Abstract: This paper clarifies the fundamental
principle of valid reasoning by dividing the principles of valid
reasoning into deductions, abductions and inductions, and providing
formalizations for each kind in terms of elementary probability theory
and elementary set theory, both of which the paper presumes. It
introduces two new probabilistic assumptions, one for abductions and
one for inductions. If accepted, these assumptions make abductions and
inductions into valid deductions depending on these assumptions in
probability theory.
Sections:
(0) Introduction
(1)
Deduction
(2) Abduction
(3) Induction
(4) Discussion of the abductive and
inductive postulates
See also: Fundamental principles of invalid
reasoning
Internet
note: The fonts used are Verdana and StarMath.
"Probability
is the guide to life".
Bishop Butler
(0)
Introduction: In this paper I shall understand
by reasoning any verbal inferencing using assumptions and
conclusions, and by valid reasoning any instance of reasoning
with a conclusion that is true if the assumptions given for it are
true.
It is true 
I shall assume  that there is more to human reasoning than verbal
inferencing using assumptions and conclusions, and it is also true that
reasoning may be very useful when it is not valid.
The reason to
restrict myself to valid reasoning as defined is that most
human reasoning can be cast in the form of verbal inferencing using
assumptions and conclusions, and that valid reasoning has the very
useful property of always leading from (presumptive) truths to
(presumptive) truths.
This property
guarantees that as long as one does use valid reasoning, one's
conclusions are true whenever one's assumptions are true, and therefore
also that one's assumptions are false if one's validly inferred
conclusions are in fact false.
Thus valid
reasoning gives us a way to find further truths from true premisses,
and to find that assumed premisses are false if they validly imply
conclusions that are false.
There are three basic
kinds of reasoning, where reasoning involves argumentation of any kind
using assumptions and inferences of conclusions:
1. Deductions:
To find conclusions that follow from given assumptions
2. Abductions: To find assumptions from which given conclusions follow
3. Inductions: To confirm or infirm assumptions by showing their conclusions do (not) conform to the observable facts.
Normally in reasoning all
three kinds are involved: We explain supposed facts by
abductions; we check the abduced assumptions by deducing the
facts they were to explain; and we test the assumptions
arrived at inductively by deducing consequences and then revising the
probabilities of the assumptions by probabilistic reasoning when these consequences are verified or falsified.
Here
are some quotations of C.S. Peirce on the subject of abduction:
"Abduction.
(..) "Hypothesis [or abduction] may be defined as an argument which
proceeds upon the assumption that a character which is known
necessarily to involve a certain number of others, may be probably
predicated of any object which has all the characteristics which this
character is known to involve." (5.276) "An abduction is [thus] a
method of forming a general prediction." (2.269) But this prediction is
always in reference to an observed fact; indeed, an abductive
conclusion "is only justified by its explaining an observed fact."
(1.89) If we enter a room containing a number of bags of beans and a
table upon which there is a handful of white beans, and if, after some
searching, we open a bag which contains white beans only, we may infer
as a probability, or fair guess, that the handful was taken from this
bag. This sort of inference is called making an hypothesis or
abduction. (J. Feibleman, "An Introduction to the Philosophy of
Charles S. Peirce", p. 1212. The numbers referred to are to
paragraphs in Peirce's "Collected Papers".)
Now
it is convenient to formalize the three types of inference I have
stated in words above in the form of three general principles of
inference. This can be done by using the theory of probability
and some basic standard set theory, with both of which I assume
familiarity in this paper. [1]
Since I assume familiarity with
elementary probability theory, it should be remarked at the start that
in this paper I have
repressed all conventional "pr(.)"notation by relying on the fact that
all probabilities can be written with a conditionalprobability sign
when absolute probabilities are taken as probabilities conditional on
the whole universe K. Thus, the probabilistic notation I use conforms
to the following:
pr(sK&T) =
sK&T and
pr(s) = pr(sK) = sK = s.
The reason to prefer the
notation of the RHS (righthand side) of the equalities over that on
their LHS is that using the format of the RHS avoids many redundant
occurences of the functional operator "pr(.)". This device makes it
possible to write elementary probabilitytheory almost exactly like
propositional logic, where the formulas of probabilitytheory are
distinguished by containing the mark of conditional probability and
by being statements of (in)equalities.
It is important to see
that the resulting probabilified version of propositional logic is
considerably more subtle and has far more possibilities for different
types of logical analysis than does standard propositional logic
without probability. [2]
This last claim is
supported by the following formalizations of the three fundamental
principles of reasoning, that are given with the help of
probabilitytheory. In each case I give a definition of the general
conditions that makes the kind of reasoning valid in elementary
probability theory and use this definition to give a rule of inference
that conforms to the definition.
I start with deduction,
because this is best known and simplest, and because I shall make
abduction and induction into deductions that use special assumptions.
Also I presume in the rest of this note the following notational
conventions:
K = knowledge assumed, T
= specific theory added to K, and s = specific statement.
K and T are supposed to be sets of statements, that can be rendered as
a conjunction, and s a statement.
The main reasons to
include K are that in fact in all reasoning (presumed) background
knowledge is used, and that indeed often the (in)validity of one's
reasoning is due to one's presumed background knowledge, and that this
is very easily accomodated by the formalism of probability theory,
namely as part of one's conditional probabilities.
(1)
Deduction: To find conclusions that follow
from given assumptions:
sK=1 is a valid deduction from T given K =def
sK&T=1 & TK=1
Thus a valid deduction
(using probability theory) is a probabilistically
certain conclusion from a (presumptively) certain theory, assuming also
background knowledge K. Of course, TK=1 may be withdrawn: All one
often needs and can get is "if the theory is true, then given
the knowledge we presume, it must also be true that ...". The general
pattern of deductions is the familiar "If soandso is true, then
suchandsuch is true, and soandso is true, therefore suchandsuch
is a valid deduction."
And a deduction is
the inference (where "A = B" = "B is a valid inference from premisses
A") that conforms to:
sK&T=1 & TK=1
= sK=1
and is used in any case
where one infers a conclusion from premisses (indeed also where in fact
sK&T=1 is false, and therefore the deduction invalid, which we
shall not consider in this paper).
In a valid deduction we
infer a fact s as conclusion from a fact T, presuming K, and the fact s
follows from K&T. The deductive reason for sK=1 is
sK&T=1 & TK=1 i.e. s is true because it follows from K&T
and K&T is true, presuming K.
There is nothing new here
(if you know some probability theory and logic) apart from the facts
that everything is stated in probabilistic terms and that the statement
is explicit about the presence of some assumed background knowledge K.
One advantage of including this reference to assumed background
knowledge in a probabilistic setting is that if one has a valid
deduction as defined, it is immediately obvious that if TK=x then
sK&T=x, and thus one has a way of dealing explicitly with
uncertainty of one's premisses. It also makes sense to point out that
the assumption [(sK&T)=1] also allows for the possibility that s
is a probabilistic statement, as in [((This die has a
probability of 1/6 to fall with 3 facing)K&(This die is fair))=1].
[3]
It also is important to
see that one has in fact very great liberties in deducing:
 one can choose whatever
assumptions one pleases and
 one can assume whatever
principle of inference one pleases
 provided only that one
retains the criterion that a deduction of a conclusion from assumptions
is valid precisely if in any case that the assumptions are
true, so is the conclusion, and chooses one's principles of inference
such that they satisfy this criterion.
An example of a
deduction is: Let K be sufficient plane geometry to include
Pythagoras' Theorem, and T the statements that X is a straightangled
triangle in which the sides joined by the straigh angle have the
lengths 3 and 4, and s be the statement that the side in X that joins
the nonstraight angles has length 5.
And though one has very
great liberties in deducing, part of the reason that this is so is that
all deduction moves in the hypothetical realm of ifthen, with no
deductve possibility of testing one's assumptions other than by validly
deriving a known falsehood from them. It is this limitation that one
seeks to escape by abduction and induction: Using only deductions and
bivalent logic, one can only deduce conclusions, and make and refute
assumptions, but one cannot infer assumptions nor support or undermine
them.
(2)
Abduction:
The definition of abduction
that I give relies on the definition of "TpsK" =def "T is positively
relevant to s given K" that follows, together with two related
definitions that will be useful below:
"TpsK" =def "T is positively relevant to s given K" =def
"sK&T > sK&~T".
"TrsK" =def "T is relevant to
s given
K"
=def "sK&T <> sK&~T"
"TisK" =def "T is irrelevant to
s given
K"
=def "sK&T = sK&~T".
Now abduction is used to find assumptions from which a given conclusion follows, and so defined as
follows using the probabilistic concept of (conditional) positive
relevance:
TK&s=x is a valid abduction
from s given K =def
TpsK & sK&T>=1/2 & sK=1
& TK=x
And so an abduction is an inference that conforms to:
sK&T>sK&~T
& sK&T≥~sK&T & sK=1 & TK=x = TK&s=x
and is generally resorted
to when one seeks to explain some puzzling fact s which one
can not (one believes) derive by deducing it from one's
presumed background knowledge. The present definition and rule of
abduction claim that this is valid if in fact the theory T is relevant
to and does support the truth of the puzzling fact s, and if in fact
one has a probability for TK that may be low. Indeed, often abductions
start as theories that would explain certain facts with a high
probability if true, while the probability of the theory is not high.
In a valid abduction we
infer a theory T as a possible explanation for a newly given fact s,
presuming K, and we also infer that T has a certain probability, namely
equal to the minimum probability of its proper consequences
presuming K. The abductive reason for TK&s=x is that
trsK & sK&TS1/2 & sK=1 & TK=x i.e. that it is
true that T is a possible explanation for s precisely if it is true
that T is positively relevant to s and that T makes s more probable
than not and that s is true, all given K, while the least probable
proper consequence of T given K has probability x. The general pattern
of abductions is: "If theory T is positively relevant to s and the
probability of s is at least 1/2 if T, and s is true and the
probability of T is x, then T is a valid abduction for s, and T has
probability x given s."
Given the definition of
valid abduction and the premisses listed, it is an easy matter to
verify that then it is deductively true, in probability theory, that T
is a possible explanation for a newly given fact s presuming K. But to
establish that TK=x we need some principle to settle the probability
of a theory, since probability theory has no axioms sufficient to
assign probabilities to theories, and we used the following:
The abductive
postulate: TK=x IFF
x=q_{i}K&T & q_{i}Є{q: K&T=q &
~(K&~T=q) & (s)(K&T=s & ~(K&~T=s) >
sK&T≥qK&T)}
This principle says that the
probability of a theory T on presumed knowledge K equals x
precisely if x is the minimum of T's proper
consequences given K, where q is a proper consequence of
K&T if it follows deductively from K&T but not from K&~T.
(One needs proper consequences to avoid problems with just any
statement with a low probability that happens to be true, for such a
statement is a logical consequence of any theory.) It is a postulate
because while probability theory implies that in the stated conditions
TKRx it does not imply that TK=x. [4]
What is new here apart
from stating everything probabilistically is that in fact two
statements are inferred, of which one is deduced and one assumed. First
it is deduced that it is true that T is a possible explanation of s,
for which it is sufficient by the given definition that
sK&T>sK&~T & sK&T≥~sK&T & sK=1. Next an
assumption is made to settle the probability of a theory, namely that
TK equals the minimum probability of its proper consequences,
all presuming K. This is motivated by the facts that this minimum
probability is the upper boundary of TK on probability theory; that
probability theory by itself provides no consequences other than this
to settle the probability of a theory; and that we need to assign a
probability to theories to reason probabilistically with them. [5]
It should be noted that
TK&s may be low, and that it still it may be a valid abduction,
for this depends not on how low it is, but on T being positively
relevant to s given K and making s given K at least 1/2. The
possibility of 1/2 is included here to take care of whatever has that
probability 1/2. Thus, that this coin fell heads half of the
times it was thrown is explained by the valid abduction that this coin
is fair (i.e. unloaded and with a heads and a tails side). But often
abductive inference is used to find a theory to account for a
surprising fact.
Also,
if TK&s=x is small this suggests to do one (or more) of three
things:
(1) Find another valid
abduction for s with a higher probability or
(2) deduce a consequence of T and test T, in the hope to inductively
confirm T and increase the probability of T or
(3) revise K so that the minimal proper consequence of T gets higher.
An example of an
abduction is: Let K be elementary physics; let s the statement that
a star S_{1} is not precisely where it should be if elementary
physics is true; and let T be the theory that there is a hitherto
unknown interstaller object S_{2} that effects where S_{1}
precisely is.
The basic weaknesses of
abductions are that (1) the only reason to infer their theoretical
conclusions is that they are relevant to some puzzling facts and that
(2) the theoretical conclusions inferred by abduction often have a low
probability. This last weakness is addressed by induction.
(3) Induction:
To confirm (or infirm)
assumptions by showing their conclusions do (not) conform to the
observable facts.
TK&s=TK*[sK&T:sK] is a valid
induction
from s given K =def
sK=1 & qK=1 & sK&q=sK &
(q)(s)(TrsK > sK&q=sK IFF sK&T&q=sK&T)
And the induction is
the inference
sK=1 & qK=1 &
(q)(s)(TrsK > sK&q=sK IFF sK&T&q=sK&T) =
TK&s=TK*[sK&T:sK]
although it may be clearer
to write it like so, explicitly listing the probabilities that must be
presumed:
sK=1 & sK&T=s_{1}
& sK=s_{2} & TK=t_{1} & qK=1 &
sK&q=sK &
(q)(s)(TrqK > sK&q=sK IFF sK&T&q=sK&T) =
TK&s=t_{1}*s_{1}:s_{2}
In a valid induction we
infer that the probability of T given K&s equals
TK*[sK&T:sK] from a true fact s that is relevant to T, while T
satisfies the inductive postulate that any fact that is in
fact irrelevant to anything T is relevant to remains irrelevant if T is
true, all given K. [6]
What is new here is
The inductive
postulate: (q)(s)(TrsK > (qK&s=qK IFF
qK&T&s=qK&s))
This principle enables one
to abstract from any fact that is not relevant when testing a theory.
The way this works can be easily shown when we abstract for a moment
from K. Then the principle turns into: [Trs > (qs=q IFF
qT&s=qs)] and allows us to abstract from any q not relevant to
any s that T is relevant to and that happens to be the case when T is
tested, for now one can reason as follows: Ts&q =
qT&s*T&s : s&q = qs*T&s : qs*s = T&s:s =
Ts. Each step is deductively valid in probability theory, but the
second step involves the inductive postulate to abstract from q. [7]
So in induction too there
is an assumption involved, namely that the theory one uses has the
property that any fact that is irrevelant in fact to any of the
predictions of the theory is irrelevant also if the theory is true and
conversely, all given K. This can be seen equivalently as the claim
that the theory T is relevant in theory to everything it is
relevant to in fact and nothing else. Indeed, the equivalence
in the inductive postulate (qK&s=qK IFF
qK&s&T=qK&s) can be conveniently read from left to right
as: what is irrelevant in fact, also is irrelevant according to theory
T, and from right to left as: what is irrelevant according to theory T
is irrelevant in fact, all presuming K (i.e. sofar as we know).
It should be noted that
TK&s may be any probability, including what T was before s came to
be known, namely TK, if T is irrelevant to s given K. What matters for
an inductive inference to be valid is the assumption that the
inductive assumption is satisfied by T, for it is this that allows one
to make an inductive inference about T given a new fact s.
An example of an
induction is: Let K be standard medical science and s be the
statement that test S is positive and T be the theory that you have
cancer, and suppose sK&T=s_{1}=0.9 and sK&~T=s_{2}=0.3
and TK=t_{1}=0.01 then TK&s=t_{1}*s_{1}:s_{2}=
0.03. If the test is positive, your chance of having cancer is trebled,
but if it was 1/100 to start with, then a positive result makes this
3/100. The
inductive postulate enters because to credit the reasoning one
must in fact assume that the test was properly done and that all manner
of circumstances that occurred when it was done either are irrelevant
to the theory or are accounted for by it. ("No madam, the lab assistant
was not drunk, and no, it does not matter the test was done on a
Tuesday in a leap year and no, your meditation exercises have nothing
to do with outcome of the test." Etc.)
(4) Discussion
of the abductive and inductive postulates:
The explanations I have
given of abductive and inductive reasoning makes these valid deductions
inside elementary probability theory, given certain postulates and
definitions, that I also have given. In this section I want to consider
briefly the postulates I made, and make a few remarks about the
interesting epistemological status they share, namely of being practically
necessary and corrigible.
A. The abductive
postulate: The basic reasons to assume the abductive postulate are
that one needs to arrive somehow at probabilities for theories,
and that one has by probabilitytheory that TKRx,
which the postulate strengthens to an identity provided
x=q_{i}K&T & q_{i}Є{q: K&T=q &
~(K&~T=q) & (s)(K&T=s & ~(K&~T=s) >
sK&T≥qK&T)}. For since whatever theories may denote
(presumably: the infinite set of logical consequences of the
assumptions of the theory), these cannot be counted in the same sense
as e.g. occurences of "heads up" in sequences of throwns with a coin
can be counted, and so there is no evident way to assign a
probability to a theory.
The abductive postulate
amounts to the assumptions that, first there is such a thing as
the probability of a theory [8], and second
that it may be initially and conveniently settled by supposing this
probability equals the maximum of what it may be given in
probability theory and such knowledge as one presumes, while one also
fully expects that this initial probability will be adjusted by further
reasoning and to be discovered new facts.
So the abductive postulate
seems not so much a truth about nature as a truth about the
ways and procedures human beings may use to discover the truth about
nature. And indeed the abductive postulate seems safe and warranted
in the sense that any probability it introduces can be  and usually
will be  rationally corrected and adjusted, and that indeed it may be
increased or decreased by inductions, and also by various
other means indicated above.
An abductive
postulate is needed because we need to have some factually based
probability for theories that we decide are good explanations, if only
to have a start to test them inductively.
B. The inductive
postulate: The basic reason to assume the inductive postulate is
that one needs some assumption to deal with the very many facts that
are true besides the theory one is interested in testing, since each of
these very many facts may be relevant to the truth of the fact
one is interested in, and so that it seems a good demand to make of a
theory to be true that it should truly and fully entail all it
is relevant to.
Also, in fact this
postulate seems to be necessarily true if human beings can come
to know nature by testing such theories as they have, for all such
tests must include knowledge of what is relevant to what is tested and
in what degree it is relevant and also of what is irrelevant to it, for
relevancies and irrelevancies are facts that are as real as the facts
they concern. In brief: one just cannot rely on any
experimental evidence if one cannot rely on one's abstraction from much
of the surrounding factual details as irrelevant, which is necessary in
any experiment. [9]
On the other hand, one
cannot normally prove in complete or even considerable detail that any
given theory that is to be tested in fact does correctly entail all
that is relevant to it and does not entail as relevant anything that is
in fact irrelevant. (Indeed, normally only a few known relevant factors
are listed in any report of a scientific experiment together with an
indication how these have been dealt with in the experimental setup.
Yet any design of experiments must involve assumptions about factors
that are relevant and that are irrelevant to what is to be tested.)
But since true theories
must properly entail the true degrees of relevancies of their
predictions, all one can do is to assume that one's theories
do so, and to take care of all relevancies one does know.
So the inductive postulate
seems not so much a truth about nature as a truth about
the ways and procedures human beings use to discover the truth about
nature, and one which is true to the extent human beings have true
theories about nature, for true theories must satisfy the inductive
postulate, even if no human being is able to survey all of the universe
and establish all its presumed relevancies and irrelevancies are
factually correct. And indeed the inductive postulate seems safe and
warranted in the sense that any probabilities it introduces can be 
and usually will be  rationally corrected and adjusted by later
evidence. Also it suggests a reason for experiments that fail or turn
out unexpected results: One may have disregarded as irrelevant some
factor that is relevant i.e. one may have falsely assumed that one's
theory T satisfied the inductive postulate. Finally, the inductive
postulate is needed because in any experimental test of a theory we
need to abstract from very many accompanying circumstances.
Summing up: It
seems to me that both the abductive and inductive postulate are used
tacitly in very much of human reasoning; it seems to me that both  or
at least: assumptions much like them  are required by human beings if
they want to arrive by reasoning at truths about nature; and it seems
to me that both postulates have the interesting property that one must
assume (something like) them in order to learn anything at all
about the natural facts, while one is able to correct such inaccuracies
as they may introduce, though this correction will take time and
trouble, as does any scientific advance.
And the reason one must assume
these principles is that in order to establish that
one's theories are about nature (and are not merely fantasy), one must test
one's theories, and to test them one needs both
probabilities for theories and make as sure as one can that all that is
relevant to what one tests is known, so that what one abstracts from as
irrelevant to what one tests in theory indeed is irrelevant
in fact.
Therefore it seems
sensible to list these assumptions of scientific procedure explicitly,
and to use them consciously wherever applicable. And finally it seems
an interesting fact that there are, then, corrigible presumptive truths
of procedure, that one needs to make in order to validly infer
intermediate conclusions that are required on the way towards new
natural knowledge.
Maarten Maartensz
Literature:
Ernest Adams: A Primer
of Probability Theory
Arthur Burks: Chance, Cause and Reason
Paul Halmos: Naive Set Theory, and Measure Theory.
Charles Peirce: Collected Papers
G. Polya: Principles of Plausible Reasoning (2 volumes)
W.G. Wood & D.H. Martin: Experimental Method
Notes:
[1]: In
fact I assume no more than standard elementary probability theory
(summarizable as: What follows from Kolmogorov's axioms without any
infinitary assumptions, or alternatively but equivalently: measure
theory without infinitary assumptions) and standard elementary set
theory. Good introductions to the former are by Adams and Burks, and a
good introduction to set theory and to measure theory are by Halmos.
Wood & Martin is a useful summary of principles involved in
scientific experimental designs.
[2]:
This is well explained by Adams and by Polya.
[3]: So
while there is no need to formalize deductions inside probability
theory it may be helpful to do so, especially when one wants to deal
with uncertainties. Also, it is well to stress that the notation I use
(and explained in the beginning) is useful,
and more elegant and easier to read and use that the ordinary format
using "pr(.)".
[4]: For
in probability theory we have for any theory T and any statement s
logically implied by T that the probability of T  assuming, as we do,
that it exists and is consistent with probability theory  cannot be
larger than the probability of s. In more or less standard notation: T
= s > pr(T)Rpr(s).
[5]:
These considerations do not logically imply the abductive postulate,
and more can be said about the reasons to assume it, some of which is
said in section (4).
[6]:
That TK&s=TK*[sK&T:sK] in fact does follow by standard
elementary probability theory and is well explained by Burks, with much
more detail than is dealt with in the present paper. In any case, the
basis of it all is the elementary theorem of probability theory to the
effect that (abstracting from K for the moment) Ts > T IFF sT >
s IFF T is positively relevant to s, while it also is a theorem that
Ts = sT.T:s. So probability theory itself easily and elegantly
enables inductive confirmation of a theory by its verified predictions,
and falsification of a theory by its falsified predictation. But 
though very interesting in principle  this is all presumed as known in
the present paper. (See e.g. Burks and Polya.)
The inductive postulate is
needed in any case s here is an empirical fact, that could be used as
confirmation or infirmation, since such a fact s always wil be
simultaneously true with very many
other empirical facts, some of which may be relevant to it, and
others which may not be relevant to it.
[7]:
There is a lot more that can and should be said about the inductive
postulate and its relations to the problems of induction as raised by
Hume and Goodman, but the present paper is not the place for it, for it
is not dedicated to the problems of induction, but to a clear and
formal statement of the fundamental principles of valid reasoning.
[8]: It
may be well to remark here that I do not presume the subjective or
personal approach to probabilities, which makes
probabilities depend on human betting quotients. These are arbitrary,
and what I want, rather, is a probability based on the facts,
in so far as these are (presumptively) known. These probabilities are
not available for theories directly, since one cannot count the cases
in which a theory is true and count the cases in which a theory is not
true, and therefore I assume a theory has the probability of its least
probable proper consequence, where this probability can be established
experimentally by counting the cases pro and con.
[9]: In
fact this is one side of the problem of induction that has been missed
by many. One of the many was Karl Popper, whose whole philosophy of
science founders on the fact that he did not see that to use
experimental evidence requires an assumption about what is and
is not relevant to it. Any experiment involves the assumption
to the effect that whatever is happening at the time that is not
considered by the theory that is tested is in fact irrelevant to
whatever the theory implies. Without this assumption anything
whatsoever that is happening may be quoted as "reason" that the
experiment has the outcome it does, whatever the outcome is. And the
assumption must be explicitly made because it may be mistaken and is
involved in all testing of theories by their empirically checkable
implications: Some of the factors that are not assumed to be relevant
in fact may be relevant.
