Wednesday, February 20, 2013

Critical Thinking: Evaluating Logical Strength of Deductive and Inductive Arguments through Relevance and Sufficiency

In the previous post we talked about logical force or logical consequence (they are interchangeable).  These terms refer to the degree to which we must accept the conclusion if we've assumed the premises to be true.  When an argument has maximum logical force we say it is valid.  Generally, there are two types of logical force: deductive and inductive.

Deductive Validity
Deductive validity means that if we accept all the premises as true, we must accept the conclusion as true.  Otherwise stated, in a deductive argument, the conclusion necessarily follows from the premises.  Think of deductive arguments as something akin to math.  If you are told x=2 and y=3 and you are adding, then you must accept 5 as the answer.   Here are a couple very simple examples to illustrate the principle:

Sample A
P1.  Bob is a man.
P2.  Bob likes turtles.
C. /.: Bob is a man that likes turtles.

Sample B
P1.  If it's raining, there are clouds.
P2.  It's raining.
C /.:  There are clouds.

Sample C
P1.   Either all dogs are pink or dogs are blue.
P2.   Dogs are not pink.
C /.:  Dogs are blue

These examples may seem trivial but I want them to be simple in order to illustrate a point about how to evaluate an argument for deductive validity.  Here's what you do:  you suppose that all the premises are true--even if they aren't--and then you assess logical force; in other words, whether you are now forced to accept the conclusion.

Take Sample B.  P2 says "It's raining".  But suppose it isn't raining and you are asked to evaluate the logical force of the argument.  What would you say?  The correct answer is that it is deductively valid.  Why?  Because it don't make no gosh darn difference if the premises are true or not when we evaluate logical force.  All we care about is if we'd have to accept the conclusion if the premises were all true.

Lets do one more.  Look at Sample C.  All of the premises are empirically false.  Now suppose you are asked to evaluate the validity of the argument.  What would you say?  Valid or invalid?  Your answer should be that the answer is valid.  Again, when assessing validity we don't care a hoot about whether the premises are true or false or ridiculous.  All we care about evaluating is whether we are logically forced to accept the conclusion if all the premises are true.

Now lets look and inductive validity

Inductive Validity
An inductively valid argument is one in which the conclusion doesn't necessarily follow from the premises, but the premises make the truth of conclusion likely.  Inductive validity has to do with probability of truth, not certainty.  Inductive arguments can take several forms and are most commonly (but not exclusively) found in science.  Here are a couple samples to illustrate:

P1.  I've seen one raven in my life and it was black.
P2.  I've seen two ravens in my life and it was also black.
Pn.  I'm on my death bed and every got-tam raven I've ever seen is black.
C /.:  All ravens are black.

Notice that regardless of how many premises I have, I'm not forced to conclude that all ravens are black.  Maybe the day after I die, a white raven flys by my room.  Or maybe in some remote part of the world someone saw a white raven but didn't tell anyone else about it.   White ravens might be possible which would negate the conclusion despite all the premises being true.  The point is this: with an inductive argument it's possible that the conclusion is false even though all the premises are true.

Notice however that given the premises, our conclusion in this example is extremely probable, so we'd say the argument is inductively valid.

Lets do one more example:

P1.  A significant portion of the US population feel strongly about maintaining the right to bare arms and are politically active.
C.  /.: If the government proposes legislation banning short-sleeved shirts, this portion of the population will react strongly and challenge the law's legality.

This is an inductive argument because the conclusion doesn't necessarily follow.  It might be the case that the bare-arm rights group doesn't challenge the law.  However, given the supporting information in P1 we can say that the conclusion is highly probable.  In this this case, we would say that the argument is inductively valid because the logical connection between the premises and the conclusion is strong.

Relevance and Sufficiency
Inductive and deductive validity are lots of fun, but there's more to the logic party than that!  When we evaluate validity we are essentially evaluation the relationship between the premises and the conclusion.  In the previous post we talked about assessing the premises in terms of acceptability; i.e., how reasonable or plausible they are.  But when we are assessing validity we also need to look at the premises from another point of view.

Can you guess what it is?  Did you guess "if they are true or not?"  I hope you didn't cuz that would be wrong.  Recall (and I will repeat this as often as necessary) validity has nothing to do with evaluating the truth or acceptability of premises; when we evaluate validity we automatically assume the premises are true...remember? Good.

Ok, back to our discussion of other ways to evaluate logical force.  We can decompose logical force into two separate elements:  relevance and sufficiency.  Logical relevance is the degree to which the premises increase the likelihood of the conclusion being true.  For example,

Sample D (Inductive Arg)
P1  Bob likes cheese
P2  Bob likes ice cream
P3  Bob likes milk
P4  Bob likes sour cream
P5  Bob likes turtles
C /.:  Bob likes dairy products

We can ask of each of the premises in Sample D if the premises support the conclusion (or the degree to which they support it).  In other words, we can ask how relevant each of the premises are to the conclusion.  Our aggregate evaluation of each will bear on our assessment of the argument's overall logical force.

In Sample D we can say that P1-P4 are relevant to the conclusion but P5 is not.  However, in this case, P5 doesn't diminish the strength of the argument.  The logical force doesn't change whether P5 is there or not.  In some arguments, however, the (ir)relevance of the premises will bear on the logical force of the conclusion.

Consider another argument:

Sample Argument E

P1  I like turtles
P2  My shoes are black
C  The chemical composition of water is H2O.

What is the logical force of this argument?  In Sample E the premises are not relevant to the conclusion yet the conclusion is true.  What should we say?  Here's what:  it doesn't matter one fig that the conclusion is true when we are evaluating an argument for logical validity.  Recall that in this phase of evaluation, we assume all premises to be true.  So, lets do that.  Now, to assess logical validity we next look at the relevance of the premises to the conclusion.  Are the premises relevant to the conclusion?  I.e., do they increase the likelihood that the conclusion is true?  Nope. Therefore this argument is logically invalid.

But, you cry (tears streaming down your face), the conclusion is true!  Yeah, I know, but as you should well know by now, when assessing validity (logical force) we don't care two hoots about truth.  Alz we care about is the logical relationship between premises and conclusion--in this case, relevance.

Final note on relevance: When you evaluate an argument for relevance you have to evaluate each premise individually. Why?  Because some of the premises might be relevant while others aren't.  You can't treat them as all relevant or all irrelevant until you've looked at each one. 

Unlike relevance, we don't evaluate the sufficiency of each premise, we evaluate the sufficiency of the combined force of the premises.  Sufficiency refers to the degree to which the stated premises give us enough information to accept the conclusion as true or highly likely.  In other words,  since we can't know every relevant fact in the world (past, present, and future), are the facts contained in the premises enough on their own without any further reasons or evidence for us to reasonably accept the conclusion?   Think of sufficiency as the "enough-ness" of the total evidence presented for the conclusion.  

As you might expect, because sufficiency is about the logical relationship between premises and conclusion when we evaluate sufficiency we are assuming the premises are true.  We ask, given that all these premises are true, is this enough information on its own to force us to accept the conclusion?  I.e., is there a way for the premises to all be true, yet the conclusion false?  

Lets look at an example:

P1  Children are generally diagnosed with autism 6 months to a year after they get the vaccination for MMR.
C  /.: Therefore, the MMR vaccine causes autism.

Is P1 sufficient to accept C?  How do we evaluate this? We can approach this problem a couple of ways.  In all of them, begin by assuming P1 is true.  
Heuristic 1:   Ask yourself, does P1, on its own, guarantee the truth of C?  
Heuristic 2:  Counter-examples:  A counter example is a case where all the stated premises are true but the conclusion turns out to be false.  To construct a counter-example you try to find additional facts, reasons or evidence that would make it so the stated premises stay true but the conclusion is false or unlikely.  So... ask yourself if there any facts that would allow us to continue accepting P1 as true yet would led us to a conclusion that implies C is false.

Consider this:  The time in at which children are diagnosed with autism is the same time which important developmental changes take place in children's brains.  Due to genetic and environmental factors, these changes can manifest as autism--regardless of vaccine administration.  That is, the symptoms of autism become most easily diagnosable at the same time vaccines are typically administered--regardless of whether you actually do administer the vaccines.  

This information allows us to continue to accept P1 as true, yet conclude something different (I.e., autism naturally manifests or become easily diagnosable at the same time children get MMR shots).  So, P1 on its own is not sufficient for accepting C.  So, we'd say the premises are not sufficient to accept the conclusion and therefore, the logical force of the (inductive) argument is weak.

Consider one more example:

P1  There are clouds
C   /.: It's raining

Is P1, if true, sufficient to accept C?  No, because it's possible for P1 to be true and for C to be false.  That is, it can be cloudy without raining.  Again, we'd say the premises are not sufficient to accept the conclusion and therefore, the logical force of the (inductive) argument is weak.

We can evaluate validity (i.e., logical force) from a couple of points of view, however in all of them we assume the premises to be true.  These points of view, when combined, contribute to our total assessment of the logical force of a particular argument.

One way to distinguish types of validity is according to whether the argument is deductive or inductive.  In a deductive argument, if the premises are assumed to be true you must also accept the conclusion as true (no matter how outrageous it is and even if the premises are actually false).

With an inductive argument, validity is a matter of degree. We evaluate the degree of logical strength by assuming the premises to be true and deciding whether this is compatible with the conclusion being false.  If it is unlikely that the conclusion is false then the logical strength is strong.  If there are many other likely conclusions to the argument that we could accept without questioning the truth of the premises, then the logical strength is weak.

We can further decompose the notion of logical strength (i.e., validity) into two sub-elements:  premise relevance and sufficiency.  When we evaluate relevance we assume the premises are true and assess how whether they impact the likelihood of the conclusion being true.  In other words, we look at how well each particular premise supports the conclusion. 

Sufficiency refers to whether the premises, when taken in toto are enough on their own to guarantee the truth of the conclusion (if we assume the premises to be true).  One way to test for sufficiency is to try to come up with counter examples, that is, cases that bring in additional premises, but preserve the truth of the existing premises, and show that a different conclusion could follow from all the new premises.  A counter example shows that there's other relevant information out there that might allow us to accept the premises as true, yet reject the conclusion.

Finally, when you are asked to assess logical force/validity/strength/consequence understand that this evaluation is made up of two separate criteria (i.e., to be evaluated independently of each other):  relevance and sufficiency.

When you give your final assessment of an argument's strength, refer to both aspects (as well as premise acceptability).

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