Tag Archives: Law of Excluded Middle

Logical Gal and what makes a good definition

8 Oct

Stop

Have you ever felt sheepish after jumping down someone’s throat….

for incorrectly assessing your conversational partner’s point of view?  Come to find out, you had been thinking about X and they had meant Y.  And it all hinged on a TERM!  Before you and I engage in a discussion, we need to clarify and clear away the fog by defining our terms precisely!

That’s rule # 1

And we always follow the rules, right? Smiley Face

Sometimes though, it is easier said then done.  Especially when we aren’t sure what makes for a good definition.

There are several guidelines, but today I want to talk about 2 characteristics in particular which assist in obtaining clarity. Communicating a definition means that your audience is left better off after listening to you – there is LESS confusion.

First things first!

When you create a definition, you need to include both a genus (the big category to which it belongs) and a differentia (the ‘what about it details’ that make IT different from other members in the category/genus).  Hence, a GooD definition – capital G for genus and capital D for differentia, includes both parts, the genus and the differential.

For example, what is a car?

Car

The ‘genus’ is the large family of motorized vehicles.

And what distinguishes a car from a truck from an airplane could be stated like this:

a member of the motorized vehicle genus which has 4 wheels and has as its primary design function that of transporting 1-6 people on a road.

What’s highlighted in red makes up the differentia.  Now here is the tricky part and what I want to convey in this blog post.  When we attempt accurately to describe the differentia, we have to be both inclusive AND exclusive in a sufficient manner so that:

  • the differentia could not apply to more than one member of the differentia

For example, if we had suggested a differentia that went like this….which has as its primary function that of transporting people and not cargo on a road..then how would we tell whether this included or EXcluded a van or  a bus or even a motorcycle?

A definition has to be ‘mutually exclusive’, meaning that there be NO confusion whether something fits into a car slot or a bus slot or a motorcycle slot or a pickup truck slot.

  • AND the genus and differentia have to provide for all potential members so that you don’t exclude a possible member, leaving them unassigned.

For example, if I had not intentionally selected the number ONE as the lower end of passenger carrying capacity, I might have left out the possibility of a really tiny one-person vehicle from being considered a car.

One passenger car

What got me thinking about the importance of ‘GooD’ definitions and the proper use of categories was the opening of a sermon in which the pastor was describing Mexican food.  He had grown up in California and argued that when you boiled down this style of cuisine, there were basically MEAT tacos and BEAN tacos.  The varieties are endless, but by his thinking, a GooD definition of Mexican food would be this:

Mexican food is a style of cooking (genus) using tacos as a base for filling with either meat or beans in various combinations combined with many possible add-ons. (differentia)

Mexican Food

After tantalizing his congregation with mouthwatering thoughts of a tasty lunch, he delivered his punch line: 

There are only 2 kinds of people in the world:

  • those who live according to the Holy Spirit and are therefore born-again Christians with a happy future
  • those who live according to their own agenda and are therefore enemies of God with a depressing future

If you think about it, every single person on Earth who ever has lived, is living or will live fits into either one or the other group. Therefore, this categorization is considered ‘jointly exhaustive’.

And no one can have a foot in one category AND the other at the same time. Therefore, this description is ‘mutually exclusive’.

It’s sobering, isn’t it!

But truth is like that.  I’m reminded of another area where there are only 2 categories:  a woman is either pregnant or she is not pregnant.  The logic law that states that limitation of reality is called the Law of the Excluded Middle.

Even though Truth can be as bracing as a bucket of ice water, there is Good News.  And it is this:

  • Although EVERYONE starts off in that 2nd category (or trajectory toward remaining an enemy of God), no one is doomed to stay there.  God’s offer of salvation by grace alone through faith alone in Christ alone is available to all those who are willing to accept the gift.

Question: what are other categories with only 2 possibilities? 

And for an explanation of how you can be adopted into God’s forever family, How to become a Christian

 

 

 

Logical Gal extols Excluded Middle Law

12 Mar

Excluded Middle

Tools are for the using.  And I was reminded yesterday, listening to a discussion, how helpful the laws of logic can be!

The conversation centered on the origins of the universe.  It seems that everyone has a theory or possible explanation for how our universe came into being or whether it has always existed. So how are we to judge?

First of all, to even hold a rational discussion presupposes that words have meaning.  And when we put words together we make truth claims that are either true or false. We all use and therefore at least implicitly rely on logic, whether we acknowledge it or not.  So to deny the laws that are present is folly.  Humans didn’t INVENT these laws, we simply have discovered them.  They are part of the universe.

rational v. irrational

And if they do NOT exist and if words hold no meaning that we can all agree on, then there is no point ever having ANY conversation.  We just end up sharing gibberish.

Back to the discussion I followed this week.  One man laid out a way of thinking about origins that relied on the Law of Excluded Middle.  This law states that there is no 3rd or middle option given A or nonA.  For example, a woman is either pregnant or NOT pregnant. There is no other possibility.

Regarding origins, the reasoning expounded proceeded this way:

  • The universe either had a beginning or it didn’t.
  • If it did have a beginning (and due to the articulation of the Big Bang Theory, most people accept that the universe is NOT eternal), something caused it or something did not.  
  • (If something did NOT cause the universe to begin, then it had to cause itself.  But that is irrational because then something would have to BE (to cause the universe) and NOT BE at the same time.  And that is impossible because of the Law of Non-Contradiction.

Law of Non-Contradiction

  • If something caused matter, time and space (MTS) to appear, it itself is MTS or it is not.  (If matter, time or space created the universe, then we are stuck with a problem of infinite regress, i.e what caused the PRE-matter, time and space that created the universe)
  • Whatever caused the universe to come into existence is either immaterial or it is nothing.
  • and finally….we are left with the conclusion that what caused the universe has to be immaterial, because nothing cannot create something.

Nothing produces nothing

The man pushing back against this line of thinking balked when pressed by this series of either/ors.  When the Law of Excluded Middle was articulated , all he could repeat was the assertion of a possible 3rd option.  I held my breath, curious to learn something I hadn’t previously considered.  But all he offered was that ONE DAY, scientists might come up with a different explanation.

I am VERY thankful for tools that help me sort out, categorize and think logically through complicated matters.

Question:  How does the Law of Excluded Middle help you?

Logical gal takes on claim: “Truth is relative”

1 Nov

I teach a few precocious 8th graders who take pleasure in striking the contrary pose.  The other day a lively discussion erupted at the end of the period. What got us started was the conclusion in a simple French reader, “Il n’y a pas de familles parfaites”  – there are no perfect families.  One boy disagreed saying, ‘En fait’  /actually that there ARE perfect families!  I responded with incredulity, “Really?  for how long?”   He backed down and said that his family could be perfect for ……. half a day.  But when pressed to admit the enormity of 7 family members actually ‘being perfect’ with one another for that many hours, we slipped OVER into the bunny trail of TRUTH.

Since there were only 2-3 minutes left in class, I allowed us to converse en anglais in this French 1 class.

So, what was the connection between the concept of perfect and the concept of truth?  It all started when I asked the students how they defined ‘perfect’.  I think someone piped up about perfection being relative, like truth.  I then humorously asked, “So the Law of Gravity is relative?”

Class ended as the boys were affirming that “All truth is relative”.  Had we been able to pursue this chain of thinking, I would have led them to define truth.  Defining one’s terms is always the hinge on which statements or propositions rest.

As Bill Clinton might have expressed and Pilate thought, “It all depends on what truth means!

So if truth is defined as “that which corresponds to reality“, then relativity has NO bearing on the definition.  Truth doesn’t change according to the one who is looking at it.

For example: Terminating the life of a person or animal is the act of killing.

Whether that action is justified or not, is morally good or not is another question.  Good or bad, it is still killing due to how the term is defined.  It doesn’t matter that in some cultures people are exhorted to Love their Enemies where in other societies members are taught to Eat their Enemies.  Those are moral values which DO change according to how and in what/in whom they are grounded.

Back to truth. Truth either conforms to reality or it doesn’t.  The ‘discovered’  Law of Excluded Middle tells us as much. There is NO middle possibility.

Only pseudo-sophisticated modernists claim that Truth depends on the eye of the beholder.   And my 8th grade boys, as advanced as they are for their age, have yet to be  grounded in philosophy and critical thinking.   Nevertheless, I was encouraged to witness their grappling with important ideas.   Logical thinking can be found in all disciplines, even in French class!  It’s just part of ordinary life!