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:

• a collection of terms and rules that come with Natural Language that allows us to reason and argue in it.

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 so-and-so is such-and-such, and this is a so-and-so, then one may write down that this is a such-and-such" (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: Bi-valence.

Rules for not:

not-1: If .not X. is true, then .X. is not true.
not-2: If .not X. is not true, then .X. is true.
not-3: If .X. is not true, the .not X. is true.
not-4: 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:

and-1: If .X and Y. is true, then .X. is true.
and-2: If .X and Y. is true, then .Y. is true.
and-3: If .X and Y. is not true, then .X. is not true or .Y. is not true.
and-4: 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:

or-1: If .X. is true and .Y. is any English statement, then .X or Y. is true.
or-2: If .Y. is true and .X. is any English statement, then .X or Y. is true.
or-3: If .X or Y. is not true, then .X. is not true and .Y. is not true.
or-4: 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

       or-D: 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 counter-example is e.g. if .Y. = .not X. and 0<pr(X)<1. See: Disjunction.

Rules for if-then or implies:

implies-1: If .X implies Y. is true, then .not X. is true or .Y. is true.
implies-2: If .not X. is true or .Y. is true, then .X implies Y. is true.
implies-3: If .X implies Y. is not true, then .X. is true and .Y. is not true.
implies-4: 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 one-word 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:

iff-1: If .X iff Y. is true, then .X implies Y. is true and .Y implies X. is true.
iff-2: If .X implies Y. is true and .Y implies X. is true, then .X iff Y. is true.
iff-3: If .X iff Y. is not true, then .X and not Y. is true or .not X and Y. is true.
iff-4: 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 one-word 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 F-1: If .x is F. is not true, then .x is not F. is true.
not F-2: If .x is not F. is true, then .x is F. is not true.
not F-3: If .x is not not F. is true, then .x is F. is true.
not F-4: 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 =:

Id-1: It is true that .x = x. is true.
Id-2: If .x = y. is true, then .y = x. is true.
Id-3: If .x = y. is true and .y = z. is true, then .x = z. is true.
Id-4: 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 one-word 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 Id-4 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:

some-1: If .some x is F. is true, then .one x is F. is true.
some-2: 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 so-and-so 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:

every-1: If .every x is a F. is true, then .any x is a F. is true.
every-2: 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 every-2 a similar restriction applies as to the rule some-2.
See: Universal Generalization.

Rules for any (arbitrary):

any-1: If .any x is P. is true, then if .y is a constant. is true, then .y  is P. is true.
any-2: 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 any-1 rule makes this explicit, and the any-2 rule has again a condition that relates to earlier occurences of the term. See: Variables.

Rules for one (specific):

one-1: If .one x is P. is true, then .any x is not P. is not true.
one-2: 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 any-2 rule. See: Variables.

Rules of inference:

ergo-1: If .X. is true and .X implies Y. is true, then one may write .Y. is true.
ergo-2: 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 ergo-1 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 ergo-2 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 ergo-2. See: Inference.

Rules for necessary:

nec-1: If .X is necessary. is true, then .X. is true.
nec-2: If .X is neccesary. is true, then .X is possible. is true.
nec-3: If .X is necessary. is not true, then .not X is possible. is true.

One often claims so-and-so is necessary and such-and-such is possible. These are much like quantifiers, but with some extra twists that are not discussed here. See: Modal Logic.

Rules for possible:

pos-1: If .X. is true, then .X is possible. is true.
pos-2: If .X is possible. is true, then .not X is necessary. is not true.
pos-3: If .X is possible. is not true, then .not X is necessary. is true.

These supplement the rules for necessary, which involve them via nec-2 and nec-3. 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 noun-like 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.