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Logical Gal and the Law of Excluded Middle

2 Oct

Have you ever been frustrated by someone’s quibbling?

Mom – “Who ate my chocolate?”

Kid – silence….Shrug of shoulder

Mom – “Listen, you either KNOW or you DON’T KNOW, which is it? “

Kids – “Maybe it just vaporized?”

Mom – “So you know!”

Kid – “I didn’t say I know.”

Mom – “So you don’t know!”

Kid – “I didn’t say that either,” whining because he knows he’s been caught by a logical mom!

Mom – “Based on the Law of the Excluded Middle, you have only 2 choices:  Either…….A – you know what happened to my chocolate .  Or……B – you don’t know…..Which is it, Buddy?”

Mom is relying on the 2nd of the 3 basic laws of Classical Thought – the Law of Excluded Middle.  (last time we talked about the Law of Identity)

This is very useful because it forces you to categorize an issue.   Something is either A or non-A.  There are no other choices.  This isn’t rocket science, just common intuition!

  • You’re either pregnant or you’re NOT pregnant (no such thing as a little pregnant)
  • My car is painted either black or non-black
  • You either speak Chinese or you don’t.  (What about if you speak just a little?  Then you speak Chinese!)

Turning to a deeper issue that often comes up, do you see how this law could help in a discussion about the existence of God? He either exists, or He doesn’t.  We might not know, but it’s either one or the other.

What other possibility could there be?  The picture below is TRYING to get you to think that there might be a 3rd choice, but there isn’t!   (the supposed 3rd option of ” Probably”  is meaningless!)

Here’s one guideline in organizing your either/or statement: 

  • Whatever predicate you are using, you have to frame it as A or non-A

You can’t say:  Either my car is blue or white.  That’s not true, for there are other possibilities.

So you would have to claim: “ Either my car is blue or it is non-blue”

What issues does the Law of Excluded Middle simplify for you?

“That’s a contradiction!” – are you sure?

30 Jul

So…how does knowing the Law of Non-Contradiction help in real life?

Remember we said that according to this DISCOVERED law (it’s built into the fabric of our universe by God as opposed to invented by culture):

 A & non-A cannot both exist at the same time and in the same way.

Consider this pair of statements:

  • ·         Susie is pregnant
  • ·         Susie is not pregnant 

Now we have to be careful and not automatically ASSUME that this is a contradiction. Two propositions that LOOK contradictory could in fact be explained…….

1.    If we mean that Susie Jones is pregnant, but Susie Smith is NOT pregnant (2 different Susies)

2.    Or if we mean that Susie is pregnant with many good ideas, but Susie is NOT pregnant with child (pregnant as an analogous term – referring to different but related concepts)  

But if we are talking about the one and only Susie Smith and we understand the predicate term ‘pregnant’ to indicate about to have a baby, then….

·         They cannot both be true OR false at the same time and in the same sense.

In Christianity this law of logic helps me sort out my theology.

My favorite attribute of God is His sovereignty.  When we say that God is sovereign, we understand God to be 100 % in charge of all that happens, the good and the bad.  I’m not saying that I understand this characteristic of God, but I am comforted by it!  (If God allows suffering and evil, then He must have a good purpose for it even if I can’t see that…yet!)

Therefore, because of the Law of Non-Contradiction, when I assert that God is always sovereign I cannot say:


God is sovereign

But

God had no control over that deadly train accident in Spain.     

That would be saying:  God is sovereign over all/ God is NOT sovereign over all

Either God IS sovereign or He is not, if I take sovereign to mean that He controls all molecules in the universe.

What we have to do when hit with confusing statements that seem irreconcilable is to ‘translate’ them, if possible, into A and non-A forms.  Then we can evaluate them clearly.

I say, ‘if possible’ with this caveat in mind – you might run across an either/or claim –

·         God is either all-loving or He is a God of wrath.

·         You’re either pro-choice or you are anti-women.   

If you can’t ‘translate’ the 2 predicates into an A and a non-A term, then you might be facing the Fallacy of Bifurcation (aka ‘false dilemmas’).  We’ll talk about that on our next Fallacy Friday!

Back to the above assertions – If we wanted to deal with that first claim, we’d have to re-frame it and then discuss terms.

·         God is either all-loving or He is not all-loving

·         You’re either pro-women or not pro-women

 Your HW for the next few days is to keep an ear out for ‘either/or’ claims and try to determine if they are in fact contradictory or perhaps examples of the False Dilemma fallacy or actually TRUE!      

When a valid argument feels wrong – Logic to the rescue!!

29 Jul

So what do you do when someone’s argument is in the correct form, but you know that there’s still a problem?  

In a previous post I asked you to ‘draw’ out this syllogism:

All roads lead to Rome

Old Cabin Cove is a road

Therefore, Old Cabin Cove leads to Rome

Here’s what it should look like where BOTH the outer red square and the blue circle represent P1, and P2 is represented by the red X within the blue Roads circle.  We can CLEARLY see with our eyes that Old Cabin Cove is situated within the larger red square, “Things that lead to Rome”

Things that lead to Rome

As you can tell visually, the conclusion does not overreach the scope of the two premises P1 and P2. The syllogism IS, therefore, in the correct form and is considered VALID.  But our work does not end there.  You can FEEL that something else is wrong.

Anecdotally, I live on the gravel road, “Old Cabin Cove” in Western NC and I can attest that it does NOT lead to Rome.  It leads up a forested hill to our house and stops there!

What do we do then, with this valid syllogism?  We examine the truth of each of the 2 premises.

  • Let’s start with P2: Is ‘Old Cabin Cove’ a road?  YES! – no problem there.
  • Now for P1:  Do all roads lead to Rome?  NO!  Here’s the problem.  You already knew that, but what is illustrative in our simple example is this:  to DISPROVE an ALL or ‘A’ statement (also called a Universal Affirmative)  find ONE counter-example.  If there is JUST ONE single solitary road in the universe that does NOT lead to Rome, then the statement, “All roads lead to Rome” is false.
  • Why?  Thanks to the Law of Non-Contradiction which states that “A and non-A cannot both be true in the same way at the same time”.  Therefore we can’t say:  All roads lead to Rome and Some roads do NOT lead to Rome.
  • But we CAN say that Some roads lead to Rome and have that be a true statement.  (By the way, it takes only ONE road leading to Rome to make it true that ‘some roads lead to Rome’)

Back to our syllogism – if we want true premises, then we have to modify them to reflect reality:

P1   Some roads lead to Rome

P2   Old Cabin Cove is a Road

Tf……NOTHING!!!! –  we CAN’T conclude that Old Cabin Cove leads to Rome. It might and it might NOT.

Just like in our previous ‘cat and cuddly pets’ syllogism, our conclusion cannot reach further than P1 and P2, even if both of the premises are TRUE.  Here’s the sketch of what that would look like. We simply do not know where to place our X representing Old Cabin Cove.

Old Cabin Cove and Some roads

In our next post, I will share some real life examples of how knowing the Law of Non-Contradiction can help evaluate an argument you might read or hear.

Arguing about cuddly cats and ‘going TOO far’

25 Jul

Random question:   Is it true that some cats make good pets’?  This was a claim or CONCLUSION from last time when we were examining an argument.  We had started labeling & analyzing the argument about ‘roads that lead to Rome’ and got side-tracked by CATS! (see to the right:  ‘Spotting errors in arguments, beginning steps’

Your HW was to practice LABELING the following syllogism:

All animals that make good pets cuddle well

Some cats cuddle well

Tf, some cats make good pets  

 

Do you remember the steps?

1.   Put each proposition in ‘logical form’

All animals that make good pets are animals that cuddle well (needed a copula and we CLARIFIED terms)

Some cats are animals that cuddle well

Tf, some cats are animals that make good pets

2.   Start labeling the terms ‘bottom- up’, beginning with the Conclusion

–      Subject term is:   cats

–      Predicate or MAJOR term is:  animals that make good pets

–      Middle term (what’s left over) is: animals that cuddle well

 3.   Evaluate the terms with some quick questions

–      Are there 3 and only 3 terms?  YES

–      Is the Middle term in just the P1 and P2? (a rule new to you today) YES 

–      Is the Major/ Predicate term in P1? (the major premise -can’t be in P2) YES

 4.  Here’s a new step – draw out the syllogism to see if we have enough info to come to the conclusion legitimately

 cats as good pets

Because we have a question of where to place that subset of ‘cats that make good pets‘ (in the blue circle or out of the blue circle), we CANNOT legitimately reach the conclusion that Some cats are animals that make good pets.   Visually we can SEE that the syllogism is NOT valid…so there is no point in continuing  to debate with a cat/cuddly pet disputer whether the argument is true, because he/she has NOT correctly formed a syllogism.

Had the syllogism BEEN valid, then we would have continued on to examine the truth of Premise 1 and Premise 2.  There is a logic law that states, “In a valid argument, if the 2 premises are true, the conclusion MUST be true.”  That IF is the crucial two-letter word. Today’s argument was NOT valid, for there was insufficient information in the 2 premises to determine if in fact SOME CATS ARE ANIMALS THAT MAKE GOOD PETS.

So, for next time, practice with the argument from our previous post, the one below.  See if you can draw it out like I did with the cat argument.

All roads lead to Rome

Old Cabin Cove is a road

Therefore, Old Cabin Cove leads to Rome

What is Chili?

11 Jul

    

Terms matter! 

Once when we were engaged to be married and were visiting my parents, Mom prepared a tried and trusted entrée she thought Mike would like – chili!  The only problem was that she left out the kidney beans!  My fiancé, who was less than tactful, remonstrated, “This isn’t chili – it has no beans!”

Was he correct?  It depends on how you define ‘chili’.  (Why did she omit the beans?  She was probably distracted by the presence of her articulate and handsome future son-in-law!)

One of the fundamental laws of thought is called the Law of Identity.  It’s pretty intuitive: a thing is the sum of its component parts, characteristics (or ‘predicates’ to be technical) and NOT something else.

Let’s suppose that you define ‘chili’ like this:

·         Chili is a thick soup or gravy composed of meat, beans, tomatoes and seasonings

What happens, then, if you leave out or add something to this ‘soup or gravy’?  According to the Law of Identity, it is NOT ‘chili’.

Such a rule or guideline is useful in all kinds of conversations.  When we talk with people, our first responsibility is to get clear what they have in mind when employing a term.  So we ask them:

 What do you mean by ‘chili’?”

          or

 “What exactly is ‘chili’?

Being clear and precise about the definition is important. If you add or take away a defining component, you have changed the thing whether it is material (concrete) or immaterial (an idea).

When I engage in conversations about controversial items like prayer, God, marriage or even Christianity, I CANNOT assume that the person with whom I am talking defines the term in question the same way as I do.  This Law of Identity is basic!

For example, if I mean by marriage the historical and biblical definition:

a covenantal relationship recognized and supported by society between one qualified man & one qualified woman for the purposes of companionship, love, mutual support and the raising of children if possible

….then taking away or adding a part changes its identity.  It is no longer ‘marriage’ but something else!  That is fine, but we should be honest.  Fuzziness doesn’t help anyone.  Such equivocating with terms allows people to hide.  No progress can be made toward establishing meaningful conversation and/or solving problems.   

What concepts do YOU see in everyday life that are ‘allowed’ to stay fuzzy?