1 Reasoning

inferences for decisions or persuasion

Introduction

Who should I vote for? Why won’t the car start? Are these symptoms serious? Take a moment to think it through. Connect relevant information to make a decision or decide what to believe. That’s reasoning.

Goals

• Describe the nature of reasoning and how we communicate reasoning
• Identify the kind of expressions used in constructing arguments
• Distinguish between inductive and deductive types of arguments
• Identify and evaluate the use of enthymemes in everyday usage
• Use connectives to create complex statements from simple statements

Wonky Thinking

There is a lot of wonky thinking.

• Politicians, social media. TV commercials, second cousins. They all make claims.
  • Some are true; some, bogus.
  • Sometimes it is opinion and other times, a good argument.
• Learning how to separate the good from the wonky is the point of these exercises.

If you enjoy these exercises, you might enjoy these classics.

• There Are Two Errors in the the Title of This Book, by Robert Martin.
• To Mock a Mockingbird and Other Logic Puzzles, by Raymond Smullyan.

Good reasoning helps decide which claims make sense.

• By learning how to separate good reasoning from the wonky thinking, you will learn the nature of an argument and how to make inferences. You will develop communication and critical thinking skills.
• Soon you’ll be catching others in wonky thinking (be forewarned).

Good reasoning can make a practical difference in life.

• A detective inspects the clues to figure out who is the culprit. A doctor checks symptoms in order to make a diagnosis.
• Figuring out whether that new iGadget worth the price. Deciding whether to buy that house, take that job, have a second date.

Reasoning is making inferences to make a decision.

• An inference connects information to make a decision or to consider whether there are persuasive reasons to support a particular belief or claim. Reasoning is connecting evidence to make a conclusion.
• These exercises put good reasoning into practice. Welcome to Reason Right, a set of exercises …
  • where you choose your path and progress at your pace,
  • as you learn how to separate right reasoning from wonky thinking,
  • in order to develop communication and critical thinking skills,
  • so you can find truth, be persuasive, and identify wonky thinking when you encounter it.

We communicate reasoning by means of arguments.

• In an argument, some statements [called premises] are reason to believe another statement [the conclusion].
  • To give reason to believe is to infer.
  • An argument is a group of statements that have an inferential relationship.
• For instance: it rained yesterday [premise] and it is raining today [premise], so it will likely rain tomorrow [conclusion].
  • The premises are evidence.
  • Taken together, they are reason to believe the conclusion.

Arguments use statements. A conclusion is a statement. Each premise is a statement.

• A statement, also called a proposition, is what a sentence asserts and is either true or false.
  • For instance ‘all humans have genes’ is a true statement
  • ‘Pigs can fly’ is false statement.
• ‘Je t’aime,’ ‘I love you,’ and ‘I am in love with you’ are different sentences, but they express the same statement.

Statements can be connected into more complex statements.

• A simple statement asserts a fact or truth, such as ‘fire requires oxygen’ or ‘my Goldie used to be a puppy.’
• The truth value of a complex statement depends on the truth value of its simple statements and the type of connective.

Common connectives are: not, and, or, if-then, if and only if.

• ‘Not’ is negation.
  • Not A = it is not true that A = A is not true.
  • When A is true not A is false; when A is false not A is true.
• ‘Or’ is disjunction.
  • A or B = A is true or B is true (A and B are called disjuncts).
  • A or B is true as long as one or both are true.
• ‘If, then’ is implication.
  • A implies B = if A is true, then B is true (A is the antecedent, B is the consequent, and the expression is called a conditional).
  • If A is true, then B is true is true except when A is true and B is false.
• ‘If and only if’ is equivalence.
  • A is equivalent to B means A has the same value as B.
  • A is equivalent to B is true when A and B have the same value (that is, both true or both false).

Not all communication is an argument.

• An argument reaches a conclusion. An expression might still be informative (e.g., an example or explanation) or persuasive (e.g., a command or wish), even if not an argument.
• Some sentences communicate, but do not declare a fact.
  • Some sentences express a command or request.
    • Leave the cat alone.
    • Please pass the prune juice.
  • Some ask a question.
    • What does it cost?
    • Where is the bathroom?
  • Others express an idea or opinion.
    • Pluto is a planet.
    • Blue and green should not be seen outside a washing machine.

Practice

Sentences can declare, ask, or command. Why do arguments use only declarative sentences? Whose thinking makes sense and whose is wonky?

• ANDY. Commands tell someone what to do and whether they do it is in the future, so the command can’t be true or false yet. As for asking, it is difficult to tell whether someone is sincerely asking and so it is difficult to measure whether the request is true. That leaves declarative sentences.
  • Can there be future facts—that which is true, but has not happened?
• DANY. A declarative sentence makes a statement about how the world is. Reasoning has to do with what is true, which needs sentences that declare a fact true or false.
  • Doesn’t a command declare what a person wants? Doesn’t a question declare what they want to know?

What are the two errors in the title of Robert Martin’s book?

• AJAY. One error is double “the” in the text of the title. The other error is that there are two errors.
  • So one error is in the title and the other is about the title? Is an error about the title still in the title?
• JAYA. The word “are” should not be capitalized and the year it was published should have been included. Other than that, there are no errors in the title; it is a bit of a trick question.
  • Should? Does ‘should’ make the claim of an error into a fact?

Everyday Arguments

In some arguments, the form or structure of the premises guarantees the conclusion.

• For instance, (1) the baby is a boy or girl and (2) the baby is not a boy. Those two premises, by their form or structure (A or B, not A), guarantee that (3) the baby is a girl. The following argument has the same form. (1) The electricity is on or off; (2) it is not on; (3) so, it must be off. Both arguments have the form: A or B; not A; therefore, B. This is one example of conclusion by form. Other forms occur in deduction.
• The point is, evidence (premises) that have a formal connection to the conclusion is reason to believe the conclusion. For instance, in a valid deductive argument the premises have a formal connection to guarantee the conclusion. 

In other arguments, the method of examining the evidence is reason to draw a conclusion.

• For example, a trial jury considers the evidence as a whole and concludes innocent or guilty. That method, preponderance of evidence, is similar to how a meteorologist forecasts the weather. This is one example of conclusion by a method. Other methods occur in induction.
• The point is, evidence that follows a method to reach a conclusion is reason to believe the conclusion. For instance, in a strong inductive argument the premises are evidence in favour of the conclusion.

In a valid deductive argument, true premises guarantee that the conclusion is true.

• Squares are rectangles; rectangles are quadrilaterals; so, squares are quadrilaterals.
• All humans have genes; I am human; therefore, I have genes.

In a strong inductive argument, relevant premises support the conclusion.

• It rained yesterday; it rained today; so, it will probably rain tomorrow.
• You were seen fleeing the scene of the crime and the stolen jewels were found in your apartment, so you are likely the thief.

Deductive arguments are based on logical form.

• If an argument has one of the following deductive forms, then that argument is valid.
  • Affirm the Antecedent (A ⊃ B), A, ∴ B
  • Conditional Proof presume A, B follows, ∴ A ⊃ B
  • Conjunctive Addition A, B, ∴ A & B
  • Constructive Dilemma (A ⊃ B) & (X ⊃ Y), A | X, ∴ B | Y
  • Deny the Consequent A ⊃ B, ~B, ∴ ~A
  • Destructive Dilemma (A ⊃ B) & (X ⊃ Y), ~B | ~Y, ∴ ~A | ~X
  • Disjunctive Addition A, ∴ A | B
  • Disjunctive Syllogism A | B, ~A, ∴ B
  • Hypothetical Syllogism A ⊃ B, B ⊃ C, ∴ A ⊃ C
  • Simplification A & B, ∴ A
• These forms guarantee that if the premises are true, the conclusion must be true.

Inductive arguments are based on effective methods.

• An inductive argument is a method for inferring a generalization from particular instances or premises, provided there is no decisive, overriding reason to the contrary. The following are effective methods.
  • Analogy
  • Corresponding Cause
  • Fair Sample
  • Statistical Syllogism
• The premises support, but don’t guarantee, that the conclusion is true. The strength of an inductive argument depends not on its form or structure, but on the relevance of the premises to the conclusion.

In everyday arguments, some parts are implied and not stated.

• You likely will not hear “the mower is running; running requires fuel; therefore, the mower has fuel.” 
• More likely, the argument would be abbreviated to “there must still be fuel in the mower since it started.” The part left out might be one of the premises or the conclusion. 

An argument in which some part is understood, but unstated is an enthymeme.

• All insects have six legs, so all wasps have six legs. [Unstated premise: all wasps are insects.]
• Your editorial is racist and racism is wrong. [Unstated conclusion: your editorial is wrong.]

Everyday reasoning is often abbreviated into enthymemes.

• I like you, so I will give you a discount on the subscription.
• You’ve got your hands full, so let me hold the door.

An enthymeme is useful when obvious, but it can conceal false claims.

• For instance: boy applies body spray [minor premise]; boy gets adoring girls [conclusion].
• This omits the false premise: body spray gets adoring girls.

To evaluate an enthymeme: state the implied parts, then assess the argument.

• Darling, I’m sorry. Busy people tend to forget such things as anniversaries. [I did not saying that I am busy, but you may presume as much if it excuses me for forgetting the anniversary.]
• I hope to repay you soon. My late aunt said she would leave a reward to everyone who had looked after her. [I actually neglected my aunt, but will allow you to presume that I looked after her.]

Practice

Are enthymemes lazy or efficient? Whose position makes more sense?

• ALAN. Enthymemes are mostly lazy, because people are in a hurry and want to say the least to get on with it. In some ways they are efficient, because shorter is always better.
  • People in a hurry. Shorter is always better. How relevant are these reasons?
• LANA. Enthymemes are kinda lazy, but people aren’t robots who need everything spelled out; some things are obvious. Yeah, kinda lazy but efficient if that which is left out won’t be misunderstood.
  • Would enthymemes not be efficient even if what is left out is misunderstood?

How does a valid deductive argument guarantee that the conclusion is true? Who is right here?

• ALEX. The form of the conclusion is in the premises. The rules of deduction transform or extract the conclusion from the premises. It is like saying: this or that, it is not this, so it is that. A or B; cover up A, what’s left is B.
  • The form of the conclusion is in the premises. If the conclusion is in the premises, isn’t that the fallacy of begging the question?
• LEXA. Any deductive argument guarantees the conclusion because it is deduction, like Sherlock Holmes used. A deductive argument cannot be invalid. Only an inductive argument can be invalid.
  • The fictional character Sherlock Holmes says deduction, but didn’t he use induction?