logical proofs demand that you painstakingly write out your premises, as well as every step you take to move from those premises to your conclusion (Note: For my readers with math anxiety: Don’t worry. It won’t be as bad as math). And just as showing your work in math class tended to look more or less the same for every problem on a given assignment, the steps of proving also have a certain structure.

Consider the following arguments:

Argument 1
1. If Max is a poodle then Max is a dog.          premise
2. Max is a poodle.                                             premise
3. Therefore Max is a dog.                                from 1 and 2

Argument 2
1. If religion is a good thing, then government should sponsor it.    premise
2. Religion is a good thing.                                                                       premise
3. Therefore the government should sponsor it.                                  from 1 and 2

Argument 3
1. Animals can think.                                                          premise
2. If animals can think then they should have rights.    premise
3. Therefore animals should have rights.                         from 1 and 2

All of these arguments are valid. (Yes, even #2. Remember, logic does not ask “are the premises true?” Logic is “truth preserving” meaning “if all of the premises were true, the conclusion would HAVE to be true also”, so Argument 2 is perfectly valid.)
Also all of these arguments have a justification column on the right that tells us if they’re premises, or – if not – what lines we used to reach that conclusion so that our thought process can be traced (this will be more important later).

Hopefully, you also noticed that all three of those arguments have the same basic layout:
1. If [A] then [B].     premise
2. [A].                        premise
3. Therefore, [B].    from 1 and 2

Sure, for #3, lines 1 and 2 are switched, but the general layout is the same. That is what it means to say that they have a valid structure. They are built in such a way that a valid inference rule can be readily applied to them. This particular inference rule is called “Modus Ponens” (“Modus” being Latin for “mode” or “way”, and “Ponens” being Latin for “affirming.” So modus ponens is “the way of affirming”.)

*****

To give an example of an INVALID inference rule, consider the following:

Argument 4
1. If Obama was president, then Biden was vice-president in 2010    premise
2. Biden was vice-president in 2010                                                          premise
3. Therefore, Obama was president in 2010                                             from 1 and 2 (Invalid)

This argument follows a superficially similar structure to modus ponens:

1. If [A] then [B].        premise
2. [B].                          premise
3. Therefore [A].        from 1 and 2 (Invalid)

Now, I readily grant that the conclusion is true. The problem is you can’t reach that conclusion from the premises. Premise 2 does not speak of Obama at all, and Premise 1 only states that IF Obama was president, then Biden must’ve been the VP. It makes no claim that Obama actually was the president, and it makes no claim about “if Biden was vice president…”

This is what is called “The fallacy of affirming the consequent” (and is sometimes lovingly known as Modus Morons). I bring it up now partially because it is an INVALID inference rule, but also because it is a TEMPTING inference rule. If you come across someone who knows what Modus Morons is and who says that they have never used it by accident, they are LYING.

Why does Modus Morons seem to be so tempting if it’s not valid? My personal hunch is that it’s because Modus Morons is safe. Consider:

You’re stranded in the woods and you see your friend Becca eat a handful of berries. You think to yourself “If those berries are poisonous, Becca will die”. Becca dies shortly thereafter. You can’t prove it was the berries… it might well have been a heart attack from the stress of being stranded in the woods, but it’s SAFE to NOT eat those berries. You and I are the descendants of people who instinctively used Modus Morons and may or may not have survived because of it. So, from time to time it will feel instinctive to use, it but keep your guard up because it is NOT valid and will lead you astray.

*****

I know this post is getting long, but very quickly, I want to cover the last of the “modus”-es: Modus Tollens (way of denying). Modus Tollens is a valid inference rule that takes the shape of:

1. If [A] then [B].                premise
2. Not [B].                          premise
3. Therefore Not [A].       from 1 and 2

Consider the following example:
1. If my kitty likes being pet then he will purr when I pet him.     premise
2. my kitty does not purr when I pet him.                                        premise
3. Therefore my kitty does not like being pet.                                from 1 and 2

Remember, premises are assumed to be universally true. If my cat likes being pet, then he WILL purr. Every day, rain or shine, twice on Sundays. If he’s being pet, and NOT purring, the only explanation that is possible (assuming the premises are true) is that he doesn’t like it.

*****

Practice

Determine if the following arguments are valid or invalid (remember, “valid” does not necessarily mean “true”).

A)
1. All cats are green.                          Premise
2. All green things are immortal.      Premise
3. All cats are immortal.                  ____________

B)
1. No Russians are wealthy.                              Premise
2. No wealthy people have gone to Mars.      Premise
3. No Russians have gone to Mars.                  ____________

C)
1. If that tress is a maple, it has acorns         Premise
2. That tree has acorns.                                    Premise
3. That tree is a maple.                                  ____________

D)
1. If dogs don’t bark, then cats meow.          Premise
2. Cats do not meow                                       Premise
3. No Russians have gone to Mars.             ____________

E)
1. If Texas is not on the west coast, then it must be Oregon.      Premise
2. Texas is not on the west coast.                                                    Premise
3. Oregon is on the west coast.                                                      ____________

F)
1. All cats are furry mammals                   Premise
2. Some furry mammals are black.          Premise
3. Some cats are black.                          ____________


Answers:

A) Valid. Modus Ponens. Remember the premises do not need to actually be true. Ask yourself “If they WERE true, would the conclusion also have to be?”

B) Invalid. Premise 2 states that no wealthy people have gone to Mars, but says nothing about people who are not wealthy. Premise 1 states that no Russians are wealthy, so there’s nothing in the argument preventing Russians from having gone to Mars.

C) Invalid. This one is an example of Modus Morons.

D) Invalid. The conclusion is unrelated from the premises.

E) Valid. Modus Ponens.

F) Invalid. This one may be a bit tricky, but imagine a world where the only creatures with black fur were dogs. Premise 1 would still be true, as would Premise 2, yet the conclusion would be false. As such this proof is not “truth preserving” and cannot be valid.


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