Tag Archives: Quantifiers

Logical Gal, Pinterest and the Hebrews

6 Dec

Today, let’s talk about conversions of propositions that begin with the quantifier ‘all’: 

Here’s a true or false question about a favorite hang-out place of many women: If it’s true that all classy gals who decorate their homes stylishly spend time on Pinterest….is it ALSO true that all those who spend time on Pinterest will decorate their homes stylishly?

No….this is because you cannot convert A statements

All S is P does NOT equate to All P is S

At best you can say: Some P is S so that in the above example, it would be true that:

Some of those who spend time on Pinterest are classy gals who decorate their homes stylishly

Don’t forget – it only takes ONE to have a SOME as a quantifier!

I read another useful example of the trap of converting A statements in an explanation about the name, “Son of Man”.  This was Jesus’ favorite title for himself.

The title actually comes from the book of Daniel in the Old Testament. Here was the statement that I read:

“(the Son of Man)..could represent Israel but Israel itself cannot be the son of man” (Tabletalk Magazine, devotion dated  5 Nov 2013) 

Apparently Jewish scholars like to plug in the nation of Israel as that which the Prophet Daniel referred to.  But given the limitations of conversion of propositions, one cannot do that.  Putting the above quote in logical form we have this:

All Son of Man is Israel

All Israel is the Son of Man

Just by looking at the 2nd proposition you can see the ‘switcheroo’ doesn’t work. 

Question:

Where have you encountered advertising or an argument whose weight rests shakily on an invalid conversion like these? 

Vitamins DO make a difference – creating our first valid syllogism

9 Sep

So, have you taken your vitamins yet?  Are you convinced that some taking of supplements is a habit that improves one’s health?

Last time we set out our conclusion by identifying 2 of the 3 necessary terms.  And we narrowed down our quantifier to SOME vitamin taking, not ALL.

Today we need to finish fleshing out the syllogism by adding a 3rd term.

You will most likely think that our syllogism doesn’t communicate a strong and complete case in support of the conclusion.  You will be right!  This syllogism is just a 1st step.  The 2 premises that we write will simply show your thinking process, how you are arriving at that first conclusion.  An entire argument involves a series of syllogisms.  By focusing on just this ONE LITTLE step, we are staying ‘ honest’ in our reasoning.        

Think about Math Teachers whose litany rings in our memories, “You must show ALL your work to get full credit!”  

Here is our conclusion from last time, properly labeled:

I statement – Therefore, some taking of supplements (Su) is a habit that improves one’s health (Pu)

By the end of our session, we had established the following information about our syllogism:

  • S term of the syllogism (aka Minor Term)  = taking of supplements
  • P term of the syllogism (aka Major Term)  = a habit that improves one’s health

Today we have to come up with our 3rd term (Rule 1), which will be the M or middle term.  This term will LINK the other two terms (major & minor terms), enabling a conclusion.

After playing around with some terms to determine the IDEAL one, I think I found the one that can link the other ideas.  What we are talking about are those daily activities that make a difference in one’s health.   Thus I chose the following for a Middle Term:

Doctor-endorsed daily practices

Next I had to choose the affirmative quantifier.  Did I intend the term to be UNIVERSAL as in ALL or particular as in SOME?

For argument’ s sake, let’s suppose that I happen to think that ALL doctor-endorsed daily practices are habits that improve health (we’ll talk about TRUTH later)

Here is what our syllogism looks like:

All  doctor-endorsed daily practices (Md)  are habits that improve one’ s health (Pu)

Some taking of supplements(Su)  is a doctor-endorsed daily practice(Mu)

Tf, some taking of supplements (Su) is a habit that improves one’s health (Pu)

 

Let’s go through our checklist to see if the syllogism is at least valid.  Remember that we haven’t even addressed the truthfulness of each premise.

1. 3 and only 3 terms? YES
2. Does the Middle term illicitly show up in the conclusion? NO
3. If a term is distributed in the conclusion, is it Distributed at least one other place NA (both terms in the conclusion are Undistributed)
4. Middle term Distributed at least once? YES (in Premise # 1)
5. Are Premises 1 & 2 negative? NO
6. If Premises 1 & 2 are affirmative, is our conclusion also affirmative? YES
7. If either of the 2 premises negative, is the conclusion also negative? N/A

Therefore, we have written a VALID syllogism!  Yay!

Once you have a valid syllogism, THEN you can look at the truth/falsity of each premise.  But that’s another discussion!

The takeaway?   Those little quantifiers REALLY make a difference.  Be precise with your words.

If all gals are pretty, then are all pretty people gals?

14 Aug

Being precise matters! “But Mom, ALL my friends get to do it….”

The question of how far a term applies is called the ‘distribution’ of a term.  Terms are either ‘distributed’ or ‘undistributed’.

And to answer the question – no – pretty is NOT JUST referring to gals, but to other members of the pretty set.

When we make a universal affirmative claim (an A statement) : All gals are pretty, we are talking about the subject term gals. And, YES, since we have the quantifier ‘all’ ,then gals IS distributed because…… we are talking about every single member of the set of gals.

What about the predicate term of pretty?   All gals are pretty

As you can see, it makes sense that there are other people/things that are pretty besides gals!  So pretty is undistributed in this A (all) statement.

If you scroll to the end of the blog you will see a chart that summarizes the nomenclature for both Subject and Predicate terms in each of the 4 propositions. Once I explain it, it’s much easier to just remember the pattern by its nickname.  Scatological references being the source of humor for 13 year old boys, my 1st crop of 8th graders called it the DUDUs and UUDDs chart.  And I have found that easy to remember and draw out myself.  

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How about a particular affirmative claim, (the I statement)?  Are the S and P terms undistributed or distributed?

This one is easy – Some books are boring.

Since we are only talking about a partial group of books, then books is undistributed.  And just as obvious, there are other things besides books that are boring, so boring as the predicate term is equally undistributedSome books are boring

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No guns are safe is our universal negative, (the E proposition).   According to our chart, the S term and the P terms are both distributed.  It’s easy to see why it if we draw it out.  No guns are safe

Are we talking about every single member of the gun category?  Yes, so guns is distributed.  Are we saying about the safe things category that all of them do not  (or none of them) apply to guns?  Yes – so safe is distributed.

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Finally, let’s look at a particular negative (the O proposition): Some homework is not difficult.  Homework will be undistributed because  clearly we are not talking about every member of the homework class.  But what about difficult?  That is ‘more difficult’ to see in our mind’s eye, so let’s look at a drawing to understand why the predicate term difficult IS in fact distributed. Some HW is not difficult

We conclude that everything that belongs IN the set of difficult things has nothing to do with the ‘some HW’ that is shaded yellow.  You can see that we are making that predicate term distributed in this O proposition.

Next week we will use this concept of distributed/undistributed terms when we pick up with Rule # 3 for evaluating the validity of a syllogism.

Here’s our challenge – keep working on being precise with your language. In other words, “ Mean what you say and say what you mean!”

Here’s the infamous DUDUs and UUDDs chart: (warning – you have to remember to write the 1st vertical column of Quantifiers in the correct order:  A,I,E,O)

Subj Pred
A(all) D U
I (some) U U
E (no) D D
O (some…not) U D

True or false: Bananas are the most popular fruit.

8 Jul

Well, what do you think? Can you even say?  (yes, you can!)  Isn’t that being….. narrow or intolerant to say?  Besides who are YOU to say that!  Horrors!

Most Westerners in the 21st century don’t like to come right out and make categorical statements.  They are afraid of being labelled the J-word!  judgmental

Put your mind at ease:  saying that something IS or IS NOT is perfectly logical!

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We’ve looked at terms, one or more words used to describe a concept (something one can either see or picture in one’s mind) and discussed how terms are either CLEAR or UNCLEAR.  (not T/F, not  logical/ illogical not  right/wrong)  This is the Question you should be asking as you choose your terms:  Does your language recipient understand what you mean, you the originator of the spoken or written term?  That’s clarity.

Next in constructing a logical argument come the propositions.  A subject term, a copula (is/am/are) and a predicate term are the parts of a proposition.  There are 4 possible propositional forms:

  • All S is P  – All girls (subject)  are people who wear skirts(predicate)
  • Some S is P – Susie is a person who wears a skirt or  Some girls are people who wear skirts  (1 or more exemplars of your subject , but not all)
  • No S is P – No boys are people who wear skirts
  • Some S is not P –  Some girls are not people who wear skirts

What all these have in common is that they are either True or False statements/sentences.

That is how we evaluate propositions.

For example:   All killing is condemned by God   (as in, ‘Thou shall not murder.)

We can’t say:  That statement isn’t logical….that statement isn’t clear.  But we CAN say:  That statement is true or it’s false.

So how do we prove the proposition true or false? :  All killing is condemned by God.

This is what is so cool.  All we have to do is find ONE SINGLE SOLITARY CASE where a killing is NOT condemned by God. That one case will make our proposition false.  What we DON’T have to show is:  No killing is condemned by God.   That is TOTALLY and extravagantly unnecessary.  Let’s take just one example of God-approved killing, say defending your family. THAT would render the proposition false.  The proposition that IS true, i.e that corresponds to reality  would be this:  Some killing is condemned by God.  (or equally true:  Some killing is NOT condemned by God)

This principle is called the Law of Non-Contradiction.  It goes like this:  Contradictory statements cannot both be true or both be false at the same time and in the same way.  This specific stipulation means that you can’t equivocate (change the meaning) of the term, “Killing”.  You have to be referring to the exact same concept in both propositions.  (All killing is condemned by God and Some killing is NOT condemned by God)

Truth (aka ‘reality’) is so clean and precise, if you handle it correctly. 

PS:  there is one other contradictory pair – and we’ll talk about that next time.

HW: Listen for and notice a ‘general statement’ that are not true and see if you can come up with the counter-example that proves it false.

Whatta you sayin’? – or how to form a proposition

2 Jul
  1. Cats show affection.
  2. Ice cream makes me fat.

What you just read are ‘propositions’.  These statements or sentences are essential building blocks in a logical argument.

Who said anything about an argument?   Actually every time you assert something and give a reason for it, you’re making an argument.  The statements are the ‘propositions’.

Propositions comprise two terms.  One is the subject – what you’re talking about.  The other is what you are saying about the subject!  That term is the predicate.

So in our first proposition, ‘ice cream’ is the subject term and (a food that) makes me fat is the predicate term.

“Hold on a minute”, you say, “where did those 3 other words in italics come from?

Good question!  To look at a proposition clearly, we need to isolate the terms. Therefore, we rearrange it a bit and force a ‘copula’ to emerge.  The ‘copula’ is the neutral verb ‘to be’, but in one of 3 conjugated forms: IS, AM or ARE.

Dogs bark becomes Dogs are animals that bark.

Boys stink becomes Boys are kids who stink.

I sing becomes I am a person who sings.

How do you figure out the word after the copula?  Just ask what kind of set or category of ‘things’ your subject term might belong?  You can choose from several appropriate categories.

Ice cream is a……. food/dessert/snack/treat/concoction/item……. that makes me fat. YOU pick what you think communicates effectively, given the context.  Some fancy logicians might say, “Ice cream is THAT which makes me fat” (Whatever works!)

Something else to consider when looking at propositions – we need to add something to our subject term for accuracy.

Let’s go back to our example – Cats show affection.

  1. 1.   First – add the copula (the ‘is, am, are’ form of the verb TO BE) Cats are animals that show affection.
  2. 2.   Now – ask yourself this:  do ALL cats show affection?  Do SOME cats show affections?  Do NO cats show affection or do SOME cats NOT show affection?

What I’m asking you to discern is called the quantifier.  Too often we inaccurately misrepresent someone, a group or something by asserting falsely, “All teens text while driving.”    

-do we truly mean ALL?

-how do we know that EVERY SINGLE teen in the universe texts while driving?

Unless we can document that claim, we owe it to our audience to pull back and say SOME teens text while driving.  That is truthful.  As long as we can prove that ONE SINGLE INDIVIDUAL teen somewhere DOES indeed text while he or she drives, then we are safe to use the quantifier SOME.

Your homework for the week:

1)    Practice ‘translating’ statements/sentences into propositions with a copula (is/ am/ are)

2)    Question yourself when putting FORTH a proposition without a quantifier.  Can you spot how misleading that communication could be?   As my dad used to say, “You can’t make a general statement, you’re not a General!”  He had the correct intention, but not the correct reason.