# 6.2: Venn Diagrams

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A Venn Diagram in some ways improves on an Euler diagram by having a blank form that we can shade and X to create each of our propositions. We’ll see later how this allows us to do our analysis of inferences.

Blank form:

We have two overlapping circles, labels for each circles, space outside of both circles, and room for putting X’s and shading in all of these regions.

Let’s put number labels on each of these regions:

Let’s make this more concrete so we can discuss each region.

1) Is the region of Shrimp that are not People.

2) Is the region of things that are both Shrimp and People (you won’t find any).

3) Is the region of things that are People and not Shrimp.

4) Is the region of non-shrimp and non-people (Tubas, Cars, Bananas, etc.)

Okay, so how do we diagram each of our standard form categorical propositions?

The first rule to note is that all universal propositions are diagrammed using **Shading** and all particular propositions are diagrammed using **X’s**.

We shade in region 1 because there are no Chickens that are not Winged (at least that’s what our proposition claims).

We shade in region 2 because the proposition claims that there are no Robots that are also Conscious Beings.

NOW WE SWITCH TO X’S INSTEAD OF SHADING

We put one X in region 2 because the proposition claims that there is at least one Donkey that is also a Stubborn Thing.

We put one X in region 1 because the proposition claims that there is at least one Horse that is not a Tame Animal.

## Using Venn Diagrams for Inferences

Any immediate inference will be apparent in a Venn diagram.

What’s an immediate inference, you ask?

It’s an inference from one proposition directly (immediately) to another inference.

Let’s look at the following inference:

*No dogs are cows, so it follows that *

*it is false that there is some dog that is a cow. *

Compare our two diagrams:

Do you see how, if we overlapped them, they wouldn’t fit together?

This is how we know that if the first proposition is true, the second one must be false.

These are **Contradictories**, so of course if one is true, then the other is false.

This inference, therefore, doesn’t work. It’s an **Invalid** inference.

Let’s try another:

*Some dog is a mammal, *

*so it follows that some mammal is a dog *

Notice how the two diagrams are identical:

If they’re claiming the exact same thing, then obviously one implies or entails the other. What it means for each proposition to be true is for the same state of the world to obtain (to be the case).

Therefore, this inference is **Valid**.

This type of inference (switching the subject and predicate) is called **Conversion** and it only works for E and I Propositions. Try to do it for an A or O proposition and you’ll find that it doesn’t work.

And another type of inference that often arises:

*Some paintings are beautiful, *

*so it follows that some paintings are not beautiful. *

Let’s look at the two diagrams.

Notice how the information contained in the second diagram *is not contained in the first diagram*. Immediate inferences only work (and in fact all deductive inferences only work) if the information contained in the conclusion is already contained in the premises.

*If in the move from premises to conclusion, we have to change our diagram (add new information), then the inference is invalid *

*(at minimum there is a hidden assumption, *

*at worst the inference simply doesn’t work).*

So, to recap, there are **three kinds** of inference here:

1. Invalid inferences because the diagrams are incompatible.

2. Valid inferences because the diagrams are the same (or sometimes because the conclusion is represented in the diagram in some way).

3. Invalid inference because the conclusion introduces new information (changes the diagram).

## Conversion, Obversion, Contraposition

Let’s talk briefly about three kinds of immediate inferences one can draw in Categorical Logic. These are in addition to the inferences we can draw using the square of opposition.

### Conversion

Conversion is the switching of the subject and predicate.

So, “No Apples are Bananas” converts to “No Bananas are Apples.”

And, “Some Frisbees are round things” converts to “Some round things are frisbees.”

This only works for E and I propositions. We can remember this by looking at the middle vowels of the word Conv**E**rs**I**on.

### Obversion

Obversion is when you **change the quality** of the proposition from negative to affirmative or affirmative to negative, and then you **replace the predicate with its complement.**

So, “No Apples are Bananas” obverts to “All Apples are non-Bananas.”

And, “Some Frisbees are round things” obverts to “Some Frisbees are not non-round things.”

Obversion works for every form of categorical proposition.

What’s a complement, you ask?

A complement of class/kind/category X is whatever isn’t a member of X.

The complement of **birds** is the set of all **non-birds**.

The complement of **people identical to Lakshmi** is the set of all **things not identical to Lakshmi.**

The complement of **animals that are going to get eaten** is the set of all **things that aren’t animals that are going to get eaten**.

We form complements by introducing a negation (non, not, n’t, etc.) into the term. Usually, you can just put a “non-” in front of the term, but, as you can see, that isn’t always easy.

The important thing to remember is that your description should cover *everything* that isn’t in the group (all non-existent things, all imaginary things, all real things, everything). So don’t make the complement of “people identical to Sven” by writing “People not identical to Sven.”

### Contraposition

Obversion is when you **switch the subject and predicate** of the proposition, and then you **replace each with its complement.**

So, “All Apples are Bananas” obverts to “All non-Bananas are non-Apples.”

And, “Some Frisbees are round things” obverts to “Some non-round things are non-Frisbees.”

This only works for E and I propositions. We can remember this by looking at the middle vowels of the word Contr**A**p**O**sition.