Natural
Logic: A collection of
terms and
rules that come with Natural Language that allows us to
reason
and argue in it.
1. Introduction
2. Rules of Natural Logic
3. Brief assessment of the Rules of Natural Logic
4. A realist context for Natural Logic
1.
Introduction
In any Natural Language there are the elements of what may be called
its Natural Logic:
Examples of
such logical terms are: "and", "or",
"not", "true", "false", "if",
"therefore", "every", "some",
"necessary", "possible", "therefore", "is the same as",
"any (arbitrary)" and "one (specific)", and
quite a few more. Examples of such
logical rules, that are here formulated in terms of what one may write
down on the strength of what one already has written down (pretending
for the moment that natural language is written rather than spoken)
are: "If one has written down that if one statement is true then
another statement is true, and if one has written down that the one
statement is true, then one may write down (in conclusion) that the
other statement is true" (thus: "if it rains then it gets
wet and it rains, therefore it gets wet") and "If one has
written down that every soandso is suchandsuch, and this is a
soandso, then one may write down that this is a suchandsuch"
(thus: "if every Greek is human and Socrates is a Greek,
therefore Socrates is human").
We presuppose
Natural Logic in much the same way as we presuppose Natural Language:
as something we have to start with and precisify later, and that may
well come to be revised or extended quite seriously, but also as
something that at least seems to be in part given in more or less the
same way to any able speaker of a Natural Language: In it there are a
considerable number of terms and  usually implicit 
rules which
enable every speaker of the language to argue and reason, that every
speaker knows and has extensive experience with.
Again, it
does not follow that these rules and terms are clear or sacrosanct.
All that I assume is that they come with Natural Language and are to
some extent articulated in Natural Language and understood and
presupposed by everyone who uses Natural Language.
Three very
fundamental assumptions about the making of assumptions that come with
Natural Logic are as follows  where it should be noted I am not
stating these assumptions with more precision than may be supposed
here and now:
1. Nothing can be argued without
the making of assumptions.
2. An assumption is a statement
that is supposed to be true.
3. Human beings are free to
assume whatever they please.
These I
suppose to be true statements about arguments and people arguing,
where it should be noted that especially the third assumption,
factually correct though it seems to be, has been widely denied in
human history for political, religious, philosophical or
ideological reasons: In
most places, at most times, people have not been allowed to speak
publicly about all assumptions they can make.
Four other assumptions about
argumentation that should be mentioned here are:
1. Conclusions are statements that are inferred in arguments from earlier
assumptions and conclusions by means of assumptions called
rules of inference, that state which kinds of statements may
be concluded from the assumption of which kinds of statements
2. Definitions of terms are assumptions to the effect that a certain term may
be substituted by a certain other term in a certain kind of
arguments
3. Rational
argumentation about a topic starts with explicating rules of
inference, assumptions and definitions of terms, and proceeds
with the adding of conclusions only if these do follow by some
assumed rule of inference.
4. A statement is true precisely if what it says is in
fact
the case.
The first two
assumptions need more clarification than will be given here and now,
but, on the other hand, again every speaker of a Natural Language will
have some understanding of setting up arguments in terms of
assumptions, definitions and rules of inference, and drawing
conclusions from these assumptions and definitions by means of these
rules of inference.
The third
assumption, when compared with the normal practice of people arguing,
entails that mostly people do not argue very rationally, at least in
the sense that all too often they rely in their arguments on rules of
inference, assumptions or definitions they have not explicitly assumed
yet have used in the course of the argument. (Often such assumptions
are made because of wishful
thinking.)
The fourth
assumption is in fact a definition of the term "true" that expresses
an idea that is older than Aristotle, who seems to have been the
first to formulate it clearly and stress its central importance. It needs also more explanation than
will be given here and now, but it seems to clearly express the
meaning of "true" people use when they discuss ideas about reality
that are personally important to them.
2. Rules of Natural Logic
There are the following Rules of Natural Logic one may propose
for the logical connectives I mentioned above:
: "and", "or", "not", "true", "false",
"if", "therefore", "every", "some", "necessary", "possible",
"therefore", "is the same as", "any (arbitrary)" and "one (specific)".
To formulate them, I in fact extend English with variables:
Terms that are not words of the language, but which may be substituted
by any arbitrary term of a certain kind of the language, in certain
conditions.
Also, the spirit in which I propose and state these rules is both
tentative and confident  tentative, in the sense that I believe
many speakers of English, when thinking about it, will agree they often
reason and argue according to such rules for the logical connectives as
follow, if possibly not quite the same, while I know this is an
empirical generalization; and confident in the sense that all of
the rules I propose have more precisely formulated and conditioned
counterparts in formal logic, where these
counterparts are provably valid in the
formal semantics for these logical
rules.
In what follows, capital letters X, Y and Z are variables for English
statements, and undercast letters x, y and z are variables for English
terms. A formula in what follows is an English statement in which one or
more terms have been replaced by variables. For clarity's sake I
surround the formulas that are claimed to be true or not true by dots,
which may be taken for parentheses  which I avoid to make things look
less hairy.
General rule of evaluation
It is true that if .X. is any English statement, then .X. is true or
.X. is not true.
This is a somewhat precise form of the notion that English statements
are true or not true, that shows how "true" is used. See: Bivalence.
Rules for not:
not1:
If .not X. is true, then .X. is not true.
not2:
If .not X. is not true, then .X. is true.
not3: If .X. is not true, the .not X. is true.
not4: If .X. is true, then .not X. is not true.
The formula .not X. corresponds to English expressions like "It is
not so that X" that may have quite a few forms in English that also have
some ambiguities that are not further considered here. See: Negation.
Rules for and:
and1:
If .X and Y. is true, then .X. is true.
and2:
If .X and Y. is true, then .Y. is true.
and3:
If .X and Y. is not true, then .X. is not true or .Y. is not true.
and4: If .X. is not true or .Y. is not true, then .X and Y. is not
true.
It is not intended to state a minimal number of rules, though it is
intended that all rules that are stated are both intuitively
valid and formally valid when made more
precise in formal logic. See: Conjunction.
Rules for or:
or1: If .X. is true and .Y. is any English statement, then .X or Y.
is true.
or2: If .Y. is true and .X. is any English statement, then .X or Y. is
true.
or3:
If .X or Y. is not true, then .X. is not true and .Y. is not true.
or4: If .X. is not true and .Y. is not true, then .X or Y. is not true.
Note that here and in the rules for and the term "true"
distributes nicely over its components.
Also, it is noteworthy that the following rule
orD:
If .X or Y. is true, then .X. is true or .Y. is true.
is not valid probabilistically, though it is valid in standard
binary logic. A counterexample is e.g. if .Y. = .not X. and 0<pr(X)<1.
See: Disjunction.
Rules for ifthen or implies:
implies1: If .X implies Y. is true, then .not X. is true or .Y. is true.
implies2: If .not X. is true or .Y. is true, then .X implies Y. is true.
implies3: If .X implies Y. is not true, then .X. is true and .Y. is not true.
implies4: If .X. is true and .Y. is not true, then .X implies Y. is not true.
The reason to write "implies" is to have a oneword counterpart for "if
.. then ". Some readers may have some intuitive difficulties with the
proposed rules for implies, and they are right in the sense that these exist.
Even so, under the assumption that all statements are true or not true, and one
states no further conditions, the rules stated for implies preserve most
intuitions and validate most arguments involving them one considers intuitively
valid. See: Paradoxes of implication, Implies.
Rules for if and only if or iff:
iff1: If .X iff Y. is true, then .X implies Y. is true and .Y implies X. is
true.
iff2: If .X implies Y. is true and .Y implies X. is true, then .X iff Y. is
true.
iff3: If .X iff Y. is not true, then .X and not Y. is true or .not X and Y. is
true.
iff4: If .X and not Y. is true or .not X and Y. is true, then .X iff Y. is not
true.
The reason to write "iff" is to have a oneword counterpart for "if
and only if". According to the rules proposes, statements involving "iff"
can be equivalently replaced by statements involving "and" and "implies". See:
Equivalence.
Rules for not and predicates:
not F1: If .x is F. is not true, then .x is not F. is true.
not F2: If .x is not F. is true, then .x is F. is not true.
not F3: If .x is not not F. is true, then .x is F. is true.
not F4: If .x is not not F. is not true, then .x is F. is not
true.
This concerns new notation: ".x is F." is a formula for any English
statement in which occurs "x" while the rest of the statement  all of
it except all occurences of "x" in it  are referred to by "F", which is
termed a predicate while "x" is termed
a subject. The "is" locution is meant as
a help for intuition and provides one instance of the type of statements
covered.
Rules for is the same as or =:
Id1: It is true that .x = x. is true.
Id2: If .x = y. is true, then .y = x. is true.
Id3: If .x = y. is true and .y = z. is true, then .x = z. is true.
Id4: If .x = y. is true and .x is a F. is true, .y is a F. is true.
The reason to write "=" is to have a oneword counterpart for
"is the same as" or "is identical to", which also has the
merit of being widely known from arithmetic and algebra.
Also, there is new notation: ".x is a F." is a formula for any
English statement in which occurs "x" while the rest of the statement 
all of it except all occurences of "x" in it  are referred to by "F".
The "is a" locution is meant as a help for intuition and provides one
instance of the type of statements covered.
The rule Id4 in which it is used formulates a form of the rule of
substitution of identities, which in general terms must have some
restrictions, that are not further considered in the present context.
See: Identity.
Rules for some or there is:
some1: If .some x is F. is true, then .one x is F. is true.
some2: If .one x is F. is true, then .some x is F. is true. (*)
The expressions "some" and "there is" are called existential quantifiers.
In standard logic, they are taken to be synonymous, and here they are given with
the help of the idea of "one" or "one (specific)", the force of which is that if
 say  some persons are English, then one specific person must make the claim
true.
(*) It should be mentioned that in formal logic there are some conditions here
to be added, usually to the effect that the term for the one specific soandso
one introduces is new to the argument (for else one might make claims that
combined with earlier claims are false). See: Existential Generalization.
Rules for every or all:
every1: If .every x is a F. is true, then .any x is a F. is true.
every2: If .any x is a F. is true, then every .x is a F. is true. (*)
The expressions "every" and "all" are called universal quantifiers. In
standard logic, they are taken to be synonymous, and here they are given with
the help of the idea of "any" or "any (arbitrary)", the force of which is that
if  say  any person is English, then any arbitrary person must make the claim
true.
(*) For the rule named every2 a similar restriction applies as to the rule
some2.
See: Universal Generalization.
Rules for any (arbitrary):
any1: If .any x is P. is true, then if .y is a constant. is true, then .y
is P. is true.
any2: If .any x is P. is not true, then .one x is not P. is true. (*)
These rules concern in fact the usage of variables and constants, and seem to
be a little more general or basic, intuitively, than the quantifier rules, in
that they also apply to context in which occur variables without quantifiers.
The any1 rule makes this explicit, and the any2 rule has again a condition
that relates to earlier occurences of the term. See: Variables.
Rules for one (specific):
one1: If .one x is P. is true, then .any x is not P. is not true.
one2: If .one x is P. is not true, then .one x is not P. is true.
These rules supplement those for any, which involve them via the any2 rule.
See: Variables.
Rules of inference:
ergo1: If .X. is true and .X implies Y. is true, then one may write .Y. is
true.
ergo2: If one may write .Y. is true, then there is a .X. such that .X. is true
and .X implies Y. is true.
These rules concern the act of inference, which is here taken to be the form
of writing down a statement. The general idea of ergo1 is that ergo
conforms to implies with the added permission that one may write down
what is implied by an implication if the antecedent of the implication is true.
The general idea of ergo2 is that anything one may write down (in this fashion,
according to logical rules) must be such as to be provable from some true
hypothesis. See abduction, and note that this
does not mean one needs to shift back to ever further assumptions, since if .Y.
is an initial assumption one may use the provable formula .Y implies Y. to
conform to ergo2. See: Inference.
Rules for necessary:
nec1: If .X is necessary. is true, then .X. is true.
nec2: If .X is neccesary. is true, then .X is possible. is true.
nec3: If .X is necessary. is not true, then .not X is possible. is true.
One often claims soandso is necessary and suchandsuch is possible. These
are much like quantifiers, but with some extra twists that are not discussed
here. See: Modal Logic.
Rules for possible:
pos1: If .X. is true, then .X is possible. is true.
pos2: If .X is possible. is true, then .not X is necessary. is not true.
pos3: If .X is possible. is not true, then .not X is necessary. is true.
These supplement the rules for necessary, which involve them via nec2 and
nec3. In standard modal logic either is definable in terms of the other, in
terms of theses that may here be summarized as: "It is necessary that X" =def
"It is not possible that not X" and "It is possible that X" =def "It is not
necessary that not X". See: Modal Logic.
3. Brief assessment
of the Rules of Natural Logic
As remarked, the rules are meant to schematize rules of reasoning with
logical connectives that for everyone who speaks English must seem familiar and
sensible, if perhaps not fully precise or adequate for one's personal use in
some arguments.
Furthermore, they are informed by formal logic: In fact, such rules as I
proposed cover the fundaments of Propostional Logic, Quantified
Predicate Logic, the theory of identity, and Modal Logic. And
if the reader is interested, there is much more to find out about these topics,
in more precise forms and terms, in mathematical logic and
philosophical logic.
Also, anyone is free to set up one's own set of rules for logical connectives
or indeed any term in English. Two reasons to try to do so are that it may
clarify one's intuitions and help to explicate and abide by rules one in fact
uses.
There are also a number of things left out, in particular rules for set
theory, and rules for propositional attitudes. The intuitive basis of the rules
for set theory in English are English arguments with nounlike expressions and
with common and proper names. The intuitive basis of the rules for propositional
attitudes in English are English arguments with attitudinal terms like
"believes", "desires", "knows" and many more.
4. A realist context
for natural logic
It
seems that most users of most natural languages presuppose a
metaphysics I shall call Natural Realism.
This also provides the context for the above Rules of Natural Logic, or
indeed for other or more such rules, though it is not really necessary,
since on may differ about metaphysics while agreeing about the rules of
logic and language that one uses to argue about metaphysics.
However, one of the things which
makes it intuitively easier to make
sense of assignments of the term "is true" to formulas is the assumption
of some reality in which are the things
(and possible properties, qualities, relations, structures, numbers,
fields, situations ...) that make one's formulas
true if they exist and false if they
don't.
Logically speaking though, even this
may be replaced by a mere hypothesis, inroduced to clarify one's terms
and statements in logic: One speaks as if they concern some
Universe of
Discourse, that if made an explicit assumption may contain or lack
anything one pleases, at one's own discretion, limited only by the
demands that the rules be precise enough to unambiguously assign true or
not true to formulas, that are either provably
consistent, also with respect to other
rules, or at least not known to be inconsistent.
