Tag Archives: Formal Logic

Logical Gal and an argument against God

13 Jun

Problem of Evil

The most oft-cited reason for why God cannot exist is the fact of evil in the world.  At least since the Enlightenment.

It goes like this:

Premise 1:  If God exists, then he would not allow suffering and evil in the world

Premise 2: Suffering and evil DO exist

Conclusion:  Therefore, God must not exist

When we run into a hypothetical argument like this, it can be valid without being true.

The above form of this particular conditional syllogism is ‘MODUS TOLLENS’ and it is valid.

The way we can see that this argument is valid, is to focus on the 2nd premise and see whether it does one of two things:

  • it can either affirm the antecedent (the clause preceding the comma in the 1st or major premise , i.e. – “God exists“)
  • or it can deny the consequent (what follows the comma in the 1st or major premise, i.e. – he would not allow suffering and evil in the world)

If the 2nd premise (the minor premise) affirms the antecedent, we call its form of hypothetical syllogism ‘Modus Ponens’.  If instead it denies the consequent, then we call this form of the valid argument ‘Modus Tollens’.

*

If you are a biblical Christian and not an adherent to Enlightenment thinking, then you can quickly spot the false premise.

Bingo!  The first premise IS false.  Only when humans started to look to their reason and perceptions as arbiter of what was TRUTH, did philosophers begin to craft God in their own image.  As Tim Keller suggests in his latest book on suffering,

Tim Keller's book on suffering

Link to book here on Amazon

 

…God might actually have a reason for allowing suffering.  But post-enlightenment man reasons this way: If I can’t see a good reason for evil and suffering, then there must not be one!   And that in itself is ANOTHER hypothetical major premise to examine.

If we are truthful, this line of thinking sure makes us seem pretty self-centered and self-referential.  Did it not occur to modern man that he might not actually know ALL the details regarding our universe?  Where is the humility???

So back to Modus Ponens and Modus Tollens, what was the point of that little Latin-flavored Logic exercise?  Just to reenforce that there are several steps to examining an argument.  We look at clarity of terms, the form of the syllogism and then the truthfulness of the premises.  Before you jump in to either congratulate someone who shares your wisdom OR to beat them over the head verbally for espousing nonsense, do your homework!  You’re less likely to come across as a fool!

Even fools are thought wise when they keep silent; with their mouths shut, they seem intelligent.   Proverbs 17:28 –

monkey with mouth shut

 

Logical Gal and Syllogisms everywhere

7 Apr

Bird Syllogism

I don’t always see what is right in front of my eyes.  It’s that common experience that occurs when you notice what was present  all along.

Today I was reading the last sentence of the book of Judges in the Bible.  It goes like this: There was no king in Israel.  Everyone did what was right in their own eyes.  (Judges 21:25)

The commentary mentioned that the author of the book meant to communicate Israel’s need for a king who was powerful enough to enforce God’s laws.  Had I not read that explicit conclusion, I would NOT have connected the 2 sentences: a) no king b) chaos as people live life as they please.

My first response to the commentary is ” Well, duh!  Thanks for bring this to my brain’s attention, because my brain (ever in self-protective filter mode) had not seen the connection.”  As it turns out only when there is extra LIGHT given to some facts do we notice the details, the nuances, the relationships.  I had read that passage numerous times.  It is also an oft-quoted observation of errant people wandering far from God.

I mentioned that thought to my husband and he responded, “Well, there’s a syllogism, for you!”  And his addition of critical thinking brought more light!  Here was a 1st degree enthymeme.  What was missing was the major premise.  The Judges passage stated the following:

  • the 2nd or minor premise – No king in Israel was reigning and enforcing the Mosaic Law
  • the conclusion – Therefore, no people followed the Mosaic Law but did what they wanted

It was then easy to construct what HAD to be the 1st or major premise:

  • All kings in Israel reign and enforce the Mosaic Law

Mosaic Law (10 Cs)

Once you know the structure of a syllogism,  you can spot arguments in toto or in fragments.  Understanding the structure not only gives you a sense of where an argument should go, it also provides NEW information that might not have been so apparent at first glance.

Ah, the usefulness of logic!

 

 

Logical Gal spots 2 fallacies in one syllogism

4 Oct

The other day I was listening to a radio program recounting a debate that had taken place in Australia.

One of the two debaters apparently resorted to name-calling and sought to be clever by doing so within a verbal syllogism.

And in the ensuing radio discussion about the quality of the debate, a listener pointed out there was a logic error within the syllogism.

Here is the syllogism (unfortunately it was intended to demean)

All mammals exhibit homosexual behavior

Joe is a mammal

Tf, Joe exhibits homosexual behavior

Can you identify the error?  You don’ t have to know anything about Joe’s sexual preferences to notice the problem.

Remember that a syllogism is limited in form by the requirement to have exactly 3 terms.  What are the terms in this one?

  1. mammals
  2. (that which)  exhibits homosexual behavior
  3. Joe

“ So…….??”  you say.

Here’s the problem: the term ‘mammals’  is actually used equivocally to mean two different concepts.

In premise 1, the term mammals really means species of mammals

So then in premise 2, mammals is used as a particular MEMBER of a species of mammals.

If we were to accurately state the premises and conclusion of the one advancing the argument, we would quickly see that he has used clever wording to cover up his Fallacy of Equivocation.  To reach his conclusion, he has to employ more than the 3 terms. (I’ve colored each term – only ‘ Joe’  is used twice.  We actually have 5 terms in this syllogism.

P1 – All species of mammals are species that have members that exhibit homosexual behavior

P2 – Joe is a member of a species of mammals that exhibits homosexual behavior

Tf – Joe is a mammal that exhibits homosexual behavior

And if that weren’t enough, he also commits the Fallacy of Division.  This happens when we assume that a quality of the group applies equally to every member of a group.

If we say “ Texas A&M sure is a passing team” in the sense that they pass the ball  a lot, it does not follow that every member of that team is a higher-than-average passer!

It may be that the Jones family is very artistic.   But Billy Jones is not necessarily artistic himself.  He might take after his great-great grandfather who played for Texas A& M!

A cake may be tasty, but each ingredient is not.  Have you ever snacked on butter? Do you see the fault in the reasoning?

Back to Joe and the name-calling debater.  Not only did his accuser have to cobble together multiple terms and then hide them, he also committed the Fallacy of Division and presumed to announce something about Joe that stretched beyond the known facts.

Remember, whoever makes the claim has to be able to defend his thinking!

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.

Who gives a darn about distribution?

21 Aug

   Distribution of terms matters….

even if YOU don’t care about distribution, the logic police do!

If we want to be logical and hold others gently to the same standard, we have to follow some rules.  Today we are talking about Rule # 4 – the one smack-dab in the middle of all 7 rules for writing a syllogism in its correct form.

Here is a synopsis of the 3 previous rules

# 1 – Three and only 3 terms are allowed in a syllogism

# 2 – The middle term can’t be in the conclusion

# 3 – If a term is distributed (applies to all in the set) in the conclusion, then it must be distributed in the premises

Today we look at # 4 – The middle term must be distributed at least once.  Since this term connects both the major and the minor terms, then it has to be as ‘ broad ‘ as possible to apply to the major and the minor terms.  We follow the technical drill of labeling the terms in the syllogism. We visually check to see if the middle term is distributed at least once. If not……then we shout FUM!!! (aka – Fallacy of undistributed middle)

                   Some chocolate is dark

                   All yummy foods are dark

                   Tf, some yummy things are chocolate

Types of Propositions Subject Terms Predicate Terms
     
A D U
I U U
E D D
O U D

When we label terms, we start with the conclusion  ‘at the bottom’ and label up.  (the term IN FRONT OF the copula is the subject or minor term…..the term AFTER the copula is the predicate or major term)

Tf,  (an I statement) some yummy things (Su) are chocolate (Pu)

We spot  ‘ yummy things’, then we notice that it is in the ‘ subject position of the proposition’ and write S.  Looking at the chart, we see that for an I statement the term in the subject position is undistributed, hence we add a ‘ u’.  The term ‘ chocolate’ is located in the predicate position of this I proposition; we write P and seeing that in an I statement, a predicate term is ALSO undistributed, we add a ‘ u’ next to the P.

Having identified the Major and Minor terms (also called the Predicate & Subject terms), the ‘ leftover term’  in the syllogism defaults to being the Middle Term (labeled M).  We can now finish labeling Premises 1 & 2.

P1:  (an I statement) Some chocolate (Pu) is dark (M u)

P2: (an A statement) All yummy foods (Sd) are dark(Mu)

So the whole syllogism looks like this:

        Some chocolate (Pu) is dark (Mu)

        All yummy foods  (Sd) are dark(Mu)

        Tf, Some yummy things (Su) are chocolate (Pu)

Is the middle term distributed at least once?  NO!!!

Therefore, we can say to the person making the argument:

“ We can’t even DISCUSS whether your case is sound until your syllogism is in the correct form!  And your middle term of ‘ Dark’ is not distributed even once!  Your conclusion assumes too much, given the data in premises 1 and 2.  You have committed…..FUM – the Fallacy of the Undistributed Middle Term.

  Off to Logic Prison with you!         

How is this useful?  I find that knowing the 7 rules of validity is a quick way to assess a syllogism when I sense that something isn’t quite right. The logic error emerges quickly when I run the argument through this checklist.

Keep an ear open for a conclusion that seems far-reaching and let me know if you’re stumped.  We’ll practice together.

Illicit logic

19 Aug

Now that I have your attention, on with validity!

Last time we chatted about ‘distribution’ of terms.  If a term is distributed, then what we mean is that we’re referring to ALL members of the subject or ALL those the predicate could possibly address.

For example in the proposition No carrots are sweet, we are saying something about ALL carrots and something about ALL ‘sweetness’ as a predicate.  So carrots and sweet are both distributed. 

If we posit……. Some boys are strong, then the terms boys and strong are undistributed because we are talking about only some of the set of boys and only some of the set of strong things.

Why do we care whether a term is distributed or undistributed?  I’m glad you asked!

Remember that we must be precise with our words.  We must not give the impression of ALL if we mean only SOME.  To say that ALL pre-teens get to stay up until midnight is a lot different than SOME do.  Since terms and their quantifiers build propositions which in turn build arguments, accuracy is important.

Often people over-generalize in order to make a point.  We, the recipient of the argument, need to be aware of quantifiers (the all, some, no, some..not) or we’ll be HAD!!!

On to rule 3 of how to test whether a syllogism (argument) is valid (i.e. in the correct FORM):

Rule 3 – if a term in the conclusion is distributed (applies to ALL of a term) , then it also must be distributed in the premises.  This prevents over-reaching conclusions.

To determine whether a term is distributed/undistributed we label our terms by the position they occupy in each of the 3 propositions and in the syllogism itself.  Here is our ‘DUDUs and UUDDs’ chart again from last time.

Subj

Pred

A(all)

D

U

I (some)

U

U

E (no)

D

D

O (some…not)

U

D

 

Some satisfying relationships are happy

Some satisfying relationships are marriages

Tf, all marriages are happy

 

Labeling our terms, starting ‘bottom up’ with the conclusion, we get:

 

Premise 1 –     Some satisfying relationships(Mu) are happy (Pu)

Premise 2 –     Some satisfying relationships (Mu) are marriages (Su)

Conclusion –   Tf, all marriages (Sd)  are happy(Pu)

 

S = subject term is marriages

P = predicate term is happy

M = middle term is satisfying relationships

U = undistributed

D = distributed

 

Rule # 3 states that if a term is distributed in the conclusion, then it has to be distributed in the premises.  We find that marriages IS distributed in the conclusion; however, where the subject term marriages is located in P2, it is NOT distributed because Premise 2 is an “ I” statement (see chart above).

Therefore, we say that the syllogism is INVALID because it violates rule # 3 (of 7 altogether), committing the Fallacy of Illicit Minor (one can violate the minor or the major term.)

Just pronounce the word ‘illicit’ in a class of 8th-graders and you have their instant attention as they wait to hear about SEX!!! 

So I have explained to my rapt class that the term ‘illicit’ means NOT allowed or unlawful.  What we are NOT allowed to conclude is that every single marriage is happy JUST because SOME satisfying relationships are happy and marital ones.   That conclusion goes FARTHER than the information given in premise 1 and premise 2.

Next time we will talk about a fallacy called FUM, where the middle term is undistributed.

In the meantime, as you read and listen to arguments, ask yourself if the conclusion drawn is valid or invalid according to Rule 3.   If you run across an egregious and interesting example, please share! 

Let’s get tough! Analyzing those arguments

12 Aug

Let’s imagine you’ve heard an argument that just doesn’t sound right,

but you can’t put your finger on the reason.  The major and minor premise are even

true statements!  So what could be wrong?

Formal Logic rules to the rescue!  Applying a few simple tests to an argument can help you determine if it is indeed ‘valid’, that is in the correct form.  (remember that formal logic doesn’t deal with the truth of propositions, but the structure of an argument)

Today we’re going to look at the first 2 of 7 rules that are easy to use in analyzing the structure of an argument.

Rule # 1 – Three and only 3 terms

       Some boys are strong

       My brother is a baseball player

       Therefore, my brother is strong

Let’s count the terms.  Remember that a term is the number of words necessary to describe a concept. Terms must contain at least one word and can have several (mint chocolate chip ice cream is one term containing 5 words).   

When we identify and label terms, we start at the bottom of the syllogism and label the terms in the conclusion

Our conclusion above is:     Tf, my brother (minor term) is strong (major term)

Next, we label the same terms elsewhere in the syllogism.  The unlabeled term will then be the middle term


As we look for that middle term, we see our problem, which term do we label as the middle term?  We have two remaining terms and they are different!

      Some boys ( ? term)  are strong (major term)

      My brother (minor term) is a baseball player ( ? term)

      Tf, my brother (minor term) is strong (major term)

You can see our problem: we have 2 terms, both different (boys, baseball player) so we don’t know WHICH one will be the middle term (the 3rd official term after we have identified the minor and major terms).

So we can say with assurance, this syllogism is NOT valid because it has 4 terms.

*

Rule # 2 – the middle term must not be in the conclusion

Again, we start to label ‘bottom up’. (this takes a while to become automatic for we are conditioned to start at the top and label down )

      Some baseball players are strong

      My brother is strong

      Tf, my brother is strong and a baseball player

We barely get started  labeling the conclusion and we see that we have a problem.  Not only are there 3 terms in that one proposition (brother, strong, baseball player), but we have a term, ‘strong’, that shows up 3 times.  That is the tip-off that our middle term strong’ is in the conclusion.  The entire syllogism is convoluted.  So we shout out: “INVALID!!”

Next time, when we look at Rules 3 & 4, we will measure how far an attribute or term extends.  We will be asking questions like,

  • Are we talking about the category or set of ALL baseball players?
  • Are we talking about the category or set of ALL that which is strong?

If we say ‘yes’, then we say that a term is ‘distributed’ – that the quality in question applies to ALL, or that we are addressing ALL the members of a set.

In the meantime, watch your words and how others use words.  We must strive to be precise with our language if we intend to communicate clearly and with as few words as possible.

Excessive and unclear verbiage is wearying!