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An Introduction To Reasoning

Real-World Reasoning

Everyday Inferences – Syntax & Logic

1 Introduction

2 Comparison Of Properties: Sameness, Difference, Change, &

Degree

3 Part-Whole Relationships

4 Reasoning With Relations

5 The Tricky Verb 'To Be'

6 Reasoning With Categorical Generalizations

7 Mixing General & Particular Propositions

8 Reasoning With (Particular) Conditionals

9 Elimination

10 Generalization

11 Fallacious Reasoning & Degree Of Difficulty

12 Concluding Remarks*

A star (*) indicates that there are exercises covering this section and previous unmarked sections.

This piece in relation to others: This chapter uses some terminology from Classifying & Analyzing Reasoning. It would be helpful to have covered Basic Evaluation – Two Criteria prior to this one. Some of the types of reasoning introduced here reappear elsewhere in the book and are treated here cursorily.

Cathal Woods 2007-2017

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Everyday Inferences – Syntax & Logic

1 Introduction

1. The kinds of inferences in this chapter are ones humans do, and typically do

well, on an everyday basis, because they are based on an understanding of basic logical

relationships. They are simple enough that they can be done in real time without

needing a pencil and paper. Some of them have more complicated versions or other,

related, forms of inference (both good and bad) which are covered in more detail

elsewhere in the book.

There are lots of names of types of inference in this chapter, but the names aren't

terribly important. What's important here is that you recognize, perhaps with a little

thought, how each kind of inference works, that is, the structure or pattern of the

inference. Once you see the features at work in each kind of reasoning, and how they are

different in each case, you can come up with your own names if you like.

2. In the exercise sets for this chapter, you will be asked to identify and evaluate

examples of the inference patterns introduced here. Many of the exercises will present a

complete match for an inference pattern but in some cases the speaker will leave out one

or more of the reasons and match a pattern only partially. In these cases, you should

supply the missing reason(s) to your standard form, with an asterisk. You can then

evaluate the strength of the reasoning and the truth of the reasons.

There are three available strengths of reasoning: (i) valid – the conclusion would

have to be true, assuming that the premises are true; (ii) strong – the conclusion would

very likely be true, assuming that the premises are true; (iii) weak – the conclusion is

only somewhat likely to be true, or not even likely to be true, assuming that the premises

are true. For more on these terms, see the chapter Basic Evaluation – Two Criteria.

2 Comparison Of Properties: Sameness, Difference, Change, & Degree

1. Entities (things, objects) have various properties. By keeping track of these

properties you can make inferences about the entities.

You can track a single entity over time, and infer that it has changed or that it

remains the same, whether with respect to place (i.e. it moves or stays still) or any other

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quality (e.g. it has changed color), or with respect to its existence (it dies or continues to

exist) or its nature (it changes what it is or remains the same kind of thing). For

example:

1 [Henry was in the kitchen earlier.] 2 [Now he is in the living-room.] (So,) 3 Henry has moved.

1 [Henry's skin was pale before his holiday to the Mediterranean island of Majorca.] 2 [When he came back it was brown.] (So,) 3 Henry got a tan.

Each of these passages presents two propositions expressing reasons-for-

believing and, in the last sentence in each passage, the proposition that is believed on

the basis of those reasons. Both of them work by comparing properties of Henry at

different times. The first involves his location at two different times, the second involves

the darkness of his skin at two different times.

The reasoning in these cases is valid: it is impossible for a thing to have and not

have the same property at the same time.

2. Based on a comparison of the properties of two different entities, you can

make a judgment that they are different, or (if a distinctive property or set of properties

is examined) are the same. For example:

1 [This mourning dove has a dark spot, that looks permanent, on its left leg.] 2 [The one last week did not have a mark.] (So,) 3 this bird is not the same as that bird.

1 This mourning dove has a dark spot, that looks permanent, on its left leg. 2 The mourning dove last week did not have a mark.

-------------------------------------------------------------------------------------- 3 This bird is not the same as the bird from last week.

In order to conclude that two things are different, you can compare the properties

of the two different entities, either at the same time or at different times, as long as you

can be sure that the second thing is not just a changed version of the first one. In the

passage above, the mark on the first bird was thought to be permanent, and so the

second bird, which lacked it, must be a different bird.

If you are sure that you are not looking at a later version of the same thing, this

inference pattern is valid. If you are reasonably sure that it is not a later version of the

same thing (but not certain that it isn't), this inference pattern is strong.

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3. By comparing the properties of entities on a single scale you can infer that they

have the property in the same or different amount or extent. For example, with respect

to height, you can use information about two entities to conclude that one is taller than

the other or the same in height; with respect to time, you can use information about two

entities to conclude that one happens earlier than another or at the same time; with

respect to number or amount, you can compare how many are in one group and how

many in another and conclude that one is more numerous (has a greater number) than

the other or that they are equal; and so on. Here is an example involving speeds:

1 [Bill ran the race in 10.5 seconds.] 2 [Henry ran the race in 11.8 seconds.] (So,) 3 Bill was faster (by 1.3 seconds) than Henry.

1 Bill ran the race in 10.5 seconds. 2 Henry ran the race in 11.8 seconds.

----------------------------------------- 3 Bill was faster (by 1.3 seconds) than Henry.

This inference pattern is valid. Comparisons of degree are comparisons of

quantities, which means that the reasoning is strong (assuming that the quantities are

on the same scale and to the same level of precision).

3 Part-Whole Relationships

1. The properties of an entity's parts sometimes transfer and often do not transfer

to the entity (a.k.a. "the whole") (and the same for the relationship between the whole

and the parts). In other words, you have to decide whether a part-whole or whole-part

inference is strong or weak on a case-by-case basis. Consider the following examples:

1 [Jim has a white paw.] (So,) 2 Jim is white.

1 [Jack's hand smells of the garlic he has been chopping.] (So,) 2 Jack smells of garlic.

Does Jim's white paw mean that he is white? It's not clear. Judging the quality of

the inference in the first passage is difficult because it depends on how you construe the

word "is" in the conclusion: What exactly does it mean for Jim to be white? Imprecision

is a BIG and CONSTANT problem in the study of reasoning. (See section 5 below for "is"

in particular, and the chapter on Problems With Meaning for other problems.) The

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second example seems (do you agree?) more straightforward: if Jack's hand smells of

garlic then Jack smells of garlic, unless, again, we want to be very careful in

distinguishing what it is for Jack to give off a smell, rather than his hand.

Here is an example of an inference from a property of the whole to a property of

the parts. For example:

1 [FC Milan is a great football team.] (So,) 2 each player on the team is great.

2. Going from the whole to the parts is called division. Division says that all of

the parts have a property that the whole has. Sometimes reasoning by division is strong

and sometime it is not; it depends on the type of thing and the parts.

Going from the parts to the whole is called collection. Collection says that the

whole has a property that all of the parts have.

There are no special names for the pattern which argues that because the whole

has a certain property, some part of it also has the property, or for the pattern which

argues that because some part has a certain property, the whole also has that property.

4 Reasoning With Relations

1. Some propositions express how two entities are related. Some examples are

"Jack is opposite Jill.", "Jack is taller than Jill.", "Jack dislikes Jill.".

2. Some relations between entities are symmetrical and some are not. Thus, you

have to decide whether such an inference is strong or weak on a case-by-case basis. For

example:

1 [Jack is next to Jill.] (So,) 2 Jill is next to Jack.

The relation "is next to" is symmetrical; if Jack is next to Jill, Jill has to be next to

Jack: the reasoning in this inference is valid. "Is behind", however, is not symmetrical

and so it doesn't follow (validly or strongly) from the fact "Jack is behind Jill." that "Jill

is behind Jack.".

Occasionally, a person might be confused about symmetricality and try to draw

an inference incorrectly:

1 [Jack likes Jill.] (So,) 2 Jill likes Jack.

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The speaker here thinks that "likes" is symmetrical – it appears in both

propositions, with "Jack" and "Jill" in both, though in reverse order in the second – but

this is a mistake.

Finally, the relationship between an entity and its properties is obviously not

symmetrical; the terms in the proposition cannot be reversed. For example, the

following inference makes no sense:

1 [Jack is (= has the property of being) tall.] (So,) 2 tall is (has the property of being) Jack.

3. A very familiar form of reasoning involves making a chain.

Chain Reasoning with entities links together two entities (A and C) by relating

them to a common entity (B, the "link" in the chain).

Two types of basic relationship that are easily formed into chains are their

relative location in space (expressed with words such as "next to", "behind" and so on)

and in time (expressed with words such as "before", "after" and so on). For example:

1 [Jim is in his kennel.] 2 [Jim's kennel is in the back garden.] (So,) 3 Jim is in the back garden.

"Jim's kennel" acts as a link so that "Jim" and "the back garden" can be joined.

This passage uses the relationship "is in" in all three propositions. However, just

using the same relationship throughout is not a guarantee that the reasoning will be

good. Consider the following pair:

1 [Seattle is west of Chicago.] 2 [Chicago is west of New York.] (So,) 3 Seattle is west of New York.

1 [Seattle is close to Eugene.] 2 [Eugene is close to San Francisco.] (So,) 3 Seattle is close to San Francisco.

Both of these examples involve the same relationship throughout: in the first

inference, the relationship used is 'west of' and 'Chicago' acts as a link; in the second, the

relationship is 'close to' and 'Eugene' acts as the link. (All three are cities on west coast

of the U.S.) In the first, the reasoning is good because the relation 'west of' is

transitive—it can be used to form a chain; in the second, 'close to' does not always form

a reliable chain, and a speaker who tries to form a chain using 'close to' will often reason

poorly.

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4. From your competence in English, you understand the logic of thousands of

relations. For example, you can also easily spot that there's something wrong with the

following inference:

1 [Smith is Jones's friend.] 2 [Jones is Henry's friend.] (So,) 3 Smith is Henry's friend.

You know that, although the speaker is attempting to construct a transitive

inference, the conclusion could well be false because you know that 'is a friend of' does

not necessarily transfer from one pair to another. Smith might not be Henry's friend,

even though Smith and Jones are friends and Jones and Henry are friends.

5. Now consider an example of chain reasoning which does not involve the same

relationship throughout and so is not an example of transitivity:

1 [Jim is in his kennel.] 2 [Jim's kennel is green.] (So,) 3 Jim is green.

This is chain reasoning because there is a "link", an entity that appears at the end

of one proposition and at the beginning of the other premise. In this case, it is Jim's

kennel. But the chain here is not an example of transitivity because the first relation is

"is in" and the second is "is". The reasoning in this example is poor because the

relationship "is in" cannot reliably be linked with "is" (in the sense of "has the

property"). We could correctly infer that Jim is in something green.

As has already been said, you already have an intuitive understanding of the

reasoning involved in the examples above. Even though you already understand the

reasoning at some level, it is worth studying the types of reasoning involved more

closely so that you can articulate what is going on. This is particularly true of the verb

"to be", which we turn to now.

5 The Tricky Verb 'To Be'

1. The example just above of Jim in his kennel, which is green, raises an

important point. The word "is" (and its other forms such as "are", "be", "was" and so on)

in English has many different meanings and is used in many different types of

predication. Consider the following inference:

1 [Smith is white.] 2 [White is a color.] (So,) 3 Smith is a color.

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If this example gives you pause, it is probably because you are trying to work out what

meanings of the word "is" in each of the three propositions will make the best sense of

the inference. The first proposition uses "white" as a property, while the second uses

"white" as a general term. The two types of predication are different, even though the

same word "is" appears in both of them. And the conclusion too is open to different

interpretations, depending on how you understand the "is". In order for this to be a

good inference, you might need to employ three different meanings of "is":

Smith is white. Smith has the property white. White is a color. White things belong to the class of colored

things. Smith is a color. Smith is colored (i.e. is a thing that belongs

to the class of colored things).

You intuitively feel the difference between different kinds of "is" propositions, but

it can take some work to articulate exactly what kind of relationship is being mentioned

in each proposition. There are different types of predication, many of which English

expresses with the same word "is". Here are only some of the different types of

predication using "is":

Jack is. Jack exists. Jack is white. Jack has the property white. Jack is a human. Jack belongs to the class human. Jack is Captain Arnold. Jack and Captain Arnold are identical.

6 Reasoning With Categorical Generalizations

1. Above, we saw chain reasoning applied to relationships (especially spatial and

temporal relationships) between entities. Chain reasoning can also be applied to

relationships between classes (also called categories).

We use classes when we say that some proportion of the members of one class are

also members of another. Here are some examples:

Anything blue is colored. (Every member of the class of blue things is also a member of the class of colored things.)

No human is greater than 10 feet tall. (No member of the class of human things is also a member of the class of things greater than 10 feet tall.)

Most dogs have tails. (Most of the members of the class of dogs are members of

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the class of things with tails.)

Some dogs are brown. (Some of the members of the class of dogs are members of the class of things that are brown.)

The first two examples are universal generalizations; they use words like "all" or

"any" and "no" or "none". Another way you can think of universal categorical

generalizations (though it rarely occurs in regular speech) is in terms of a conditional

with a variable "x". For example, "Anything blue is colored." can be thought of as "If x is

blue, then x is colored.". "x" stands for any entity. The third and fourth examples are not

universal. They use "most" and "some" as quantities.

2. Generalizations, both universal and non-universal, can be linked together in a

chain. For example:

1 [All humans are mammals.] 2 [All mammals are animals.] (So,) 3 all humans are animals.

1 [All fish live in water.] 2 [Some pets are fish.] (So,) 3 some pets live in water.

The first example involves three "all" propositions. The second example involves

one "all" and two "some" propositions. They are both types of chain reasoning because

they each have a common class or category; in the first example the common class is

"mammals", in the second it is "fish".

P&C – The Venn Diagram Method takes a brief look at chain reasoning involving

classes.

7 Mixing General & Particular Propositions

1. A familiar type of reasoning mixes categorical generalizations and propositions

about specific entities. For example:

1 [Almost all dogs have tails.] 2 [Jim is a dog.] (So,) 3 Jim is a dog.

Here, Jim is a specific member of the category 'dog', and since the generalization

tells you that if some thing is a dog, it has a tail, you can infer that Jim also has a tail.

This kind of reasoning is called instantiation (or instantiation syllogism or quasi-

syllogism).

Instantiation properly includes an additional premise, that the specific case is a

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typical one. In our example, a careful speaker might have included the premise "Jim is

typical dog with respect to having a tail.". In other words, the speaker believes that there

is nothing unusual about Jim that would suggest that he does not have a tail. (See I&S —

Induction for a full discussion.)

One familiar use of instantiation is reasoning with cause-and-effect

generalizations. You are constantly making instantiation inferences about what is

happening or going to happen based on a generalization about what typically happens in

these circumstances. Or to put it simply, a prediction (also called inference to an effect)

is a kind of instantiation. For example:

1 [There is a heat wave over most of Europe.] 2 [Heat waves cause deaths.] (So,) 3 some (heat-related) deaths are occurring.

Here, the generalization in the second sentence is causal: the deaths are the result of the

heat wave. It might have alternatively been written as "Any time there's a heat wave,

people die as a result.". The first premise states that there is a (particular) heat wave,

and the conclusion is that there must be some particular (though as yet not known)

people dying.

Notice that Instantiation looks like, but is different from, Chain. You might think

that in our example "dog" provides a link between 'Jim' and 'having a tail', but Jim is a

specific dog while "dogs" (in the first premise) is a class or category. This mix of general

and particular is distinctive of Instantiation. By contrast, in Chain the term used for the

link will be identical in both premises.

8 Reasoning With (Particular) Conditionals

1. We have already seen Chain reasoning applied to relations between entities

and to classes. Chain reasoning can also be applied to particular states of affairs or

events. States of affairs are expressed in propositions such as "Jack is at the store." and

"The Chargers won by 30 runs.". For chain reasoning involving states or events, we need

the states or events to be connected using the "If … then …" construction.

Let’s look at "If … then …" propositions. Here are some examples:

If it continues to rain, (then) the river will burst its banks.

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If the Bulls win tonight, (then) they will go through to the finals.

If you don't get off my land right now, (then) I'll call the police.

If Bill is in the library, (then) Henry is too.

If-then propositions are also called conditionals. The proposition that goes with

the "if" in a conditional is called the "antecedent" and the proposition that goes with the

"then" is called the "consequent".

Note that the word "then" is often omitted in English, and that there are other

ways of expressing a conditional in English without using "if … then …" or "if …, …" at

all. For example, the first sentence above ("If it continues to rain, the river will burst its

banks.") could be expressed as "Any more rain and the river will burst its banks.".

(Various different ways of expressing a conditional are covered in P&C's chapter on

Logically Structured English.)

2. Now that we have conditionals connecting two states, we can build a chain of

conditionals. For example:

1 [If Jack arrives before noon, we will eat lunch at home.] 2 [If we eat lunch at home, we will save time.] (So,) 3 if Jack arrives before noon, we will save time.

1 If Jack arrives before noon, we will eat lunch at home. 2 If we eat lunch at home, we will save time.

---------------------------------------------------------------- 3 If Jack arrives before noon, we will save time.

In this example, eating lunch at home is the event that provides the link which

joins Jack's arrival with saving time. It is the consequent (the "then" part) of one

conditional ("If Jack arrives before noon, we will eat lunch at home.") and the

antecedent (the "if" part) of the other ("If we eat lunch at home, we will save time.").

To repeat, the conditionals in section 6 were general (the variable x is general)

while the conditionals in this section involves particular states or events: this current

rainfall; the Bull's victory in tonight's game; the departure of a particular person from

the speaker's land; Jack’s arrival, and so on.

Note that conditionals do not say how the world is; they only say what would be

true if something else were true. And similarly, the conclusion of this chain inference

says only that if Jack arrives before noon, we will save time; it does not say that we will

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in fact save time.

Chain reasoning applied to conditionals is covered in both P&C - Big 8 Method

and P&C - Method Of Derivation.

3. Besides chain reasoning, "if … then …" propositions (conditionals) are involved

in another very familiar kind of reasoning. If you believe (or grant) that a conditional is

true and also believe that the antecedent (the "if" part) is true, you can infer that the

consequent (the "then" part) is also true. Here is an example:

1 [It is seven o'clock.] 2 [If it is seven o'clock, the cafeteria is closed.] (So,) 3 the cafeteria is closed.

One name for this kind of reasoning — involving a conditional about specific

events and then asserting the antecedent — is matching the antecedent (also asserting

the antecedent or modus ponens). This type of reasoning is discussed in P&C - Big 8

Method and P&C - Method Of Derivation.

Notice that MA looks like but is different from Instantiation. Crudely, they both

seem to involve "<one thing> connected <another thing>." and then the repetition of

<one thing>. But in MA, the repetition is exact while in Instantiation, the first premise

is more general than the second premise. This mix of general and particular is

distinctive of Instantiation. By contrast, in MA the item that is "matched" will be

identical in both premises.

Matching The Antecedent Vs Instantiation

1 If <proposition-p>, then <proposition-q>. 1 % As are Bs. 2 <proposition-p>. 2 x is an A.

-------------------------------------------------- -------------- 3 <proposition-p>. 3 x is a B.

9 Elimination

1. A final very common form of reasoning concerns situations with a limited

range of options, followed by all but one of the possibilities being eliminated. Here's an

example: (The sentences in italics provides some context, but are not part of the

reasoning.)

You think to yourself: 1 [I'll get either the stir-fry or else the stuffed ravioli.] The waiter then says: Unfortunately, 2 [there is no ravioli this evening.] You think:

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(So,) I'll get the stir-fry.

This kind of reasoning is called elimination inference or argument by

elimination or disjunctive syllogism. Elimination is discussed in P&C.

10 Generalization

1. How you come to know propositions about categories of things or about cause-

and-effect relationships is another kind of reasoning, one based on generalizing from

your experience of particular things. (You can of course be told a generalization by

someone else. But that person, or somebody further back in time, used her experience to

generate the generalization.) The process of generating quantified categorical

propositions from repeated experience of particular things is called induction or

generalization. Induction is not a reasoning process that is easy for humans to do. Or

rather, it is not one that is easy for humans to do well. Humans perform generalizations

very quickly, but (as is discussed in the I&S chapter on Induction) they often do so too

quickly.

11 Fallacious Reasoning & Degree Of Difficulty

1. Many of these types of reasoning are so familiar that you can also easily tell

when reasoning involving them goes wrong. For example, you can easily spot that

there's something wrong with the following attempt at transitivity:

1 [Jack is standing shoulder to shoulder with Jill.] 2 [Jill is shoulder to shoulder with Henry.] (So,) 3 Jack is shoulder to shoulder with Henry.

You can spot the fallacious reasoning because you know that 'shoulder to

shoulder with', doesn't transfer. Now compare that with this piece of reasoning:

1 [Jack is to the right of Jill.] 2 [Jill is to the right of Henry.] (So,) 3 Jack is to the right of Henry.

Unlike 'next to', 'to the right of' is transitive, and so this is valid reasoning. "Valid"

means that the conclusion has to be true, assuming the reasons offered are true. Validity

is the highest possible strength of reasoning.

2. Telling the difference between good and bad reasoning gets more difficult as

the passages get more complex. The types of simple reasoning we have seen so far can

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quickly become difficult. First look at this example of Chain reasoning involving the

relative location of three cities in Ireland:

1 [Belfast is north of Newry.] 2 [Newry is north of Dublin.] (So,) 3 Belfast is north of Dublin.

1 Belfast is north of Newry. 2 Newry is north of Dublin.

------------------------------ 3 Belfast is north of Dublin.

No problem. You can follow this in your head and see that the reasoning is valid.

But notice that it gets a little trickier if we present the exact same information in a

different order. Consider:

1 [Newry is north of Dublin.] 2 [Belfast is north of Newry.] (So,) 3 Belfast is north of Dublin.

1 Newry is north of Dublin. 2 Belfast is north of Newry.

------------------------------ 3 Belfast is north of Dublin.

This is a little trickier because there's a longer gap between the two mentions of

Newry, which is the linking city. Your mind has to go back and get the first piece of

information about Newry (that it is north of Dublin) once it hears Newry mentioned the

second time, in relation to Belfast. We prefer the reasoning to proceed step by step, with

an obvious connection at each step.

Now, let's also increase the number of items involved (and change the example):

1 [Jack is left of Jill.] 2 [Henry is left of Smith.] 3 [Jones is left of Jack.] 4 [Jill is left of Henry.] (So,) 5 Jack is left of Smith.

The relationship here is a fairly simple one, 'left of', but the number of people

involved, along with the non-sequential ordering, makes this difficult enough that you

probably need to move very slowly, constructing a mental model of the information in

your mind, or perhaps on paper. The types of thing involved and the pattern of

reasoning itself, however, are still familiar, and thus you should feel confident that you

can, with some patient thought, judge whether or not the conclusion is made true by

the reasons. Keep in mind the general principle that any complex piece of reasoning can

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be broken down into a series of smaller steps and each small step is simple.

12 Concluding Remarks*

1. As you can see from this brief survey, reasoning is part of every human's

everyday speech. Everyone makes basic inferences about the identity and location of

objects, and about various different relationship between things (including between

things and their parts), and is able to link together information in chains, to apply

general propositions to specific cases, and to argue by elimination.

2. In the other Parts of this book we take some of these basic modes of reasoning

and expand them up to and somewhat beyond the point of most people's intuitive

reasoning. Some of the reasoning discussed in this book is very simple and the rest you

are capable of performing even if it takes not only a pencil and paper but some learning,

practice and concentration.

3. The summary on the next page provides a brief description of each of the

different patterns of reasoning you have seen in this chapter.

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Summary

● Sameness/Difference Of Properties Can be used to conclude that an entity has or has not changed (e.g. Jack was in front of the TV when I came in. He still there now. So, he hasn't moved.) or to conclude that two entities are or are not identical (e.g. Jack has dark hair (and had dark hair at the time of the theft). The thief had light hair. So, Jack is not the thief.).

Difference Of Properties 1. A has property-p. 2. B lacks property-p.

----------------------- 3. A is different from B.

Sameness Of Properties 1. A has properties-p, q,

r, s, ... 2. B has properties-p, q,

r, s, ... ----------------------- 3. A is the same as B.

Change Of Property 1. A had property-p. 2. A lacks property-p.

----------------------- 3. A has changed.

● Comparison Of Properties 'Properties have certain values on a scale and so are same/different.' A tumbler is 3.5" high. A tall-boy is 6". So, the tall-boy is taller.

Comparison Of Properties 1. A has property-p to degree-n. 2. B has property-p to degree-n+.

------------------------------------ 3. B exceeds A.

● Part-Whole/Whole-Part 'A part has a certain property. So, the whole does.' (Or vice versa.) My arm is sore. So, I am sore. (If about the relationship between all parts and the whole, or vice versa, these are called Collection and Division, respectively.)

Part-Whole 1. A's parts have property-p.

------------------------------- 2. A has property-p.

Whole-Part 1. A has property-p.

--------------------- 2. A's parts have property-p.

● Symmetricality (Usually between entities.) The order of the entities is reversed. Jack is next to Jill. So, Jill is next to Jack.

Symmetricality

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1. A is in relationship-r to B. -------------------------------

2. B is in relationship-r to A.

● Chain Reasoning involving … ● Three Entities: Jack is taller than Jill, who is taller than Jim. So, Jack is taller

than Jim. (Transitive if same relationship used throughout.) ● Three Classes: Anything with kidneys has a liver. All humans have kidneys. So,

all humans have a liver. ● Three States/Events: If the Chargers win, they are champions. If the Chargers

are champions, I'll lose my bet. So, if they win, I'll lose my bet.

Chain Reasoning 1. A is in relationship-r to B. 2. B is in relationship-r to C. ------------------------------- 3. A is in relationship-r to C.

Chain Reasoning using the if-then relationship: 1. If <proposition-p>, then <proposition-q>. 2. If <proposition-q>, then <proposition-r>.

-------------------------------------------------- 3. If <proposition-p>, then <proposition-r>.

● Matching the Antecedent A conditional (about particular states/events) plus its antecedent yields the consequent. If the Chargers win, they are champions. The Chargers win. So, the Chargers are champions.

Matching the Antecedent 1. If <proposition-p>, then <proposition-q>. 2. <proposition-p>

-------------------------------------------------- 3. <proposition-q>

● Instantiation A generalization is applied to a particular entity. Dogs love hot dogs. Jim is a dog. So, he loves hot dogs.

Instantiation 1. % of As are Bs. 2. x is an A.

----------------- 3. x is a B.

● Elimination

'One or other. Not the one. So, the other.' Jack will go to see either Snakes On A Plane or Mission Impossible. Snakes On A Plane is sold out. So, he will see Mission

17

Impossible.

Elimination 1. <proposition-p> or <proposition-q>. 2. not <proposition-p>.

-------------------------------------------- 3. <proposition-q>.

Exercise Set (1) | Exercise Set (2)