Venn Diagrams and Syllogisms

​
number of sets
2
3
valid/not valid syllogistic form
valid
1
not valid
1
language
Traditional
Reset
move '+' sign
show '+'
move '×' sign
show '×'
Contradiction!
Ex(
S
)
existential support
M a P
major premise
U
S a M
minor premise
U
S a P
conclusion
U
This Demonstration lets you verify 24 valid syllogisms using Venn diagrams with only one element in the domain. The domain only needs two elements, denoted by "+" and "×", to show that a syllogistic form is not valid.
The universal set
U
is divided into eight subsets by
S
,
P
, and
M
. If a subset is shaded, it is empty. A white subset does not guarantee that it contains an element, but if the sign "+" or "×" is in a subset, then it does have an element. If "+" or "×" is in a shaded subset, there is a contradiction. So the statement that a subset is empty is true if it is shaded, false if either "+" or "×" is in it, and otherwise the statement is undecided.

Details

A monadic formula of first-order logic is one for which all nonlogical symbols are one-place predicates.
Theorem. If
s
is a monadic sentence that is satisfiable, then
s
is true in some interpretation whose domain contains at most
k
2
r
members, where
k
is the number of one-place predicate letters and
r
is the number of variables in
s
.
Therefore there is an effective procedure for deciding whether or not a monadic sentence is valid[1, p. 250].
Syllogistic forms are monadic sentences if considered as sentences of the form
s∧pq
with predicate letters
S
,
P
, and
M
.

References

[1] G. S. Boolos and R. C. Jeffrey, Computability and Logic, Cambridge, UK: Cambridge University Press, 1974.
[2] L. Carroll, Symbolic Logic and The Game of Logic, New York: Dover Publications, 1958.
[3] I. M. Copi and C. Cohen, Introduction to Logic, 9th ed., New York: Macmillan Publishers, 1994 pp. 214–218.
[4] J. M. Bocheński, A History of Formal Logic, 2nd ed. (I. Thomas, trans. and ed.), New York: Chelsea Publishing Company, 1970 p. 235.
[5] Wikipedia. "Syllogism." (Aug 3, 2016) en.wikipedia.org/wiki/Syllogism.
[6] Wikipedia. "Venn Diagram." (Aug 3, 2016) en.wikipedia.org/wiki/Venn_diagram.
[7] J. Venn, "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings," Philosophical Magazine Series 5, 10(59), 1880 pp. 1–18. doi:10.1080/14786448008626877.

External Links

Interactive Venn Diagrams
Venn Diagrams
Square of Opposition in Aristotelian Logic
Lewis Carroll's Diagram and Categorical Syllogisms
Euler Circles for Categorical Syllogisms
The Ontological Vocabulary
Lewis Carroll's Bilateral Diagram

Permanent Citation

Izidor Hafner, Marc Brodie
​
​"Venn Diagrams and Syllogisms"​
​http://demonstrations.wolfram.com/VennDiagramsAndSyllogisms/​
​Wolfram Demonstrations Project​
​Published: August 9, 2016