Conditional Reasoning 1

Overview

Conditional reasoning involves arguments with ‘if-then’ statements. There are two valid rules of conditional reasoning: modus ponens and modus tollens. And there are two invalid forms of conditional reasoning: Affirming the consequent and denying the antecedent. Learning to distinguish the valid forms from the invalid forms is essential to constructing and recognizing good arguments.

Conditional Reasoning

Conditional reasoning is any form of reasoning that involves conditionals. A conditional is an if-then statement. For this lesson, all conditionals take the form, ‘if A then B’.

Example:

If Omar misses work then he will lose his job.

The clause that follows the ‘if’ is called the antecedent. In the above conditional, [Omar misses work] is the antecedent. The clause that follows the ‘then’ is called the consequent. In the above sentence, [he will lose his job] is the consequent.

It’s important to notice that the antecedents are determined by whether they follow the ‘if’ NOT by the order of the conditional. Let’s look at an example:

Omar will lose his job if he misses work.

In the above sentence [he misses work] is still the antecedent because it follows the ‘if’. The fact that the sentence order is rearranged has not bearing on which clause is the antecedent. The antecedent always follows the ‘if’.

Symbolization
Before we learn the rules, we’re going to learn how to symbolize sentences. This will allow us to identify patterns more easily as well as reduce the amount of writing we have to do!

To symbolize if-then statements:

a) Take the first letter or letters of a key term in the antecedent,

b) draw an arrow,

c) then write the first letter or letters of the key terms in the consequent.

d) Any time you see a negation in a clause, indicate it with a ~.

Negations are: not, can’t, won’t, couldn’t, wouldn’t, isn’t, aren’t, didn’t, doesn’t, etc..

Example:

If I’m LATE for work then I won’t GET a raise.

The conditional should be symbolized this this: L —> ~G.

The ‘L’ is for ‘if I’m late for work’; the ‘G’ is for ‘I get a raise’ the ~ negates G making it ‘I won’t get a raise’; the —> indicates it’s an if-then statement.

Let’s try one more:

If you don’t KNOW how to symbolize sentences then you should PAY attention =

The conditional should be symbolized like this: ~K—>P

Modus Ponens vs Affirming the Consequent

Suppose your friend says to you: If I don’t text you by 5pm then I’ll be at the gym. Later that day you look at the clock. It’s 5:30 and still no text from your friend. Assuming what your friend said earlier is true, where is your friend? At the gym.

Let’s formalize and symbolize the argument:
(P1). If I don’t TEXT you by 5pm then I’ll be at the GYM.
(P2). (The friend) didn’t TEXT you by 5pm.
(C). Therefore, the friend is at the GYM.

Symbolized:
(P1). ~T—>G
(P2). ~T
(C). G

The above inference pattern is called modus ponens. Any argument that has this form is valid. The basic form of modus ponens looks like this:

(P1). If [antecedent] then [consequent].
(P2). [antecedent].
(C). Therefore, [consequent].

In textbooks, it’s typically symbolized like this:

(P1). P->Q
(P2). P
(C). Q

Let’s look at a similar form of argument called affirming the consequent:

(P1). If I don’t TEXT you by 5pm then I’ll be at the GYM.
(P2). I’m at the GYM.
(C). Therefore, I didn’t TEXT you by 5pm.

Let’s symbolize it:

(P1). ~T—>G
(P2). G
(C). ~T

Recall the test for validity: If I assume the premises to be true, does the conclusion necessarily follow? Is the above argument valid? The correct answer is “no” because it’s possible for all the premises to be true and the conclusion false. That is, the premises, even when assumed to be true don’t guarantee the truth of the conclusion.

If you’re not sure why, let’s work through it. All you know from the premises is that if I don’t text you then I’ll be at the gym. But the above premises don’t preclude me from going to the gym before 5pm. They also don’t preclude me texting you from the gym either. In short, (P1) only tells you what to believe if I don’t text, it doesn’t tell you what to believe if I’m at the gym.

The above form of reasoning is called affirming the consequent (AC). I affirm the consequent when I have a conditional and as my second premise I affirm the consequent (e.g., “I’m at the gym”). Compare this with the valid form, modus ponens, where my second premise is the antecedent NOT the consequent.

Here’s one more example contrasting modus ponens and affirming the consequent:

Modus Ponens

(P1) If it’s raining then there are clouds.
(P2) It’s raining.
(C) There are clouds.

Symbolization:
(P1) R—>C
(P2) R
(C) C

Affirming the Consequent

(P1) If it’s raining then there are clouds.
(P2) There are clouds.
(C) Therefore, it’s raining.

Symbolization
(P1) R–>C
(P2) C
(C) R

In the modus ponens version, the conclusion must be true and so the argument is valid. In the second version, the conclusion isn’t necessarily true. If there are clouds it doesn’t follow necessarily that it’s raining. It’s possible for there to be clouds without rain. So, when an argument has the modus ponens form it will always be valid regardless of the truth of the premises (because validity and premise acceptability aren’t related). An argument that follows the form of affirming the consequent will never be valid.

Modus Tollens vs Denying the Antecedent

Modus tollens (MT) and denying the antecedent (DA) are two other forms of conditional reasoning are often confused. Let’s look at them both and see which is valid.

Let’s start with an intuitive example:

(P1). If it’s RAINING then there are CLOUDS.
(P2). There are no CLOUDS.
(C). Therefore, it isn’t RAINING.

Symbolization (careful of negatives!):
(P1). R—>C
(P2). ~C
(C). ~R

The above argument follows the form of modus tollens: If I negate the consequent I can conclude the negation of the antecedent.

Let’s now look at Denying the Antecedent:

(P1). If it’s RAINING then there are CLOUDS.
(P2). It isn’t RAINING.
(C). Therefore, there are no CLOUDS.

Symbolization (careful with negatives!):
(P1). R—>C
(P2). ~R
(C). ~C

Does the conclusion follow necessarily from the premises? Nope. The important thing here is to understand why not. Most of us just know from experience that there are cloudy days without rain. But what makes the argument invalid has to do with logic, not our knowledge of the world. The first premise only tells us about what follows from the fact that it’s raining. It doesn’t tell us what follows from ‘it isn’t raining’. Using only the premises we’ve been provided, we can’t draw any logical inferences from ‘it isn’t raining’.

Let’s turn to a less intuitive argument to further examine the logical relations in modus tollens and denying the antecedent.

Example:

(P1). If you don’t exercise you’ll feel weak.
(P2). I don’t feel weak.
(C). Therefore I exercised. (Technically, “I didn’t not exercise” but we eliminate the double negation).

(P1). ~E—>W
(P2). ~W
(C). E/~~E (either is fine).

The above form follows modus tollens and is valid. We negated the consequent which allows us to infer the negation of the antecedent. Notice that our original antecedent was already negated (If you don’t exercise). This means we have to add one more negation to follow the modus tollens rule. However, when we have two negations side by side, in natural language they cancel each other out…To make things simpler, that’s what we do.

Let’s look at denying the antecedent:

(P1). If you don’t exercise you’ll feel weak.
(P2). I exercised (or I didn’t not exercise).
(C). Therefore, I don’t feel weak.

Symbolization:
(P1). ~E—>W
(P2). E/~~E
(C). ~W

The above argument is invalid. The premises tell us what will happen if I don’t exercise but they don’t tell us what will happen if I do. Is it logically possible to exercise and to feel weak? Yup. Maybe you exercised but you also ate some poisonous mushrooms. You’ll feel weak even though you exercised. In fact, exercising after eating poisonous mushrooms might make you feel weaker! (Pro tip: Don’t eat poisonous mushrooms then go to the gym).

The point is that denying the antecedent isn’t a valid argument form. Even if we assume the premises to be true it’s still logically possible for the conclusion to be false.

Summary of Argument Forms

We can summarize the relationship between the four argument forms and validity in the chart below:

Argument Form Valid vs Invalid
Modus Ponens (MP) Valid
Modus Tollens (MT) Valid
Affirming the Consequent (AC) Invalid
Denying the Antecedent (DA) Invalid

 

Principles of Logic 3B: Hypothesis Testing

Introduction

Now that we know what conditionals are we can take a baby step towards learning the scientific method. In it’s most basic form, the scientific method uses modus ponens and conditionals to test causal claims. Let’s see how:

Suppose someone says echinacea cures the common cold. I can take this claim and convert it into a conditional hypothesis:

Conditional hypothesis: If someone with a cold takes echinacea then their cold will be cured.

In order to test the conditional hypothesis I’m going to get out my Bunsen burner, my beakers, my test tubes, safety goggles, and my white lab coat. But most importantly I’m going to need logic. Specifically, I’m going to need the modus ponens rule. Let me explain:

I have a conditional hypothesis:

(P1). If someone takes echinacea then their cold will be cured.

In order to test it I’m going to make the antecedent true. In other words, I’m going invite people to my lab when they get a cold and give them echinacea. Hence, the second premise of our modus ponens:

(P2). People with colds take echinacea.

Finally, I’m going to see if the conditional is true. If I give people with a cold echinacea it it true that

(C). it will cure their cold (?)

To recap, all I’m doing is making a modus ponens argument:

(P1). If someone with a cold takes echinacea then their cold will be cured.
(P2). People with colds take echinacea.
(C). Are their colds cured?

BEHOLD THE POWER OF LOGIC!!!!11!!!!1 WITHOUT IT SCIENCE IS NOTHING!!!!111!!!1

The Details: Control Groups, Natural Prevalence Rates, and Placebos

Suppose I gave a 1000 people echinacea and 7 days later their colds are completely better. Conditional hypothesis confirmed!!!! If I give people with colds echinacea they get better!!!11 I gave it to them and they got better!!!!11!! Hold the press!!!! Echinacea cures colds in just 7 days!!!!

At this point you might begin to suspect the result isn’t that impressive. Why not? Because in order to establish causation we also need to know the average amount of time it takes for someone to get rid of their cold without using echinacea. Let’s call it the natural rate of remission (or recovery). Then we’ll compare that duration to the group that gets the treatment. If there’s a difference, then we have reason to believe the causal connection.

So, what this means in terms of our experiment is that we need two groups. We need one group as the control group: that is, the group that doesn’t receive treatment. The control group will give us the average rate of recovery without intervention. The treatment group will get the treatment (duh!). Once we’ve given treatments to both the control group and the treatment group, we’ll take the average recovery time of each group. If the recovery times are substantially different then we can say there’s some evidence in favor of the conditional hypothesis.

Quick Aside on What This Lesson Doesn’t Include: Before moving forward, I need to make a few remarks about scientific reasoning concepts that we won’t cover in this particular unit: measurement errors, blinding, dosing groups, selection bias, sample bias, placebo, nocebo, degrees of latitude, statistical power, statistical significance, and many more. We’re just at the beginning stages of learning about scientific reasoning. We have a long way to go. If you’re familiar with the above concepts, hang on, we’ll get there. If you aren’t, please continue on in blissful ignorance (for now!). All this to say, for this unit, we’re just keeping things simple so we can get a grasp on the foundations of scientific reasoning. The fancy details will come later in the course.

Now, let’s finish up with the above example. Many real labs have done the above experiment. Guess what the results are? People buy echinacea all the time so it must work, amIright? AmIright? It turns out that all the highest quality studies (larger sample sizes, better controls) all report either no difference between groups or a statistically negligible effect size.1 So why do people keep buying it? Return to our lessons on confirmation bias!

The best way to learn how to use conditional hypothesis for basic scientific thinking is to run through examples…

Example 2: Thimerosal and MMR Vaccines
If you waste a few dollars on echinacea, no big deal. But sometimes failing to engage in critical thinking doesn’t just lead to false beliefs and an lighter wallet; it leads to people dying or getting sick unnecessarily. In 1998, Andrew Wakefield, a British doctor published a study purporting to show that thimerosal in the MMR (measles, mumps, rubella) vaccine causes autism. The study was problematic for a variety of reason and was eventually retracted.2 But not before the damage was done.

The media, who either forgot their college critical thinking class or were on their cell phones when they took it, reacted unskeptically to the Wakefield study. Soon Jenny McCarthy and other celebrity “experts” took up the cause, infecting the internet with false beliefs. Thousands of parents, scared by the prospect of inadvertently giving their child autism, refused to vaccinate. Over time, pockets of unvaccinated populations emerged in turn leading to outbreaks of what had previously been an eradicated disease.3

Rant over. Let’s pull out our test tubes and modus ponens and see how to test the claim.

Claim: Thimerosal in vaccines causes autism.
Conditional hypothesis: If a child receives a vaccine with thimerosal then they are (more) likely to get autism.

Notice my application of the principle of charity in formulating the conditional hypothesis. An uncharitable interpretation would be: If a child receives a vaccine with thimerosal they will get autism. That’s too strong. The charitable reading is that it increases the likelihood not that itguarantees the effect. Besides, if it were true the vaccines with thimerosal always cause autism pretty much everyone vaccinated before 2001 would have autism. I haven’t googled it yet, but I’m pretty sure that’s false.

How are we going to set up the study? If I only measure the autism rate of children who have been vaccinated with the MRR vaccine I can’t establish causation. I need to compare the autism rate of unvaccinated to vaccinated and see if there’s a difference. There are a couple of ways to do this.

The simplest way is to give one large group of children the MMR vaccine and to have another group as my control (i.e., they don’t get the vaccine). The control group establishes the natural prevalence rate. It tells me the natural prevalence of autism in a given population. Then I follow the two populations for several years. (Some of the first behavioral indications of autism are at around 4 years old but usually it’s at around 5 or 6 years old). If the vaccinated group has a statistically significant higher rate, then I have evidence in favor of the hypothesis.

Aside: Sometimes, for ethical reasons you can’t have a control group so you need to do either retroactive studies or look for what are called “natural experiments.” Vaccines are one such case. Don’t worry about this for now. We’ll cover these concepts in the unit on scientific reasoning.

So what really happened with the relation between thimerosal and vaccines? Here’s the really interesting thing. Even though many labs tried to replicate Wakefield’s results, none could. But since so many parents got scared and stopped vaccinating their children, a new MMR vaccine was developed without thimerosal. By 2001 all vaccines (except the flu vaccine) were thimerosal-free.

If thimerosal causes autism what should have happened to the autism rate after thimerosal was removed from vaccines in 2001? It should have fallen, right? Guess what happened? Autism rates rose (significantly). In the unit on scientific reasoning we’ll discuss why in more detail. The short answer is the inclusion criteria changed. The diagnosis “autistic” used to be reserved for only severe cases but it changed to “autism spectrum” and now includes even mild cases.

The removal of thimerosal is an excellent example of a natural experiment. We had a population who had vaccines with thimerosal and we had a population with vaccines without it (anyone vaccinated after 2001). If we control for the changes in inclusion criteria, we can see that autism rates remained unchanged, thereby falsifying the conditional hypothesis that thimerosal causes autism.

Your Turn:

Here are some examples you can try for yourself:

Hypothesis/Claim 1: Sour sop cures cancer!!11!!11!!
Conditional Hypothesis: ?
Construct the Experiment: Suggest how you’d test the hypothesis by using the concept of control group, treatment group, and natural rate of remission (i.e., what percentage of the population will survive the cancer without any treatment).

A note on testing treatments. Generally, treatments aren’t tested against a natural remission rate because it would be unethical not to give a patient a treatment. Instead, new treatments are tested compared to the current standard treatment. If the new treatment leads to a higher rate of remission than the current standard, then we adopt it. But even if it has some effect relative to the natural non-treatment rate we don’t care unless it’s better than the current standard. Keeping the above in mind, construct your experiment.

Hypothesis/Claim 2: Eating/drinking supplement X prevents me from getting sick.
Conditional Hypothesis: ?
Construct the Experiment: Suggest how you’d test the hypothesis using the concepts we’ve discussed.

Hypothesis/Claim 3: Weight-loss supplement “Super Quantum Lipo-Fat Burner Extra Mega Shred” causes weight loss.
Conditional Hypothesis: ?
Construct the Experiment: Suggest how you’d test the hypothesis.

 

HOMEWORK

A.

(a) Symbolize the argument (use the first letter of the underlined word and ‘~’ for negation).

(b) Is the argument valid? If it is, cite whether it’s modus tollens or modus ponens. If it isn’t, cite whether it’s denying the antecedent (DA) or affirming the consequent (AC). 

Example:

I’m not tempted to look at my cellphone when I study because if you put it in another room you won’t be tempted to look at it. And I put it in another room.

(P1). P–>~T

(P2). P

(C). ~T

This is valid because it’s modus ponens.

Be sure to keep track of the negations! (E.g., not, don’t, doesn’t, won’t, can’t, isn’t, aren’t, couldn’t, wouldn’t). E.g. If you can’t say anything nice then you shouldn’t talk = ~S –> ~T. I’ll help you by pointing them out for the first 3. 

1. If you Quong doesn’t come here right this instant, he’s gonna be in big trouble. Quong’s in big trouble. Therefore, he didn’t come right away.

2. If you try to pet Otis before he’s ready then he’ll bite you. He bit you, therefore you tried to pet him before he was ready.

Nooo-Meme-Kevin-Hart-02.jpg

3. If you bless the rains down in Africa then you’re gonna take some time to do the things we never had. You didn’t bless the rains down in Africa. Therefore, you’re not gonna take some time to do the things we never had.

4. You don’t want to be my lover because if you want be my lover you gotta get with my friends. And last night when I asked you to get with my friends, you said “no.”

5.  If you know what kind of mushroom that is then you should eat it. But you don’t know what kind of mushroom it is. Therefore, you shouldn’t eat it.

6. If you like tomatoes then you’ll like ketchup.  I like ketchup therefore I like tomatoes.

7. I have an apple and a pen. If you have an apple and you have a pen then…UH…apple pen. Therefore, apple pen.

8. I don’t eat popcorn. If you eat popcorn then you’ll have to floss. Therefore, I don’t have to floss.

9. If you have high levels of anti-bodies then you’ll test positive for the zombie virus. I tested positive, therefore I have high levels of anti-bodies.

10. If he doesn’t like the food then he doesn’t have to eat it. He likes the food, therefore he has to eat it.

More difficult:

11. If you don’t like Orange is the New Black then you care only about how many explosions and car chases a show has. You care only about how many explosions and car chases a show has, therefore you don’t like Orange is the New Black.

12. If you don’t want to people to make fun of you in the gym then don’t be a half-repper. I saw you doing half-rep squats, therefore you want people to make fun of you.

13. You won’t have to drown your sorrows if you do well on the test. You don’t have to drown your sorrows, therefore you did well on the test.

 

B. Conditional Reasoning, hypothesis testing, and Falsificationism. 

(a) Convert the claims below into a conditional hypotheses. (b) Suggest how you’d test the conditional hypothesis by employing the concepts of natural prevalence rate, natural recovery rate, baseline, and control.

Example:

Claim: Ginko biloba increases your ability to memorize.

Conditional hypothesis: If I take ginko biloba then my memory will be better.

How would you test the conditional hypothesis? I would divide people into two groups. Then I’d give the members of both groups the same memory tests to establish baseline capacities. Then I’d give one group the ginko and give the other a placebo for a month. Then I’d get both groups to take another memory test. If the ginko group’s new score increased greater to baseline than the placebo groups, then I’d have (preliminary) evidence for the claim.

1. Claim: Zika causes microencephalopathy (condition where the brain does not fully develop resulting in a smaller than normal head) in pregnant women.

(a) Conditional Hypothesis:

(b) How would you test the conditional hypothesis?

2. Claim: Essential oils cure cancer. (Yes, people actually make this claim)

(a) Conditional Hypothesis:

(b) How would you test the conditional hypothesis?

3. Claim: Students who exercise regularly perform better in school than those who don’t.

(a) Conditional Hypothesis:

(b) How would you test the conditional hypothesis?

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