17.9 Evaluating Deductive Arguments with Euler Diagrams

We can interpret a deductive argument visually with an Euler diagram, which is essentially the same thing as a Venn diagram. This can make it easier to determine whether the argument is valid or invalid.

Example 1

Consider the deductive argument “All cats are mammals and a tiger is a cat, so a tiger is a mammal.” Is this argument valid?

import matplotlib.pyplot as plt
from matplotlib.patches import Circle

# Each argument: (inner set, outer set, individual, which premise 2 we were given)
# "member of inner" pins the individual inside the small circle; "member of outer"
# only pins it inside the big one, which leaves room for doubt.
ARGUMENTS = [
    ("cats", "mammals", "a tiger",
     "All cats are mammals.", "A tiger is a cat.", "A tiger is a mammal.", True),
    ("firefighters", "people who know CPR", "Jill",
     "All firefighters know CPR.", "Jill knows CPR.", "Jill is a firefighter.", False),
]

fig, axes = plt.subplots(1, len(ARGUMENTS), figsize=(10.5, 5.0))
for ax, (inner, outer, who, prem1, prem2, conclusion, pinned) in zip(axes, ARGUMENTS):
    ax.add_patch(Circle((0, 0), 1.05, alpha=0.18, fc="tab:blue", ec="black", lw=1.6))
    ax.add_patch(Circle((0, -0.32), 0.58, alpha=0.30, fc="tab:orange", ec="black", lw=1.6))
    ax.text(0, 0.82, outer, ha="center", fontsize=11)
    ax.text(0, -0.32, inner, ha="center", fontsize=11)
    if pinned:
        ax.plot([0], [-0.62], "ko"); ax.text(0.06, -0.64, who, fontsize=11)
    else:
        ax.plot([0], [-0.62], "ko"); ax.text(0.06, -0.64, who + " ?", fontsize=11)
        ax.plot([0.45], [0.45], "ko"); ax.text(0.51, 0.43, who + " ?", fontsize=11)
    ax.set_xlim(-1.35, 1.35); ax.set_ylim(-1.35, 1.35)
    ax.set_aspect("equal"); ax.axis("off")
    ax.set_title(("VALID" if pinned else "INVALID") + f"\n{conclusion}", fontsize=12)
plt.tight_layout(); plt.show()

print("the same two arguments, checked by listing every world the premises allow\n")
for inner, outer, who, prem1, prem2, conclusion, pinned in ARGUMENTS:
    print(f"Premise:    {prem1}")
    print(f"Premise:    {prem2}")
    print(f"Conclusion: {conclusion}")
    # A world says where the individual sits. Premise 1 forbids inner-but-not-outer.
    worlds = [(i, o) for i in (True, False) for o in (True, False) if o or not i]
    worlds = [w for w in worlds if (w[0] if pinned else w[1])]
    counter = [w for w in worlds if not (w[1] if pinned else w[0])]
    for i, o in worlds:
        print(f"   world: in {inner}? {str(i):<5}  in {outer}? {str(o):<5}"
              f"  conclusion holds? {(o if pinned else i)}")
    if counter:
        print(f"   {len(counter)} of the {len(worlds)} allowed worlds leave the "
              f"conclusion FALSE -> INVALID\n")
    else:
        word = "world" if len(worlds) == 1 else "worlds"
        print(f"   all {len(worlds)} allowed {word} make the conclusion true -> VALID\n")

print("the rule: an argument is invalid if you can draw the diagram so the premises")
print("hold and the conclusion fails. Whether Jill really is a firefighter is beside")
print("the point - validity is about whether the premises FORCE the conclusion.")

Show solution

Euler diagram inside a rectangle: a large circle labelled Mammals with a smaller circle labelled Cats drawn entirely inside it, and a red x labelled Tiger placed within the Cats circle, so the tiger necessarily lies inside Mammals as well. The premises are:

All cats are mammals.

A tiger is a cat.

The conclusion is:

A tiger is a mammal.

Both the premises are true. To see that the premises must logically lead to the conclusion, we can use a Venn diagram. From the first premise, we draw the set of cats as a subset of the set of mammals. From the second premise, we are told that a tiger is contained within the set of cats. From that, we can see in the Venn diagram that the tiger must also be inside the set of mammals, so the conclusion is valid.

Your Turn

Try it Now 9

Determine the validity of this argument:

$\begin{matrix} Premise: & All cats are scared of vacuum cleaners. \\ Premise: & Max is a cat. \\ Conclusion: & Max is scared of vacuum cleaners. \end{matrix}$

Answer

Euler diagram inside a rectangle: a large circle labelled Scared with a smaller circle labelled Cats drawn entirely inside it, and a red x labelled Max placed within the Cats circle, so Max necessarily lies inside Scared as well. Valid. Cats are a subset of creatures that are scared by vacuum cleaners. Max is in the set of cats, so he must also be in the set of creatures that are scared by vacuum cleaners.

Example 2

Euler diagram inside a rectangle: a large circle labelled Know CPR with a smaller circle labelled Firefighters drawn entirely inside it. Two red marks reading x with a question mark give the two positions the premises allow for Jill, one in the Know CPR ring outside Firefighters and one inside the Firefighters circle, so her position cannot be pinned down. $\begin{matrix} Premise: & All firefighters know CPR. \\ Premise: & Jill knows CPR. \\ Conclusion: & Jill is a firefighter. \end{matrix}$

Show solution

From the first premise, we know that firefighters all lie inside the set of those who know CPR. (Firefighters are a subset of people who know CPR.) From the second premise, we know that Jill is a member of that larger set, but we do not have enough information to know whether she also is a member of the smaller subset that is firefighters.

Since the conclusion does not necessarily follow from the premises, this is an invalid argument. It’s possible that Jill is a firefighter, but the structure of the argument doesn’t allow us to conclude that she definitely is.

It is important to note that whether or not Jill is actually a firefighter is not important in evaluating the validity of the argument; we are concerned with whether the premises are enough to prove the conclusion.

Your Turn

Try it Now 10

Determine the validity of this argument:

$Premise:, All bicycles have two wheels.; Premise:, This Harley-Davidson has two wheels.; Conclusion:, This Harley-Davidson is a bicycle.$

Answer

Euler diagram inside a rectangle: a large circle labelled Two wheels with a smaller circle labelled Bicycles drawn entirely inside it. Two red marks reading x with a question mark give the two positions the premises allow for the Harley, one in the Two wheels ring outside Bicycles and one inside the Bicycles circle, so its position cannot be pinned down. Invalid. The set of bicycles is a subset of the set of vehicles with two wheels; the Harley-Davidson is in the set of two-wheeled vehicles but not necessarily in the smaller circle.

Your Turn

Try it Now 11

Determine the validity of this argument:

$\begin{matrix} Premise: & No cows are purple. \\ Premise: & Fido is not a cow. \\ Conclusion: & Fido is purple. \end{matrix}$

Answer

Euler diagram inside a rectangle holding two circles that do not touch, Cows on the left and Purple things on the right. Two red marks reading x with a question mark give the two positions the premises allow for Fido, one outside both circles and one inside the Purple things circle, so his position cannot be pinned down. Invalid. Since no cows are purple, we know there is no overlap between the set of cows and the set of purple things. We know Fido is not in the cow set, but that is not enough to conclude that Fido is in the purple things set.

In addition to these categorical style premises of the form “all ___”, “some ____”, and “no ____”, it is also common to see premises that are conditionals.

Example 3

$\begin{matrix} Premise: & If you live in Seattle, you live in Washington. \\ Premise: & Marcus does not live in Seattle. \\ Conclusion: & Marcus does not live in Washington. \end{matrix}$

Show solution

From the first premise, we know that the set of people who live in Seattle is inside the set of those who live in Washington. From the second premise, we know that Marcus does not lie in the Seattle set, but we have insufficient information to know whether Marcus lives in Washington or not. This is an invalid argument.

Your Turn

Tryt it Now 12

Determine the validity of this argument:

$\begin{matrix} Premise: & If you have lipstick on your collar, then you are cheating on me. \\ Premise: & If you are cheating on me, then I will divorce you. \\ Premise: & You do not have lipstick on your collar. \\ Conclusion: & I will not divorce you. \end{matrix}$

Answer

Euler diagram inside a rectangle with three nested circles: an outer circle labelled Divorce, a circle labelled Cheating inside it, and a circle labelled Lipstick inside that. Three red marks reading x with a question mark give the positions the premises allow for You: outside all three circles, inside Divorce but outside Cheating, and inside Cheating but outside Lipstick. Invalid. Lipstick on your collar is a subset of scenarios in which you are cheating, and cheating is a subset of the scenarios in which I will divorce you. Although it is wonderful that you don’t have lipstick on your collar, you could still be cheating on me, and I will divorce you. In fact, even if you aren’t cheating on me, I might divorce you for another reason. You’d better shape up.

Adapted from Math in Society by David Lippman, hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 3.0. Changes were made. License: CC-BY-SA-3.0.