Download Cognitive Illusions and the Welcome Psychologism of Logicist

Cognitive Illusions and the Welcome Psychologism
of Logicist Artificial Intelligence
Selmer Bringsjord & Yingrui Yang
The Minds & Machines Laboratory
Dept. of Philosophy, Psychology & Cognitive Science
Department of Computer Science (S.B.)
Rensselaer Polytechnic Institute (RPI)
Troy, NY 12180
[email protected] [email protected]
2.19.02
Abstract:
We begin by using Johnson-Laird’s ingenious cognitive illusions (in which it seems that certain
propositions can be deduced from given information, but really can’t) to raise the spectre of a naive brand
of psychologism. This brand of psychologism would seem to be unacceptable, but an attempt to refine it in
keeping with some suggestive comments from Jacquette eventuates in a welcome version of psychologism
-- one that can fuel logicist (or logic-based) AI, and thereby lead to a new theory of context-independent
reasoning (mental metalogic) grounded in human psychology, but one poised for unprecedented
exploitation by machines.
Introduction
The overall goal of this chapter is two-fold: (i) sketch a perfectly acceptable brand of psychologism that can
function as a productive foundation for logicist (= logic-based) artificial intelligence (AI); (ii) introduce a
new theory of human context-independent reasoning as the cornerstone of this foundation. To reach our
goal, we begin by using Johnson-Laird’s ingenious cognitive illusions (in which it seems -- to most,
anyway -- that certain propositions can be deduced from given information, but really can’t) to raise the
spectre of an unpalatable brand of psychologism. Once this brand of psychologism is refined, the promised
welcome version of psychologism is produced (and, in fact, in part displayed in action), as is our new
theory of human reasoning: mental metalogic, or just ‘MML,’ for short. We conclude by briefly presenting
the context of our belief that though MML is firmly grounded in human reasoning and empirical
psychology, it’s a theory poised for unprecedented exploitation by machines.
Logical Illusions and a Naive Brand of Psychologism
Due in no small part to ingenious experimentation carried out by Johnson-Laird, one of the dominant
themes in current psychology of reasoning is the notion of a cognitive or logical illusion. In a visual illusion
of the sort with which you’re doubtless familiar, one seems to see something that, as a matter of objective
fact, simply isn’t there. (In the Sahara, what seems to be a lovely pool of water is just the same dry-as-a-
bone expanse of sand -- and so on.) In illusions of the sort that Johnson-Laird has brought to our attention,
what seems to hold on the basis of inference doesn’t. Here’s a first specimen: 1
Illusion 1
(1)
If there is a king in the hand, then there is an ace, or else if there isn’t a king in the hand, then there is an ace.
(2)
There is a king in the hand.
Given these premises, what can you infer?
Johnson-Laird has recently reported that
Only one person among the many distinguished cognitive scientists to whom we have given
[Illusion 1] got the right answer; and we have observed it in public lectures -- several hundred
individuals from Stockholm to Seattle have drawn it, and no one has ever offered any other
conclusion (Johnson-Laird 1997b, p. 430).
The conclusion that nearly everyone draws (including, perhaps, you) is that there is an ace in the hand.
Bringsjord (and, doubtless, many others) has time and time again, in public lectures, replicated JohnsonLaird's numbers -- presented in (Johnson-Laird & Savary, 1995) -- among those not formally trained in
logic. (The reason for the italicized text will become clear later.) This is wrong because ‘or else’ is to be
understood as exclusive disjunction,2 so (using obvious symbolization) the two premises become
(1')
((K  A)  (K  A))  ((K  A)  (K  A))
(2')
K
It follows from the right (main) conjunct in (1') by DeMorgan's Laws that either K  A or K  A is false.
But by the truth-table3 for ‘,’ in which a conditional is false only when its antecedent is true while it’s
consequent is false, it then follows that either way A is false, i.e., there is not an ace in the hand. Technical
philosophers and logicians, in our experience, invariably succeed in carrying out this deduction. In fact,
1
Illusion 1 is from (Johnson-Laird & Savary, 1995). Variations are presented and discussed in (JohnsonLaird, 1997a).
2
Even when you make the exclusive disjunction explicit, the results are the same. E.g., you still have an
illusion if you use
Illusion 1'
(1) If there is a king in the hand then there is an ace, or if there isn't a king in the hand then there is
an ace, but not both.
(2) There is a king in the hand.
Given these premises, what can you infer?
3
Of course, a fully syntactic proof is easy enough to give; indeed we give and discuss it below (see Figure
1).
when Bringsjord presented this problem at the 2000 annual International Computing and Philosophy
Conference, two logicians rapidly not only produced the proof in question, but commented that “Of course,
this assumes that you’re dealing with an exclusive disjunction.” Predictably, both these thinkers symbolized
the problem in first-order logic (FOL).
From the standpoint of discussing psychologism -- put roughly for now, the view that correct reasoning is
reasoning in accord not with abstract formalisms, but rather in accord with how humans in fact reason -Illusion 1 is uneventful. (On the other hand, from the standpoint of those who know a thing or two about
logic, the fact that so many cognitive scientists went awry on this problem is remarkable.) But now
consider a second logical illusion, one given again by Johnson-Laird (1997b, p. 430):
Illusion 2
(*)
Only one of the following assertions is true:
(3) Albert is here or Betty is here, or both.
(4) Charlie is here or Betty is here, or both.
(5)
This assertion is definitely true: Albert isn't here and Charlie isn't here.
Given these premises, what can you infer?
Johnson-Laird says: “These premises [in Illusion 2] yield the illusion that Betty is here” (1997b, p. 431).
But upon closer inspection this pronouncement is peculiar. To see this, using A and B as obvious
abbreviations, (5) becomes A  C. We know from (*) that (3) is true, or (4) is, but not both. Suppose
that (3) is true; then by disjunctive syllogism on (3) and A we obtain that Betty is here (B). Suppose
instead that (4) is true; then by disjunctive syllogism on (4) and C we obtain that Betty is here (B). Either
way, contra Johnson-Laird, Betty is here.4
Bringsjord showed this simple proof (and other similar ones in connection with other “illusions” offered by
Johnson-Laird) to Johnson-Laird himself; Bringsjord’s assumption was that Johnson-Laird would say that
an error on his part had been made. That assumption turned out to be wrong. Johnson-Laird replied in
personal communication that people don’t make inferences as sophisticated as the ones Bringsjord had
made in dissecting Illusion 2, and that the illusion stands. Incredulous, Bringsjord pointed out that surely a
logical illusion I is classified as such because the premises  in I are taken by most subjects to entail ,
while in fact  cannot be deduced from  using laws of logic. Johnson-Laird replied that logic is not the
arbiter in illusions; the arbiter by his lights is what most logically untrained subjects believe about what
must be true given the truth of the propositions in . As we see it, this position, and the attitude toward
logic that accompanies it, is about as good a candidate for an instance of (a presumably objectionable)
psychologism as can be found anywhere.
4
In a recent presentation of Illusion 2 in a lecture by Bringsjord, the reasoning just given was provided by a
high school student. Everyone else in the audience declared that Betty is here, “Obviously” -- but they
provided, at best, exceedingly murky arguments. The same student went on to demonstrate, on the basis of
the given information, that Betty isn't here. His reasoning this time involved indirect proof: The premises
are inconsistent, as you may wish to verify. In addition, this clever student declared that he could prove that
the moon is made of green cheese, given the premises in question (reductio again, of course). Figure 2
shows a proof in HYPERPROOF, the fourth line of which confirms that a contradiction is a logical
consequence of the given info.
But let’s slow down. Is Johnson-Laird’s position even coherent? Suppose that there is some purported
illusion I with premises , and that all subjects declare that  can be deduced from . If in fact all subjects
make this declaration, then we have here a powerful illusion. (After all, that Illusion 2 is powerful is
something Johnson-Laird attempts to substantiate by pointing out that audience after audience has declared
that Betty is here can be inferred.) But if all subjects make this declaration, then by Johnson-Laird’s
psychologistic stance it follows that it’s valid to infer  from . But since by hypothesis we’re dealing with
an illusion, by hypothesis the inference isn’t valid! We thus have a contradiction: incoherence.5
Toward a Palatable Brand of Psychologism
However, there’s a reading of Johnson-Laird’s response that provides him with a possible escape, and
points the way toward a welcome version of psychologism. To begin, we need to be more precise about
what an illusion is. Let E1, E2, … denote declarative sentences in English (actually, any natural language will
do, but that needn’t detain us). (So for example in Illusion 1 Only one of the following two
assertions is true.6 could be E1.) Let Q stand for some question in English. Following the
notation we used above, we can let capital Greek letters , , … range over sets of formulas in some
logical system (e.g., first-order logic, or just ‘FOL’), and lowercase Greek letters , , … refer to
individual formulas. Next, notice that there is some variation in the questions. Sometimes the question isn’t
open-ended, as it is in Illusions 1 and 2. Instead, sometimes it is a query about whether a particular
proposition (or state-of-affairs) is “possible.” Here's an example: 7
Illusion 3
(6)
Only one of the following premises is true about a particular hand of cards.
(7)
There is a king in the hand or there is an ace, or both.
(8)
There is a queen in the hand or there is an ace, or both.
(9)
There is a jack in the hand or there is a ten, or both.
Is it possible that there is an ace in the hand?
Now obviously, at least at this point, we don’t want to call upon some of the recherché machinery of
symbolic logic (e.g., modal logic) to formally analyze the question here as one involving a possibility
operator. Fortunately, the question is clearly intended to be a question about consistency in the standard,
5
Another route to reach apparent incoherence is to simply point out that Illusion 1 is ruled an illusion by
Johnson-Laird because the reasoning we gave above yields, while subjects don't apprehend this, and the
reasoning is based on formal logic, not on what is going on in the minds of subjects.
6
While Johnson-Laird apparently likes to use the colon, if we use a period we lose nothing, and gain the
fact that we're dealing with clearly demarcated sentences.
7
From page 1053 of (Yang & Johnson-Laird, 2000).
first-order, extensional sense.8 That is, the question here is whether the declarative sentence There is
an ace in the hand is consistent with (6)-(9). (Is it?)9 More generally, we will say that the question
Q can be one of three types: Q1: open-ended (it can ask what can be inferred); Q2: a question about whether
(a) given inferential relation(s) hold(s) between the Ei; or Q3: a question about whether (a) given inferential
relation(s) hold(s) between the Ei and another sentence (such as There is an ace in the hand in
Illusion 3). Because the Qi are concerned with an inferential relation, it would seem to make sense to
stipulate that one of the arguments in this relation is the “single turnstyle,” which traditionally denotes
provability; that is, ‘ |- ’ indicates that  can be proved from . Putting together what we have so far
gives us the following scheme for cognitive illusions:
English Version
Formal Version
E1
E2

…
En
Q 1/Q2/Q 3
|-?/R(,|-)/R(,|-,
But how can we describe the structure of the illusions themselves, rather than just the presentation of them?
Well, first, to ease exposition, but without loss of generality, hereafter suppose that we have an illusion
specifically of the Q3 variety; and suppose, specifically, that the inferential relation in question is |-.
Assuming this framework, we might venture that an illusion obtains when two conditions are met:
Def1
A cognizer c succumbs to a logical illusion
E1
…
En
Q3
if and only if:
1. c believes that  an be inferred from , i.e., Belc[ |- ];
2. not  |- .
8
And the idiosyncratic use of ‘possible’ may stem from what might be called “mental model mindset,” in
which subjects are presupposed to be conceiving of possibilities in the mental model sense. Syntactically, a
set  of formulas is inconsistent if and only if a contradiction can be derived from . Semantically, a set 
of first-order formulas is inconsistent just in case there is no interpretation on which all of  are true. The
syntactic and semantic senses of inconsistency are equivalent in first-order logic.
9
If you're letting yourself function as a subject in connection with this illusion, don't read any further in this
footnote. The answer is “No.”
Unfortunately, this account doesn't help Johnson-Laird: on it, Illusion 2 isn't a cognitive illusion, since
Betty is here can be derived from the relevant knowledge (as we’ve shown). 10 However, some
additional elements can change the story dramatically -- elements that reflect sensitivity to Johnson-Laird’s
seemingly unpalatable notion that how reasoners reason should determine what’s “valid” and “invalid.”
Specifically, here’s what we have in mind: We can turn to a more sophisticated version of our first
definition, by including some indication of what system S underlies ‘|-‘:
Def2 A cognizer c succumbs to a logical illusion
E1
…
En
Q3
if and only if:
1. c believes that  an be inferred from  in S, i.e., Belc[ |-S ];
2. not  |-S .
For example, given Def2, we can speak of deduction in accordance with a full, standard set of natural
deduction rules -- a set that includes the rules of inference we used in our proof that Betty is here. In
such a set, for example in the elegantly presented set that’s part of the natural deduction system F in
Barwise and Etchemendy's Language, Proof, and Logic (1999), there is always a rule for introducing a
logical connective, and a rule for eliminating such a connective. Specifically, then, there is a rule for
eliminating ‘,’ and a rule for introducing ‘.’ The first of these corresponds to so-called constructive
dilemma, and it’s a rule that is used in our proof that Betty is here. To see this, a formal version of
the proof in is shown in Figure 1. Where  is here composed of the two Givens, this proof establishes that
 |-F B.
But now we are finally in position to see how to make respectable Johnson-Laird’s response to this result:
We can read him to be presupposing that the S in Def2 is simply indexed to the population in question. For
example, suppose that a particular implication can only be proved if rule r is employed; to completely fix
the situation, suppose that this implication goes through only if r’ =  Elimination is available. In this case,
not  |-F-r’ B,
but the subjects in question may nonetheless believe that B is provable in the relevant manner. It seems
possible that what Johnson-Laird may have in mind is that untrained reasoners at least generally believe
that Betty is here can be inferred in a system without constructive dilemma, and that this belief may
be incorrect.
10
It’s interesting to note that the account just given, with modifications of the sort we momentarily express
in subsequent definitions, apparently captures the general phenomena that have long driven psychology of
reasoning. For example, puzzles like the Wason selection task can be precisely represented in the
machinery we have introduced and crystallized in successors to Def1. Indeed, it would seem that apparently
all the puzzles, problems, and tasks at the heart of psychology of reasoning can be formalized along the
lines set out in the present chapter. See (Stanovich & West, 2000) for an excellent catalogue of these
puzzles and problems.
The general notion that untrained reasoners lack an understanding of highly specific patterns of primitive
inference is not unprecedented in psychology. Indeed, apparently the view that such reasoners lack an
understanding of reductio ad absurdum is widespread among psychologists of reasoning. Johnson-Laird, in
personal communication, has told us that that is in fact his view; many other eminent psychologists of
reasoning are of the same view (e.g., Braine). And more concretely, there is the example of what Lance
Rips (1994) seems to have done in devising his PSYCOP system, for this system includes (e.g.) conditional
proof, set out on page 116 of (Rips, 1994) as
+P
…
Q
IF P THEN Q
but not the rule which sanctions passing from the denial of a conditional to the truth of this conditional’s
antecedent (a rule that will be central to our concluding illusion, Illusion 4). For example, Rips (1994,
p. 125-126) says that this inference cannot be made by PSYCOP:
NOT (IF Calvin passes history THEN Calvin will graduate).
Calvin passes history.
At this point it may be safe to say that we have sketched a formal account of logical illusions on which
Johnson-Laird’s response to the provability of B in Illusion 2 is coherent. But how does this account relate
to the promised welcome version of psychologism? We answer this question now.
Toward a Welcome Brand of Psychologism
Jacquette anticipates an approximation of the brand of psychologism we have in mind when he writes:
In engineering one learns how to build a bridge ‘correctly’ so that it will bear its load properly,
withstand wind shear and other kinds of stress, minimize mental fatigue and the like; similarly, in
logic one learns how to reason ‘correctly’ to draw sound inferences and avoid fallacious ones.
When we learn to build bridges or perform heart surgery or reason correctly, we also learn what not
to do relative to presumed practical purposes. We want the bridge to span that gorge safely and not
collapse, the patient to recover with improved health and not die on the operating table, and our
reasoning to expand our knowledge and improve our decision-making ability, and not lead us from
truth to falsehood. Now, I think that most theorists would be reluctant to conclude that engineering
and medicine are not reducible to physics and biology, but rather that their respective practices
combine these sciences with an assumption about the goals they should help their practioners to
achieve. Why should things be different with respect to logic as an applied discipline grounded in
the psychology of reasoning? (Jacquette 1997, pp. 324-325)
The version of psychologism that we have in mind involves a relationship not between what Jacquette calls
“applied logic” and psychology of reasoning, but rather between a field that marks the marriage of
engineering and logic, and the psychology of reasoning; the field is: logicist (= logic-based) AI. Logicist
AI, or just ‘LAI’ for short, is devoted to engineering machines capable of simulating intelligent behaviors
of various sorts, where these machines carry out their relevant work via formal reasoning. (A full
introduction to LAI can be found in (Bringsjord & Ferrucci, 1998). For a book-length treatment of LAI as
an approach to machine creativity see (Bringsjord & Ferrucci, 2000).) Though LAI isn’t exactly what
Jacquette ascribed the term ‘applied logic’ to, this term is certainly an appropriate one for the kind of AI
we’re talking about. We believe that LAI is, or at least ought to be, to use Jacquette’s phrase, “grounded in
the psychology of reasoning.” On this view, the process of building the intelligent machines in question
involves looking to human reasoning for guidance; as this building proceeds, new knowledge of value to
psychologists of reasoning is produced; and the process iterates, back and forth.
Now, in order to see this looping reciprocity in action here, let’s suppose that we wish to build a computing
machine capable of solving English-expressed logic problems, including logical illusions. A bit more
precisely, we want to construct an artificial intelligent agent A that takes some such problem P as input,
and produces for output an answer that constitutes a solution. 11 We would also like to be able to modify A
so that it succumbs to logical illusions, and thereby offers a mechanical instantiation of the human
reasoning that takes place when a logical illusion “tricks” some human. What we have learned so far about
the underlying structure of logical illusions should be helpful in working toward such an intelligent agent,
but is it? Not very; for only a little reflection from the perspective of LAI and the goal of building A reveals
that Def2 is inadequate. Why? There are many reasons; here's a fatal one.
When you aim at building A you immediately see that we have swept under the rug the thorny issue of how
to go from the English to (in this case) FOL. Our artificial agent is going to scan the English in, parse it,
and represent the English in FOL. 12 But we have completely left this process out of Def1 and Def2. Suppose
(and notice now that we’re moving back for guidance to the human sphere) that a human looks at Illusion 1
and represents English sentences (1) and (2) as P and Q, respectively; and suppose that she represents
There is an ace in the hand as A. Suppose as well that the set of inference rules in question, S,
is a standard Fitch-style set at the propositional level (such as propositional F). Clearly, not {P, Q} |- A in
this set; this implies that the second clause of Def2 is satisfied. But now suppose that this human also
happens to believe that A can be derived from these premises, for (irrational) reasons having to do with the
price of tea in China. At this point, both clauses in the definition are satisfied -- but no one would want to
say that this human has fallen prey to the illusion in question. Ergo, Def 2 can't be right, and should
presumably be supplanted with:
Def3 A cognizer c succumbs to a logical illusion
E1
…
En
Q3
if and only if:
1a. c represents the Ei as  and the “target” English sentence in the query as ;
1. c believes that  an be inferred from  in S, i.e., Belc[ |-S ];
11
Such an agent would have lots of practical value, but such value isn’t our concern herein. For example, a
great many standardized tests consist in giving humans logic problems; the Graduate Record Exam (GRE)
and Law School Admission Test (LSAT) are two examples. An agent able to solve these problems would
presumably point the way to agents able to automatically generate the problems in question, which would
presumably save many institutions lots of money.
12
For coverage of how this would work, in general, see (Russell & Norvig, 1994).
2. not  |-S .
Do we have here an acceptable account? Unfortunately, no; counterexamples are still easy to come by:
Suppose that Jones “correctly” represents the premises in Illusion 1 as , believes that  |-S A, where S is
appropriately instantiated -- but: suppose that this belief, as before, has an irrational justification, namely
that because the price of tea in China has reached exorbitant levels, the implication in question is correct.
Certainly we wouldn't want to say that Jones has been affected by Illusion 1. What’s missing is that Jones
doesn’t have in mind the right sort of justification for believing  |-S A. It would seem that those who
succumb to Illusion 1 believe that A follows from the premises on the strength of a relevant argument – it’s
just that the argument isn’t sound. For example, in our experience, when interviewed subsequent to giving
their response in Illusion 1 that the presence of the ace in the hand can be deduced, most subjects give such
rationales as: “Well look, either there is a king in the hand or there isn’t, but we know that either way there
is an ace.” It would seem that subjects tripped up because they have in mind an inchoate, enthymematic
argument establishing
{(K  A)  (K  A)} |- A,
a true equation.
Let’s take stock of where we are, then, from the standpoint of LAI. Apparently there are two tasks to
consider: the representation of the English in an illusion in formal form, and the judgment as to whether
this formal knowledge can be used to (given, as you’ll recall, that we are limiting ourselves to illusions of
Q3 form) establish . Given this, there are three possibilities, in general: 13
Label
Representation Reasoning “Tricked”?
Possibility 1
correct
incorrect
yes
Possibility 2
incorrect
correct
yes
Possibility 3
correct
correct
no
Possibility 4
incorrect
incorrect
yes
Obviously, our definition needs to take account of these possibilities via suitable disjunctions. Where 
stands for a correct representation and ’ an incorrect one, we have:14
Def4 A cognizer c succumbs to a logical illusion
E1
13
We say “in general” because of such finer-grained possibilities as that the subject could correctly
represent the premises but incorrectly represent the proposition the query concerns, and so on. In the fourth
possibility, Possibility 3 in our table, the cognizer isn’t tricked, so this possibility has no bearing on the
definition we’re seeking.
14
For ease of exposition and to work within the space constraints of this chapter, we leave aside certain
niceties. For example, to be more precise, we would need to distinguish between representation of one
premise versus another, between the premises versus the proposition implicit in Q, etc.
…
En
Q3
if and only if either
1a'. c represents the Ei as  and the “target” English sentence in the query as ;
1'. c believes, on the basis of a relevant invalid argument  that  can be inferred from  in S, i.e.,
Belc[ |-S ];
2. not  |-S .
or
1a. c represents the Ei as ’ and the “target” English sentence in the query as ;
1''. c believes, on the basis of a relevant valid argument  that  can be inferred from  in S, i.e.,
Belc[ |-S ];;
2. not  |-S .
or
1a. c represents the Ei as ’ and the “target” English sentence in the query as ;
1'. c believes, on the basis of a relevant invalid argument  that  can be inferred from  in S, i.e.,
Belc[ |-S ];
2. not  |-S .
This account is perhaps approaching respectability, but it’s doing only that: approaching: lots of work
remains to be done. We can perhaps assume that standard natural language processing techniques will
secure  for A from the Ei. But what about the incorrect representation of the English (denoted by ’)?
How is this representation produced, specifically? And what about the notion of a “relevant argument”?
What does this amount to, exactly? Without answers to these questions, our agent A will not be attainable.
Though we haven’t the space to provide the answers, we can say a few words, with help from our core
examples. Take Illusion 1 first. In this case, as we’ve already indicated,
 = {(K  A)  (K  A), ((K  A)  (K  A)), K}
and
’ = {(K  A)  (K  A), K}
and the “relevant valid argument” is shown in Figure 3.
Now, what about Illusion 2? Can we analyze it in terms of Def4? Yes, as follows. First, there are two
possibilities: either Johnson-Laird didn't err, and our charitable reading of him on this illusion is correct, or
he did in fact simply slip up. Either way we can make sense of Illusion 2.
Suppose first that Johnson-Laird did make a mistake in writing that Illusion 2 is an illusion despite the fact
that B is provable in elementary logic. Then (with a standard set of inference rules) what he apparently
wanted was English that when represented would yield (A  B)  (C  B) and A  C. Our hypothetical
cognizers could for example then produce the incorrect representation: (A  B)  (C  B), along with A 
C. There would then be a straightforward valid proof of B.
On the other hand, if Johnson-Laird didn't err (because he had in mind the subtle variations on the variable
S in Def4), then B can’t be obtained because constructive dilemma is unavailable. 15 But would our cognizer
have a correct or incorrect representation? She can’t have a correct representation, and at the same time
believe that she can obtain B. The reason is that this belief would be based on an argument that itself
deploys constructive dilemma. Hence, if Johnson-Laird didn’t make a mistake, cognizers must have on
hand some ’ from which you can get B in this restricted set of inference rules.
At this point, perhaps Def4 can be said to be at least somewhat promising. But surely it’s still incomplete, if
for no other reason than that it simply doesn’t present all the permutations that arise from the formal
structure we have set out. For example, it should be obvious that we would have an illusion if we simply
“flip the switches” in Def4’s first disjunction in straightforward ways. To be more specific, we would have
an illusion of the Q3 variety if subjects were tricked into thinking that some proposition  can’t be derived
from the correct representation  of the premises, when in fact  |- . This would be the relevant variation
on Def4’s disjunction:
1a'. c represents the Ei as  and the “target” English sentence in the query as ;
1'. c believes, on the basis of a relevant invalid argument  that  cannot be inferred from  in S, i.e., Belc
not[ |-S ];
2'.  |-S .
Now, what about generating illusions? If Def4 is any good, we should be able to use it to construct an
artificial agent that generates bona fide illusions before any data is gathered. This agent should yield some
rather tricky illusions, given that we have now identified six variations to instantiate (two for each of the
disjuncts in Def4), and that individual illusions can actually instantiate at least two of these variations
simultaneously. Of course, without a specification of the relationship between correct and incorrect
representations, and of justifying arguments, we concede that we can’t literally construct the agents we
seek, but we can certainly at this point “hand simulate” the processes involved. Accordingly, here’s a new
illusion Bringsjord derived from the analysis we’ve produced to this point:
Illusion 4
(10)
The following three assertions are either all true or all false:



(11)
15
If Billy helped, Doreen helped.
If Doreen helped, Frank did as well.
If Frank helped, so did Emma.
The following assertion is definitely true: Billy helped.
There are of course other ways to obtain B. For example, ignoring every fixed proof-theoretic system S,
one could produce a “semantic” proof, one based, say, on a truth table. But we are assuming in accordance
with |- that we are restricting the space of proofs in such a way that these kinds of proofs are inadmissible.
Can it be inferred from (10) and (11) that Emma helped?
(If you're interested in pondering this illusion without knowing the answer, pause before reading the rest of
this paragraph.) Sure enough, in our preliminary results, most students, and even most professors, often
including, in both groups, even those who have had a course in logic, answer with “No.” The rationale they
give is that, “Well, if all three of the if-thens are true, then you can chain through them, starting with ‘Billy
helped,’ to prove that Harry helped. But you can’t do that if the three are false, and we’re told they might
be.” But this is an illusion. In the case of reports like these, the premises are represented correctly, but there
is an invalid argument for the belief that E can’t be deduced from the conjoined negated conditionals (while
they do have in mind a valid argument from the true conditionals and Billy helped to Emma
helped). This argument is invalid because if the conditionals are negated, then we have
(B  D)
(D  F)
(F  E)
But the first two of these yields by propositional logic the contradiction D  D. Given this, if we assume
that Emma failed to help, that is, E, we can conclude by reductio ad absurdum that she did help, that is,
E.16 Notice that this solution makes use of both the rule that Rips’ says humans don’t have (a negated
conditional to the truth of its antecedent) and the rule that most psychologists of reasoning say humans
don’t have (reductio). In light of this, it’s noteworthy that a number of those to whom we presented Illusion
4 did produce the full solution we just explained.
We next draw your attention to an interesting feature of Illusion 4, and use it as a stepping stone to our
theory of human reasoning: mental metalogic.
Mental MetaLogic: A Glimpse
Illusion 4 instantiates a variation in which cognizers conceive of disproofs. More specifically, a cognizer
fooled by Illusion 4 imagines an argument for the view that there is no way to derive Emma helped from
the negated trio of conditionals and Bill helped. Such arguments cannot be expressed in F. And when
-- in keeping with the version of psychologism that is guiding us -- we look to psychology of reasoning for
help in making sense of such arguments we draw a blank. The reason for this is that psychology of
reasoning is dominated by two competing models of human reasoning (especially deductive reasoning),
and neither one allows for the expression of disproofs. The two theories are mental logic (ML) and mental
models (MM). ML is championed by Braine (1998) and Rips (1994), who hold that human reasoning is
based on syntactic rules like modus ponens (recall yet again the rule we earlier pulled from (Rips, 1994)
and displayed). MM is championed by Johnson-Laird (1983), who holds that humans reason on the basis
not of sequential inferences over sentence-like objects, but rather over possibilities or scenarios they in
some sense imagine. As many readers will know, the meta-theoretical component of modern symbolic
logic covers formal properties (e.g., soundness and completeness) that bridge the syntactic and semantic
components of logical systems. So, in selecting from either but not both of the syntactic or semantic
components of logical systems, psychologists of reasoning have, in a sense, broken the bridges built by
logicians and mathematicians. Mental logic theory and mental model theory are incompatible from the
perspective of symbolic logic: the former is explicitly and uncompromisingly syntactic, while the latter is
explicitly and uncompromisingly semantic. Our theory, mental metalogic (or just ‘MML'), bridges between
16
A fully formal proof in F is shown in Figure 4. The predicate letter H is of course used here for ‘helped.’
these two theories (see Figure 5). More concretely, MML provides a mechanism which our agent A will
need if it is to simulate human cognizers who have beliefs about provability on the basis of valid and
invalid disproofs. In this mechanism, step-by-step proofs (including, specifically, disproofs) can at once be
syntactic and semantic, because situations can enter directly into line-by-line proofs. The HYPERPROOF
system of Barwise and Etchemendy (1994) can be conveniently viewed as a simplistic instantiation of part
of MML.17 In HYPERPROOF, one can prove such things as that not  |- . Accordingly, let's suppose that in
Illusion 4, a “tricked” cognizer moves from a correct representation of the premises when (10)’s
conditionals are all true along with (11), to an incorrect representation when the conditionals in (10) are
false. Suppose, specifically, that the negated conditionals give rise to a situation, envisaged by the cognizer,
in which four people (objects represented as cubes) b, d, f, and e are present, the sentence Billy is
happy is explicitly represented by a corresponding formula, and the issue is whether it follows from this
given information that Emma is happy. This situation is shown in Figure 6. (We have moved from helping
to happiness because HYPERPROOF has ‘Happy’ as a built-in predicate.) Notice that the full logical import of
the negated conditionals is nowhere to be found in this figure. Next, given this starting situation, a disproof
in HYPERPROOF is shown in Figure 7. Notice that a new, more detailed situation has been constructed, one
which is consistent with the original given info (hence the CTA rule), and in which it is directly observed
that Emma isn’t happy. This demonstrates that Emma’s being happy can’t be deduced from the original
given information. Note that such demonstrations can be created and processed by computer programs in
general, and hence can be created and processed by our agent A.
We will stop at this point on the road toward A. Hopefully the reader now has a sense for how the kind of
psychologism we are promoting can be constructive from the standpoint logicist AI. We end with a brief
discussion of our promised final point.
Mental MetaLogic and Machine Reasoning: Future Work for
Our Brand of Psychologism
How productive can our version of psychologism be? What grander things might it enable in LAI? Well,
our hope is that it can make “Simon's Dream” a reality. What is Simon’s Dream? It is a dream that Herb
Simon, one of the visionary founders of AI (and, for that matter, of LAI, specifically) expressed in the
Summer before he died: viz., to build an AI system able to produce conjectures, theorems, and proofs of
these theorems as intricate, interesting, subtle, robust, profound, etc. as those produced by human
mathematicians and logicians. In this talk Simon pointed out something that everybody in the relevant
disciplines readily concedes: machine reasoning, when stacked against professional human reasoning, is
laughably primitive. For example, John Pollock, before delivering a recent talk at RPI on his system for
automated reasoning, OSCAR, conceded that all theorem provers, including his own, are indeed primitive
when stacked against expert human reasoners, and he expressed a hunch that one of the reasons is that such
provers don’t use diagrammatic, pictorial, or spatial reasoning: they can all be viewed as machine
incarnations of mental logic. Simon’s “condemnation” of today's machine reasoning has also been carefully
expressed in specific connection with Gödel’s incompleteness theorems: Bringsjord (1998) has explained
why the well-known and highly regarded prover OTTER, despite claims to the contrary, doesn’t really prove
Gödel's incompleteness results in the least.
We take this quote from Jacquette very seriously: “Logic can be understood as descriptive of how some
reasoning occurs, at the very least the reasoning of certain logicians.” (Jacquette 1997, p. 324) We
specifically submit that mental metalogic, used in conjunction with our Jacquette-inspired variety of
psychologism, is AI's best bet for reaching Simon's Dream. As we have said, expert reasoners often
explicitly work within a system that is purely syntactic (and hence within a system that relates only to
17
We have our own, more sophisticated instantiations of MML, but HYPERPROOF is simpler, is widely
used in teaching philosophy and computer science, and in the present context is perfectly sufficient.
mental logic) -- but it's also undeniable that such reasoners often work on the semantic side. Roger Penrose
has recently provided us with an interesting example of semantic reasoning: He gives the case of the
mathematician who is able to see via images of the sort that are shown in Figures 8 and 9 that adding
together successive hexagonal numbers, starting with 1, will always yield a cube.
MML (Rinella et al., 2001; Yang & Bringsjord, in press), as we’ve said, is a theory of reasoning that draws
from the proof theoretic side of symbolic logic, the semantic (and therefore diagrammatic) side, and the
content in between: metatheory. Furthermore, while theories of reasoning in psychology and cognitive
science have to this point been restricted to elementary reasoning (e.g., the propositional and predicate
calculi), MML includes psychological correlates to all the different forms of deductive reasoning that we
find in relevant human experts; for example, modal, temporal, deontic, conditional, reasoning. Accordingly,
MML can serve as the basis of an automated reasoning system that realizes Simon’s Dream, and we will
attempt to demonstrate this -- not now, but in tomorrow's work arising from the psychologism we have
described today.
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