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VENN: A Generative Program for
Computer-assisted Instruction in Syllogistic Reasoning
by Dr. Walter Maner
Department of Philosophy
Old Dominion University
Norfolk, VA 23508
ABSTRACT
VENN is a instructional program which, through
generative routines, is able to provide virtually
inexhaustible resources for computer-assisted practice
in traditional syllogistic reasoning. The program
creates interesting and original syllogisms for the
student to analyze, and thereafter checks his analysis
for accuracy, providing help messages and default
responses to simplify his task. An arbitrary degree of
learner control can be exercised, overriding program
defaults, permitting the use of VENN to check homework.
A choice of interpretations (Aristotelian or Boolean)
is offered, Venn diagrams are drawn and checked, and
semantic heuristics provide counterexamples to about
two-thirds of all invalid syllogisms.
KEYWORDS: VENN, computer-aided instruction, problem
solving, logic, heuristic, artificial intelligence,
CAI, generative, SIMULA, syllogism, PDP-10
VENN.DOC 1 August 1977 (Version 1) Page 2
'
VENN: A Generative Program for
Computer-assisted Instruction in Syllogistic Reasoning
1.0 INTRODUCTION
VENN[1] is a program which, through generative routines, is able to
provide virtually inexhaustible resources for computer-assisted practice in
traditional syllogistic reasoning. Such reasoning involves the inference
of one proposition (the conclusion of the syllogism) from two other
propositions (the major and minor premises). Each premise is constructed
in such a way that it has one category term in common with the conclusion
and another in common with the other premise. Each proposition contains
exactly two terms, linked by "are" or "are not," and the first of these
terms is preceded by "all," "no," or "some." For as long as the student
wishes to work, VENN will create interesting and original syllogisms for
him to analyze in detail, e.g.,
Major Premise: No READY ROMANCERS are BACKSEAT DRIVERS.
Minor Premise: Some BACKSEAT DRIVERS are COMPULSIVE LIARS.
-----------------------------------------------------------
Conclusion: No COMPULSIVE LIARS are READY ROMANCERS.
The effect is the same as if the program were picking them at random from a
database containing some twenty-seven million different syllogistic
arguments.
In the course of generating the syllogism, VENN analyzes the argument
internally to determine (among other things) its mood, figure, validity,
diagrammatic representation, and--if invalid--the exact reason(s) for its
invalidity. It then queries the student on each of these points, providing
brief help messages and default responses to simplify his interaction with
the program.
Twice during the execution of the program, a summary of correct
answers to previous questions is displayed, but this normally occurs only
after the student has already discovered on his own which answers are
---------------
1. Named after the English logician John Venn (1834-1923), who
developed diagrams of the type used in the program. As Venn knew, these
diagrams were generalizations of algebraic processes which had already been
introduced by Boole and illustrated in Jevons' "logical alphabet."
VENN.DOC 1 August 1977 (Version 1) Page 3
indeed correct. For example, after the student has demonstrated that he
knows what kind of mark belongs in each sector of the Venn diagram ('\',
'/', 'X', '*', '?' or 'BLANKS'), the program proceeds to mark the diagram
for him:
------------------ MAJOR CLASS: READY ROMANCERS
M ! ! ! ! M MINOR CLASS: COMPULSIVE LIARS
I ! ! ! ! A MIDDLE CLASS: BACKSEAT DRIVERS
N ! ---------- ! J MOOD-FIGURE: EIE-4
O ! !**!//!//! ! O VIEWPOINT: Boolean (hypothetical)
R ! !**!//!//! ! R
------------------
! !
! !
----------
MIDDLE
Next it asks him whether or not the syllogism is valid and, after an
acceptable answer, it pauses to display all the valid inferences (if any)
which could have been made from the same two premises, e.g.,
Conclusion: Some COMPULSIVE LIARS are not READY ROMANCERS.
During this pause the student can verify for himself that those listed are
the ONLY possible valid inferences by referring again to the Venn diagram.
If the syllogism happens to be invalid,[2] the student is quizzed about the
reason(s) for its invalidity. Ultimately, a complete data summary is
output to the student:
------------------ MAJOR CLASS: READY ROMANCERS
M ! ! ! ! M MINOR CLASS: COMPULSIVE LIARS
I ! ! ! ! A MIDDLE CLASS: BACKSEAT DRIVERS
N ! ---------- ! J MOOD-FIGURE: EIE-4
O ! !**!//!//! ! O VIEWPOINT: Boolean (hypothetical)
R ! !**!//!//! ! R VALID: No
------------------ FALLACY(7): Illicit minor
! !
! !
----------
MIDDLE
Before the program recycles to create another syllogism, it tries to
generate a plausible counterexample to the syllogism just analyzed. This
involves a systematic replacement of the original terms of the syllogism by
other terms in such a way that both premises become clearly TRUE while the
conclusion becomes clearly FALSE:
---------------
2. If the student imposes no restrictions on the generation process,
the chance of his receiving a valid syllogism is about 1 in 11 under the
Aristotelian interpretation or about 1 in 17 under the Boolean
interpretation.
VENN.DOC 1 August 1977 (Version 1) Page 4
TRUE Major Premise: No FIVE-YEAR-OLD BOYS are FORMER US PRESIDENTS.
TRUE Minor Premise: Some FORMER US PRESIDENTS are LIVING THINGS.
--------------------------------------------------------------------
FALSE Conclusion: No LIVING THINGS are FIVE-YEAR-OLD BOYS.
The semantic heuristics built into the program (see discussion below,
section 2.2) succeed in providing a counterexample for approximately
two-thirds of all invalid syllogisms.
At his option, the student may skip the question-and-answer routines
and receive the final data summary at once. Since the nature of the
syllogism to be generated can be specified with any degree of completeness,
this skipping option allows the student to use VENN for something as
specific as checking homework exercises or for something as general as
exploratory learning. He may, for example, be curious to see how a
syllogism of the same mood would fare if it were cast in another figure or
if it were analyzed under the other interpretation (e.g., Aristotelian
instead of Boolean). Or he may want to try replacing elements of a
defective syllogism one at a time until it is rendered valid. Perhaps the
teacher will challenge him to get VENN's help in constructing a series of
one-fallacy syllogisms, each one exemplifying a different fallacy. The
possibilities are many, and VENN will follow cheerfully wherever curiosity
may lead.
While several programs currently exist for drawing Venn diagrams,[3]
VENN is unique in several ways. It is the first generative program
developed for teaching syllogistic reasoning. It is, moreover, the first
program of its kind which works under either the Boolean or Aristotelian
interpretations. And it is the first program to employ heuristic rules for
constructing counterexamples. The author also believes it may be the first
program which permits the user to control the nature of the syllogism
displayed to any degree he wishes, from an extreme of no control on the one
hand to an extreme of complete control on the other. It may also be the
first program to query the student so thoroughly on facts pertinent to
developing an understanding of the syllogism under consideration.
2.0 COMMENTS ON THE PROGRAM
2.1 How VENN Determines Validity
Internally, VENN determines validity in three different ways: (1) by
comparing the form of the syllogism generated with a list of valid forms,
(2) by "reading" the Venn diagram and (3) by a process of fallacy
elimination. These mechanisms are independent, but not wastefully so,
since they permit the program to monitor itself for internal errors.
Of these, (3) is the one most like the procedures human beings would
use in the same circumstances. To see how it works, let us consider the
premises of the syllogism used in the previous examples. After the Major
---------------
3. Dr. Richard Wright, Department of Philosophy, University of
Toledo, has authored one of the better programs for mid-size computers:
CATSYL.BAS
VENN.DOC 1 August 1977 (Version 1) Page 5
Premise has been generated, VENN knows that fallacies #2 (undistributed
middle term), #5 (negative conclusion from all positive premises) and #6
(illicit minor) can be eliminated from consideration. Then, after
generation of the Minor Premise, VENN realizes that fallacies #1 (premises
all negative), #4 (affirmative conclusion from a negative Minor Premise)
and #8 (the existential fallacy) are no longer possible. This leaves #3
(affirmative conclusion from a negative Major Premise), and #7 (illicit
minor) still threatening. The generation of a negative conclusion
eliminates #3, however, leaving #7 as the sole mistake in reasoning.
By a process of addition VENN, in a manner somewhat analogous to this,
determines which marks belong in which sectors of the Venn diagram. This
procedure is, however, considerably complicated by (1) the observance of a
precedence relation among the marks and (2) the need to change the '?' to a
'*' when it becomes certain that the sector has a member.
2.2 How VENN Uses Heuristic Rules To Create Counterexamples
Since counterexamples must contain plausible true and false statements
about the real world, any computer program which attempts to construct them
must be told enough about that universe of discourse to be able to generate
such statements. What little VENN knows is exhaustively summarized in the
following semantic presumptions or rules-of-thumb:
1. VENN presumes (a) that some arbitrarily selected member of the
following set 'B' of set labels
LIVING THINGS
CELLULAR ORGANISMS
MORTAL THINGS
CHROMOSOMAL CREATURES
BIOLOGICAL ENTITIES
BUNDLES OF PROTOPLASM
FORMS OF LIFE
should be used in a proposition when the objective is to make
reference to a set having a large number of members and (b) that
two arbitrarily selected members of 'B' should be used in a
proposition when the objective is to make reference to coextensive
sets (i.e., to sets whose symmetric difference is the null set).
VENN.DOC 1 August 1977 (Version 1) Page 6
2. VENN presumes (a) that some arbitrarily selected member of the
following set 'L' of set labels
WORLD BANK DIRECTORS
MULTIBILLIONAIRES
NOBEL LAUREATES
OLYMPIC GOLD MEDALISTS
SOVIET CHESS CHAMPIONS
LUNAR EXPLORERS
FORMER US PRESIDENTS
FIVE-YEAR-OLD BOYS
FIVE-YEAR-OLD GIRLS
AFRICAN HEADS OF STATE
should be used in a proposition when the objective is to make
reference to a set having very few members and (b) that two
non-identical but otherwise arbitrarily selected members of 'L'
should be used in a proposition when the objective is to make
reference to mutually exclusive sets (i.e., to sets whose
intersection is the null set).
3. VENN presumes, finally, that some arbitrarily selected member of
the following set 'M' of set labels
HOMOSAPIENS
RATIONAL BEINGS
PROMISE MAKERS
CONSCIOUS BEINGS
HUMAN BEINGS
PERSONS
POLITICAL ANIMALS
FEATHERLESS BIPEDS
should be used in propositions when the objective is to make
reference to some set (a) which is a proper subset of some
arbitrary B-labeled set and (b) which has some arbitrary L-labeled
set as a proper subset.
A list of objectives of the kind mentioned above is established for
each proposition at the time the proposition is generated. For example, if
the premise is of type 'A' (i.e., "All S is P"), then--in order to make it
true--VENN's initial objective will be to replace both 'S' and 'P' with
B-type labels. Such terms will likely be coextensive, thereby rendering
the proposition true. If that can't be done consistently given the
objectives for the OTHER propositions, then VENN tries replacing 'S' with
an M-type label and 'P' with a B-type label. Since M-labeled sets are
likely to be subsets of B-labeled sets, this kind of substitution will also
tend to make the original proposition true. If this also fails, VENN will
(for similar reasons) try substituting an L-type label for 'S', then first
a B-type and finally an M-type label for 'P'.
The following is a summary table of the sequence of objectives VENN
sets for replacing terms in order to render propositions of type 'A', 'E',
'I' or 'O' either true (when they are premises) or false (when they are
VENN.DOC 1 August 1977 (Version 1) Page 7
conclusions):[4]
True 'A' True 'E' True 'I' True 'O'
False 'O' False 'I' False 'E' False 'A'
---------------------------------------------------------
S P S P S P S P
---------------------------------------------------------
B B B B
B M B M
B L B L
M B M B
M M
M L M L
L B L B
L M L M
L L L L
---------------------------------------------------------
Mnemonic: (B)ig, (M)edium, (L)ittle
In practice, VENN first tries to make the conclusion false then, as a
subgoal to that goal, tries to make the Minor Premise true; finally, as a
subgoal to that subgoal, VENN tries to make the Major Premise true. If
these goals can be achieved consistently within the framework imposed by
the heuristic rules and the original syllogism, VENN will output a
counterexample to the student. Otherwise, VENN will merely report how
close he came to success before giving up.
2.3 Design Compromises
2.3.1 Squares, Not Circles -
In the interest of making VENN's graphic output fast and terminal
independent, overlapping squares were substituted for the more traditional
overlapping circles. Since virtually all introductory logic textbooks use
Venn or Euler circles, however, students commonly need four or five
practice runs to adjust perceptually to VENN's squares.
---------------
4. For no particular reason, VENN now prefers B-labels to M-labels to
L-labels. The author is currently investigating whether any small
advantage can be gained by resequencing VENN's path through these
objectives.
One would think that a substantial improvement could be realized by
having TWO 'M' classes, one for referring to coextensive sets (like 'B')
and another (like 'L') for referring to exclusive sets. But M-labeled sets
must each contain as subsets all L-labeled sets, hence could not have null
intersections with each other.
VENN.DOC 1 August 1977 (Version 1) Page 8
2.3.2 Marking Conventions -
It is common to use the shading characters '/' and '\' to mark those
Venn circle sectors which are declared empty by the premises. To identify
the NONempty sectors, either (a) an 'X' or (b) an '*' and a bar of variable
length and orientation is often used.
Since the '/' and the '\' occasionally intersect, howver, and since
VENN asks the student about the FINISHED diagram, it was necessary to
reject (a) and use 'X' strictly as a third shading character. Option (b),
likewise, was rejected because of the considerable inefficiency involved in
computing the length, orientation and point of insertion for the bar.
The marking conventions actually used by VENN are simple and
straightforward:
'/' = this sector must be empty (by Major Premise)
'\' = this sector must be empty (by Minor Premise)
'X' = this sector must be empty (by Major AND Minor Premises)
'?' = this OR adjacent sector must be NONempty
'*' = this sector must be NONempty
'BLANKS' = default value of sector
One might think that the '?' would be even more ambiguous than required
since it is not visually paired with a '?' in a neighboring sector. In
practice, however, this does not cause any difficulty.
2.3.3 Minimal Existential Assumptions -
Under the Aristotelian (or existential) interpretation, VENN assumes
that the subject class of each premise has at least one member, even if the
premises are universal (i.e., type 'A' or 'E'). In many systems of
syllogistic logic, however, it is assumed that EACH class mentioned--three
in all--must be nonempty. VENN's somewhat stingier outlook is partly
motivated by a desire to keep the diagramming task simple for the student.
But the main motivation lies in a desire to have a definition of
"existential import" for three-line syllogistic inferences which is
identical to the one normally used for two-line or "immediate" inferences.
Obviously, there are advantages and disadvantages to this decision.
The chief advantage, beyond that involved in not "multiplying entities
beyond necessity," is that it becomes possible to use a series of immediate
inferences to prove syllogisms valid or invalid. The method, which has
been described by Terrell,[5] involves a series of appeals to some one
previous line of proof and some principle of immediate inference (i.e.,
contradiction, conversion, obversion, contraposition or "complement
conversion", and limitation) plus an occasional appeal to some two previous
lines of proof and the "Principle of the Syllogism":
---------------
5. D. B. Terrell, LOGIC: A MODERN INTRODUCTION TO DEDUCTIVE
REASONING (New York: Holt, Rinehart and Winston, 1967), sections 4.8 ff.
VENN.DOC 1 August 1977 (Version 1) Page 9
In the first figure, when the major premise is universal and the
minor premise is affirmative, a conclusion can be drawn that has
the quantity of the minor premise and the quality of the major
premise.[6]
A subsequent version of VENN will incorporate a proof checker for such
deductions. The student will then have an opportunity to enter his proof
one step at a time, always under VENN's watchful eye, and thereby gain
experience which will be valuable later when he is asked to construct
deductions in sentential logic.[7]
The disadvantage is that syllogisms of the form AEO-4, usually
considered valid, turn out to be problematic under the stingier
Aristotelian interpretation even though such arguments don't violate any of
the traditional rules. The latin scholars would not have considered this
much of a loss, however, and it is easy to modify the "existential fallacy"
rule to outlaw AEO-4.[8] At present, however, the program does consider
AEO-4 to be valid even though it fails the second of the three tests used
for determining validity (see section 2.1, above).
2.3.4 A Proliferation Of Rules -
VENN applies eight rules to check for fallacies in syllogistic
reasoning. Several of these could be consolidated, of course, but this
would not be appropriate given the many different ways textbook authors
state "the" rules.
---------------
6. Terrell, p. 94.
7. It probably would not be too difficult to add a GPS-like theorem
PROVER to VENN for generating such proofs. Part or (finally) all of the
program-generated proof could then be displayed should the student become
stuck.
8. Instead of
(8) Both premises universal, yet conclusion is particular
VENN could simply pose
(8) Minor class nonempty by Conclusion but not by Minor Premise
as a true/false response item. Since AEO-4 is already diagram-invalid, the
only other change required in the program would be the deletion of AEO-4
from the list of valid syllogisms.
VENN.DOC 1 August 1977 (Version 1) Page 10
3.0 TECHNICAL REMARKS
1. System Description.
VENN was developed on a DEC PDP-10 computer with 256K words
of memory (36-bit words) under a KL 602A+VM monitor at Old
Dominion University, Norfolk, Virginia.
2. Source Language.
VENN is written in SIMULA67, a general purpose, high-level,
ALGOL-like programming language comparable in power and features
to PL/I, ALGOL68, ALGOLW or SAIL. Specific features worth noting
include record-oriented dynamic memory allocation, efficient
garbage collection, reference (pointer) structures, list
structures, sets and queues, text- and character-handling
procedures, sequential and direct access input/output,
quasi-parallel processing via coroutines, and process- or
event-oriented simulation capabilities.
Like PASCAL, SIMULA was designed to be not only a language
but also a tool of thought in its own right which, by its very
use, would steer the programmer toward the production of
well-structured, highly readable and secure programs. SIMULA is
especially well suited to programming methodologies which stress
"modular construction," "top-down development" or "stepwise
refinement."[9] Despite its length (see below), VENN needs only
three GOTOs.
3. Length.
VENN.SIM requires over 1600 lines of source code (including
comments), occupies about 135 blocks of disk storage, and contains
about 16000 DEC words. VENN.SAV occupies about 150 blocks of disk
storage and contains about 18700 DEC words.
4. Core and CPU Requirement.
The execution of VENN.SAV requires a maximum of 47+12 pages
(or about 30K) of DEC core. One cycle through the complete
program with an invalid syllogism, accepting default answers
wherever possible, typically requires about 1.3 seconds of CPU
time. (SIMULA text-handling is extremely fast compared to other
ALGOL-type languages.)
5. Other Software Required.
VENN.SAV, of course, requires the SIMULA run-time system. In
addition, recompilation of VENN.SIM (e.g., after modification)
would require the presence of essential compiler files, the SIMULA
libraries 'SIMLIB' and 'LIBSIM' (plus their '.ATR' files) in the
---------------
9. See O.-J. Dahl, E. W. Dijkstra, and C. A. R. Hoare,
STRUCTURED PROGRAMMING (New York: Academic Press, 1972). Part III is
entirely devoted to SIMULA.
VENN.DOC 1 August 1977 (Version 1) Page 11
'SYS:' area. Given a normal implementation of SIMULA, however, no
difficulties in recompilation should be experienced.
6. Distribution.
VENN is in the public domain and is distributed worldwide by
the DECUS Library, Digital Equipment Computer Users Society, 146
Main Street, Maynard, Massachusetts 01754.
7. Portability.
Although no particular attention has been given to making
VENN portable to other major SIMULA installations (i.e., IBM
360/370, CDC 6000 and UNIVAC 1100), VENN does adhere closely to
"common base" SIMULA and should not be too difficult to implement
on non-DEC SIMULA systems.[10] Version 2 will address portability
problems directly.
---------------
10. The SIMULA library contains programs which will perform nearly
all of the source code translation required for interchange of the software
between SIMULA installations. The author will be glad to try to prepare
exportable versions upon request.