> ,.)*+y mbjbj .{{u ?
zpP:::NNN8NJnn@9I;I;I;I;I;I;I$4NP_I:+++_I::LJDDD+x::9ID+9IDDD`2?#W/D%IbJ0JDQ?AJQDQ:DT"D`%<'{_I_ICHJ++++Q
@: 9*20*2012 How we naturally reason
fred sommers
i.
I recently asked my 8 year old grand-daughter, Eliza, whether she knew that all ponies are horses. Of course, she said.
What if I told you I know a man who doesnt own a horse but does own a real live pony? She looked quizzically at me and decided not to take me seriously. No you dont! she said. Elizas reasoning, (E) (i) All ponies are horses, so (ii) All owners of ponies are owners of horses is an example of deductive inference that children as well as adults commonly make.
I am now retired. When I taught college logic in the Sixties, I would show the class that (ii) Every owner of a pony is an owner of a horse follows necessarily from (i) Every pony is a horse.
Using the symbolic mathematical language of modern predicate logic [MPL], I would translate the premise, (i) Every pony is a horse as (1) For every thing, x, if x is a pony then x is a horse, I would translate the conclusion, (ii) Every owner of a pony is an owner of a horse as (2) For every thing x, for every thing y, if y is a pony and
x owns y then there is a thing, z, such that z is a horse and x owns z.
By applying rules of inference, I had earlier taught the class, I would proceed stepwise to carefully derive (2) from (1). Not counting the time taken for translating into the quantifier/variable notation, the MPL proof that consisted of a chain of reasoning beginning with (1) and and ending with (2), took about seven minutes of class time. Thats extremely slow compared to Eliza who moved from every pony is a horse to so every owner of a pony is an owner of a horse almost instantaneously. Of course Eliza reasons intuitively, and when I taught logic in college in the Sixties, I used MPL to show what a formal derivation of (ii) from (i) looks like.
It occurred to me even then that children must also be using some methodical way to derive (ii) from (i). A popular Speaker of the U.S. House of Representatives famously said All Politics is local. It is even more true that all Reasoning is formal. Though she is altogether unaware of how she reasons, Eliza reasons formally. Her formal way of getting from (i) to (ii) couldnt be anything like the formal process teachers of logic use to get from (i) to (ii). For Eliza knows nothing of quantifiers or bound variables and she infers (ii) from (i) in less than a second, which cant be done by reasoning in the manner of MPL. When we reason intuitively theres time for no more than one link in the chain of reasoning that derives (ii) from (i). Since (E) is a valid inference, we have to assume that people who derive (ii) from (i) in a split second, are unconsciously using some logically legitimate method in making that inference. What method could it be that enables us to derive (ii) from (i) intuitively?
Logicians dont know how children reason and Eliza herself cannot tell us anything about how she reasons. In the Meno, Plato famously asked how people, most of whom have never been taught how to reason, nevertheless reason correctly--- intuitively. One can always depend on Plato, to ask intriguing questions. Unfortunately, he begs this one by arguing that our ability to reason intuitively is evidence that we must all have been taught how to reason after all --- in an earlier incarnation before we were born.
1.1 Back in the Sixties, I spent some weeks looking for a more acceptable answer. I felt I could learn something very valuable if I could somehow figure out how people intuitively reason. For the intuitive method that one uses to get from (i) to (ii) is not something recently invented by mathematically talented 19th Century logicians; it is a method we all already possess as evolved, rational human beings. Elizas method of reasoning naturally is not generally known but it is clearly more expeditious than the slow pokey predicate logic method I used to inflict on my logic classes. That methodthe method of MPL has only recently been discovered. By contrast, Elizas intuitive and very rapid method of reasoning couldnt involve the use of a symbolic technical language that takes an average college student about week to learn.
Piqued by the problem Plato raised in the Meno, I examined many ordinary language arguments and eventually hit on the method we unconsciously use when reasoning intuitively.
Natural language, the language in which human beings normally reason, is variable-free. People dont reason in the canonical quantifier-variable notation of predicate logic taught in the universities. When I presented the pony-owner/horse-owner inference to my logic class, it was immediately obvious to all the students that (i) entails (ii). Like Eliza, they derived (ii) from (i) instantly. Nevertheless, they sat patiently sat for the seven minutes of a formal proof that (ii) follows logically from (i). They were in fact grateful to me for providing an explicit chain of technical unfamiliar argumentation connecting (i) to (ii) that officially proved that their intuitive rush to judgment, was correct. They realized that Its obvious! is not a proof; they knew that they could never, by themselves, have produced any kind of explicit proof that justified the validity of the inference they had confidently but intuitively made.
*******************************************
1.2 The moral of this little opening discussion is that in our logic classes we do not teach anyone how to reason. For, as Plato had observed, and as the example of Eliza and my students illustrates, people already know how to reason --- intuitively. That is to say, people are able to reason, even though they are quite unaware of how they do it.
Logicians too, do not know how people naturally reason; few have have the temerity to claim that we reason intuitively in the manner prescribed by modern predicate logic.
We normally reason without being aware of how we reason. For example, we arrive at the conclusion that anyone who rides (owns, feeds, ) a pony, rides (owns, feeds, ) a horse from the premise that all ponies are horses without being aware of how we inferred that conclusion from that premise. We confidently judge that two sentences are logically equivalent (e.g., that No ape is immortal and All apes are mortal say the same thing, in other words) or that some discourse is inconsistent (e.g, that some boy loves every girl and some girl is not loved by any boy cant be jointly true). We produce a constant stream of sound deductive judgments intuitively and instantly, without knowing how we do it.
Despite our intuitive rush to judgment, our reasoning is usually valid. We are rational animals after all. We reason correctly most of the time. Teachers of logic are primarily engaged in proving the validity or invalidity of deductive judgments, naturally arrived at. They are not concerned with how people intuitively reason. They do not focus on the process of reasoning but on the product: is the inference valid or invalid, and if so how can we prove it? Since the publication of Gottlob Freges logical writings and Whitehead and Russells Principia Mathematica, the canonical way to prove or disprove the validity of an inference is the way of Modern Predicate Logic (MPL).
The awkwardness of reasoning in the grammar of MPL
1.3 The quantifier-variable notation of MPL is,as Quine candidly calls it, an artificial grammar designed by logicians that we tendentiously call standard, a made for logic grammar. Utilizing the quantifier-variable notation of predicate logic, modern logic constructs many proofs of validity that seem to be beyond the deductive powers of traditional (pre-Fregean) term logic, whose proofs, in ordinary language which is variable-free, do not deploy the quantifier/bound-variable mechanisms of MPL.
Quine acknowledges that regimenting inferences in the symbolic language of predicate logic is often irksome and cumbersome. But he firmly maintains that it is scientifically necessary:
All of austere science submits pliantly to the Procrustean bed of predicate logic. Regimentation to fit it . . . serves not only to facilitate logical inference, but to conceptual clarity.
The students in my logic class are typical examples of how, in the name of austere science, we all quietly submit to the Procrustean bed of predicate logic. Predicate logic does offer a scientific method of reasoning that facilitates Elizas inference. But the method of reasoning it provides, cannot explain how Eliza facilitates in a fraction of a second an inference that takes a teacher of MPL thousands of seconds to validate. Before we agree that the method of predicate logic as the more scientific one, we ought to find out how Eliza reasons so efficiently.
1.4 Aristotles term logic which had been standard logic for millennia, has in the course of a single century been deposed and thoroughly supplanted by Freges predicate logic. Dummett calls Freges discovery of quantification, the deepest single technical advance ever made in logic. p. xxxiii It is now generally assumed that traditional (Aristotelian) term logic of natural language is deductively weak, being incapable of accounting for the validity of arguments that involve sentences with relational predicates like loves or owns that are predicated of more than one general subject. As Michael Dummett says:
Modern logic stands in contrast to all the great logical systems of
the past in being able to give an account of sentences that
depends upon the mechanism of quantifiers and bound variables. For all the subtlety of the earlier systemsmodern logic is, by its capacity to handle multiple generality, shown to be far deeper than they were able to attain. (1993: xxxii).
Referring to Freges creation of predicate logic, with its grammar of quantifiers and bound variables, and its ability to deal with inferences involving relational, multiply general sentences Dummett says, Frege solved the problem which had baffled logicians for millennia by ignoring natural language.
Though Dummett speaks of Freges discovery of quantification, it is not something one finds finds in sentences of the natural language that figures in our everyday reasoning. It is true that natural language lacks the elegant mechanism of quantifiers and bound variables that facilitates logical proofs in MPL. But it is also true that people, whose language of thought is natural language, make valid inferences involving all kinds of sentences and modern philosophers of language should be baffled by our ability to reason make such inferences intuitively. The fact that we can reason intuitively in sentences of natural language is strong prima facie evidence that the ability to reason does not depend on the mechanism of quantifiers and bound variables. I have found that confining myself to the language in which people normally think and reason --- does not condemn anyone to methods of reasoning that are deductively inferior to those afforded by the symbolic language of MPL.
Anyone who rides a pony, rides a horse is a relational, multiply general sentence with predicates that have more than one general subject. What would Eliza tell us if she somehow had articulate, conscious access to the way she derives this sentence from the premise, All ponies are horses? How would she explain why she is so confident that it follows from that premise? How would her account of how she naturally reasons, explain the ease, fluency and sheer speed of her reasoning? Above all how would her account explain why she reasons correctly much more often than not?
2 If we go back to a preFregean era when logicians regarded Logic as a Laws of Thought, the science of how we intuitively reason in the natural language we use in everyday reasoning, we find Thomas Hobbes dogmatically and enigmatically announcing that we mentally reason in an algebraic manner: By the ratiocination of our mind, we add and subtract in our silent thoughts, without the use of words. Historians of logic say that Hobbes thought of reasoning as a species of computation but they point out that his writing contain in fact no attempt to work out such a project.
Later in the 17th century, Leibniz strongly endorsed Hobbess characterization of our silent ratiocination:
Thomas Hobbes, everywhere a profound examiner of principles, rightly stated that everything done by our mind is a computation by which is to be understood either the addition of a sum or the subtraction of a difference. So just as there are two primary signs of algebra and analytics, + and , in the same way there are, as it were, two copulas.
Though Leibniz alludes to the plus/minus character of the positive and negative copulas, neither he nor Hobbes say anything about a plus/minus character of other common logical words that mentally drive our intuitive, everyday deductive judgments --- words like some, all, if, then and and, each of which, as I discovered in my efforts to answer Platos Meno question, also have a positive or negative character that allows us, in our silent thoughts to reckon with it as one reckons with a plus or a minus operator in elementary algebra or arithmetic.
Hobbes was actually right about the additive and subtractive nature of intuitive reasoning, but he did not go into details. In particular, he did not focus attention on the diverse logical constants of natural language --- specifying which are the positive ones we reason with as plus-words and which the negative ones we reason with as minus-words. As a result, his insight into the +/- character of everyday ratiocinations had little effect on the future development of logic.
The Logical and the Extra-logical
2.1 Why, Plato had asked, are uneducated children or slaves able to reason logically? This question focused my attention on the natural logical constants that drive inference in intuitive reasoning, logical words and particles and particles such as is, isnt, not, non-, all, if, and and that figure so prominently in everyday reasoning.
Scholastic logic sharply distinguishes between formative, logical, elements of a sentence and its material extra-logical elements. The material elements are the categorematic words that semantically relate the sentence to things in the world. For example, in the sentence Not every dog is friendly the terms dog and friendly are the material elements; these words carry its matter or content. The formative elements of the sentence are syncategorematic words that determine the logical form of the sentence. The logical form of Not every dog is friendly is Not every X is Y whose formative elements are the words Not, Every, and Is. (The letters X and Y mark the place where the terms go.)
Hoping to learn how we reason intuitively, I spent several weeks looking at the way the various logical formatives are facilitating everyday reasoning. At first I confined my efforts to syllogistic reasoning but eventually I discovered an algorithm that applies generally to all kinds of reasoning that reckons with the logical formatives of natural language as one reckons with the plus and minus operators in elementary algebra.
I found, for example, that is, and, some, and then, are PLUS-WORDS but that isnt, not, all, and if, are MINUS-WORDS. In 1967 I became convinced that we normally reason intuitively by unconsciously exploiting the +/- character of the logical formatives of natural language, reckoning with its plus words and its minus words as quickly and as easily as one reckons with the plus and minus signs of simple expressions in elementary algebra or arithmetic.
Plato rightly believed that all rational beings innately possess some rudiments of algebraic knowledge, so that when a seven year old child is told that x+y = y+x, it greets that truism as one greets a friend or familiar acquaintance something it is not meeting for the first time. I think he is right about that. Today we would accept that as an example of how we are genetically endowed with some algebraic and geometric know-how that has given us an enormous advantage over lesser animals. By the time I was teasing Eliza about pony owners and horse owners I had long convinced that she reasons intuitively by reckoning with the +/- character of the logical formatives of natural language in very much the way she reckons with the pluses and minuses in arithmetic or beginners algebra. Unconsciously Eliza regards and, is, and some as plus-words and all, isnt , and not as minus-words. Knowing that +x+y = +y+x, she can instantly see that Some{+} Teachers are{+} Men says the same thing as Some{+} Men are{+} Teachers,
Discursively: Some{+} Teachers are{+} Men a" Some{+} Men are{+}Teachers
( ( ( (
Logibraically : + Teachers + Men = + Men + Teachers
She also reasons that being a Boy and{+}a Redhead is the same as being a Red-head and{+} a Boy:
=
** =
I once overheard a father warning his nine year old son, Timmie, not to pet an approaching strange dog, sayiing:
Not{-}: All{-} Dogs are{+}Friendly, Timmie.
Timmie immediately transformed his fathers sentence into its logically equivalent obverse, responding:
Some{+} dogs, arent{-} friendly, dont you think I know that, Dad!
We can explain Timmies rational fluency if we assume that he unconsciously reasoned the +/- way, instinctively reckoning the formatives not, all, and arent as minus words while reckoning the formatives some and are as plus words:
Not{-} All{-} Dogs are{+}Friendly ( Some{+}Dogs arent{-} friendly
( ( ( ( (
- ( - Dogs + Friendly) = + Dogs Friendly
Reasoning logibraically with relational sentences
3.1 Being aware that distributing the external minus sign of -(-a+h+t) inward, changes the
internal signs, a teenager will immediately recognize that it is equal to +a+(-h)-t. Teenagers are just as quick to see that the discursively meaningful, multiply general, sentence Not: every archer had hit some target is logically equivalent to Some archer had missed every target; here we find the teenager unconsciously reckoning with discursively meaningful sentences as he reckons with discursively meaningless algebraic expressions, distributing the external minus-word not{-} to the right as if it were a minus-sign prefixing an algebraic expression. In this case the teenager the minus-word not changes every{-} archer to some{+} archer, hit{+}, to its logical contrary, hit{-} and some {+}target to every{-} target:
Not{-}: Every{-} archer had{+} hit some{+} target a" some{+} archer had{+}+ missed every{-} target
( ( ( ( ( ( ( ( (
- ( - archer + (+hit) + target)) = + archer + (-hit) - target)
In transforming not every archer had hit a target to some archer had missed every target we reason in just the way we reason in elementary algebra when transforming -(-a+h+t) to
+a+(-h)-t. The difference between the two moves is that when people reason with logical words like not, every, some and is or with logical contraries like hit and missed, they are completely unaware that they are reasoning logibraically, in this case, distributing a negative particle like not{-} to the right, thereby changing all the positively and negatively charged particles in its scope to their logibraic opposites.
I eventually came to believe that we reason intuitively by instinctively (unconsciously) exploiting the additive and subtractive powers of the logical words of the natural language in which we think. In my opinion, that is what accounts for speed and fluency of our daily reasoning. Hobbes had somehow divined this. I had read the Leviathan and must have read his peremptory pronouncements on how we reason, but had forgotten them. (Or perhaps I should I say, I was not conscious of the impression they had made on me.)
Two possible +/- accounts of Ponies are Horses, so Pony-owners are Horse-owners
3.2 This is the kind of inference that had baffled logicians for millennia before Frege showed us a way to handle them by ignoring natural language. Since people reason in natural language, the question to ask is: How might we instinctively be taking advantage of the +/- powers of the natural constants of natural language to infer So every{-} owner of a{+} pony is{+} an owner of a{+} horse from Every{-} pony is{+} a horse?
One way we could validate that inference is by regarding it as an enthymeme whose missing premise, is the tautology, Every owner of a horse is an owner a horse. We can then derive Every{-} owner of a{+} pony is{+} an owner of a{+} horse from the following conjunction:
(1)Every{-}Pony is{+}a Horse and{+}(2)Every{-}Owner of a{+}Horse is{+}an Owner of a{+}Horse:
Adding (2) to (1) we get , Every{-}Owner of a{+} Pony is{+} an Owner of a{+} Horse:
[ -Pony+Horse] + [-(Owner of +Horse) + (Owner of +Horse)] => -(Owner +Pony)+(Owner of +Horse)
1. Zoological Premise: Every{-} Pony is{+} a Horse
2. Tautological Premise: and+ Every{-}Owner of a{+} Horse is{+} an Owner of a{+} Horse
3. Conclusion: Every{-}Owner of a{+} Pony is{+} an Owner of a{+} Horse.
A second way is by reasoning indirectly, arguing that it cant possibly be true that some owner of a pony isnt an owner of an horse animal. For we should then have:
(i) Every{-} pony is+ a horse{+} and{+} (ii) Some{+} owner of a{+} pony doesnt{-} own a{+} horse.
From these two premises, it would follow that
(iii) Some{+} owner of an{+} horse isnt{-}an owner of an{+} horse:
Logibraically: (i) -P+H + (ii) +(O+P) -(O+H)] => (iii) +(O+H) (O+H)
(i) -P+H
+(ii) +(O+P) - (O+H)
( (iii) +(O+H) (O+H)
We know that (i) is true. If (ii) were also true, (iii), an absurd conclusion, would follow. To avoid that reductio one must deny (ii). But the negation of (ii) is
is [+(O+P) -(O+H)] or -(O+P)+(O+H):
-(O+P)+(O+H) ( Every owner of a pony is an owner of a horse.
Eliza possesses the resources to use either one of these two ways to get from all ponies are horses to All pony owners are horse owners.
Plato believed that we reason intuitively because we have been taught logic in a previous incarnation. There is a more acceptable, modern, explanation for our ability to reason that does not beg the question he had originally raised.
We are rational social animals whose ancestors had naturally evolved to the point of possessing the ability to communicate in a descriptive language and to reason in that language. In a Pickwickian sense we are reincarnations of the myriads of increasingly sapient primates from whom we are descended, including those ancestors who acquired some rudimentary algebraic know-how and the advantageous ability to use natural language not only to describe but also to reason --- --- by instinctively exploiting the plus/minus character of its logical formatives.
Term Logic and Propositional Logic
4. Logic, the science of reasoning, distinguishes two main branches. Aristotles logic of terms preceded the Stoics logic of propositions by some two hundred years. In Aristotles day and for more than two thousand years thereafter, term logic was primary logic. Indeed Leibniz looked for a way to incorporate propositional logic as a special branch of term logic:
If as I hope, I can conceive all propositions as terms, and hypotheticals as categoricals, and if I can treat all propositions universally, this promises a wonderful ease in my symbolism and . . will be a discovery of the greatest importance.
Until the advent of modern predicate logic as inaugurated by the Gottlob Frege in 1879 and codified in Whiteheads and Russells Principia Mathematica (1905), Aristotles logic of Terms was primary logic and the Stoic logic of propositions was secondary. After Frege and the Principia, the primacy was reversed. As the Kneales say of the Stoics:
The logic of propositions which they studied is more fundamental than the logic of general terms which Aristotle studied in the sense that it is presupposed by the second [which] is primary because it must come at the beginning of any systematic development. If we adopt this practice, we reserve the title, general logic for the study in which we are concerned not only with the notions of negation, conjunction, disjunction and disjunction but also with the notions of generality expressed by every and some. General logic, so defined, includes primary logic and cannot be developed without it Within this scheme Aristotles syllogistic takes its place as a fragment of general logic, in which theorems of primary logic are assumed.
4.1 In a general logic that includes both the logic of terms and the logic of, neither logic is primary. Leibnizs ideal of a unified symbolism for the logic of terms and the logic of propositions is achieved the moment we recognize the +/- character of all the logical constants of natural language, whether they be propositional connectives like if{-} then{+} and both{+}and{+} or term connectives like every{-} is{+} and some{+} is{+}.
Some natural constants belong both to the logic of terms and to the logic of propositions. And conjoins terms (gentleman and scholar) as well as propositions. Not has both term and propositional scope (not happy, un-happy; Not: a creature was stirring). And is literally additive and plus-like. Not is literally subtractive and minus-like. Formatives like every and some, if, and then dont strike us as being minus-like or plus-like. Nevertheless they too behave like plus or minus operators.
How to determine the plus/minus character of the natural logical constants
5. Consider the propositional conjunction both p and q. Intuitively, the logical words and and both behave in a plus-like way. Their logibraic, plus-like character is evident in the commutative character of the conjunction both p and q, where both as well as and behaves like the two plus signs in the algebraic expression +x+y.
Both p and q ( Both q and p
( ( ( (
+ p + q = + q + p
The equivalence shows that both, like and, is a plus-word .
Having determined this, we may logibraically transcribe Not: both p and not-q as
(+p+(q)).
Not: both p and not-q
( ( ( (
( + p + ( q))
Now consider the propositional connective if...then. Neither if nor then strikes us as plus-like or minus-like. However, we know that If p then q can be defined as Not: both p and not-q, which transcribes as (+p+(q)). We may therefore determine the +/- character of If ..then by setting If{?} p then{?} q equal to Not{-}(both{+} p and not{-}q) and seeing what that tells us about the positive or negative character of if and then :
If? p then? q =def. Not-: both{+} p and{+} not{-}q
( ( ( ( ( (
p + q =def. ( + p + ( q) )
Expressed logibraically, the definitional equivalence shows that if is a minus-word and then is a plus-word:
If{-} p then{+} q =def. Not{-}: both{+} p and{+} not{-}q
The +/- character of if and then is manifest in contraposition:
if p then q = if not-q then not p
( ( ( ( ( (
- p + q = - (- q) + (-p)
Note the difference between representing the logical equivalence of if p then q to Not: both p and not-q conventionally as p ( q ( ~(p&~q), and representing it logibraically as p+q = (+p+(q)). One proves that p(q and ~(p&~q) are logically equivalent by truth tables. Represented as p+q = (+p+(q)), no proof is needed; the equivalence is perspicuous as an algebraic truism.
Among the basic inference patterns in standard propositional logic are the principles knows as modus ponens, modus tollens and hypothetical syllogism. All three principles are perspicuous in the logibraic notation:
Modus Ponens Modus Tollens Hypothetical Syllogism
p+q p+q p+q
p q q+r
( q ( p ( p+r
Determining the plus/minus character of the formatives in the logic of terms
5.1 The natural formatives is, isnt, some, and every drive inference in the logic of terms. We think of the positive copula is as a plus-word and of the negative copula isnt as a minus-word. Although we do not think of some, and every, as plus-like or minus-like, they too have a plus or minus character. How can one show that they too are +/- formatives?
Starting with is and and as plus-words and with not as a minus-word, we can determine the +/- character of some and all.
That some is a plus-word
The commutative equivalence of some{?} A is{+} B to some{?} B is{+} A shows that some A is B is to some B is A as +A+B is to +B+A:
Some{?} A is{+} B ( Some{?} B is{+} A
( ( ( (
+ A + B = + B + A.
The parallel equivalences reveal that some is a plus-word.
That all is a minus-word
Here is what Aristotle says of propositions that begin with every or all: We say that one term is predicated of all of another, whenever no instance of the subject can be found of which the other term cannot be asserted. (24b 29-30)
Aristotles treatment of All M are P as the denial of Some M are not-P, is analogous to the way modern logicians define if p then q as the denial of both p and not q.
Having fixed the +/- character of are, some, and not; we may proceed to determine the plus/minus character of all :
All{?} M are{+}P ( def. Not{ }: some{+}M is not{ }P
( ( ( ( ( (
M + P ( def. ( + M + (-P))
The logibraic equivalence shows that all is a minus-word.
The minus-character of all is exemplified in BARBARA syllogisms and other common inferences:
Barbara:
\ All{-}M are{+}P and{+} All{-} S are{+} M, hence All{-}S are{+} P
( ( ( (
[-M+P] + [-S+M] => -S + P
Obverse equivalence:
Not{-}: all{-} M is{+}P = Some{+}M isnt{-} P
-(-M +P) = +M-P
It is in this fashion that one specifies the plus/minus character of the syncategorematic elements that drive deductive reasoning which it observes in play in everyday reasoning .
6. The following is a very partial but representative list of natural language formatives that figure in our everyday intuitive reasoning:
some(a..), is (was, will be, etc.), both, and, and then are plus- words;
Every,(all,any..),not, (no, aint, un-, etc.), and if are minus-Words.
Russell somewhere says that a good notation is like a live teacher. Here are three sample lessons that the +/- notation teaches about the logical constants of natural language:
I. Consider the categorical forms Some S is P, All S is P, and No S is P. Unlike some and all, which are words of quantity, no is not a word of quantity (in the sense that zero is a number). No is a denial of propositional scope whose meaning is it is not the case that. Logibraically, the No of No- S is+ P denies a proposition that follows it. We transcribe no{-} S is{+} P as -(S+P), an abbreviated form of -(+S+P) [( not{-}: some{+S is{+} P].
II. Why Or, does not appear on the list of primitive plus/minus words.
Unlike and and if, or is neither a plus-word nor a minus word but a composite of two minus words. Logibraically, p or q is p, if{-} not{-}q. If English had a contraction for if not, or would literally have the meaning ifnt. p or q logibraically transcribes as p (-q), Or not like other composites such as isnt and wont, which, being contractions of +, - can be safely be transcribed as minus words. Because Or is a logical diphthong that irreducibly transcribes as - it belies the saying that two negatives always make a positive.
p or q =def. p if{-} not{-}q ( p (-q).
III. Self-contradictory propositions of form Some X isnt X and p and not-p give zero information about the world. This is perspicuous in the forms +X-X and p+(-p). The same is true of tautological propositions. Every- X is+ X and if- p then+ p also give zero information about the world: -X+X =0; -p+p = 0. Contradictions say nothing falsely. Tautologies also say nothing but say it truly.
This is true not only of simple tautologies like if p then p but also of interesting tautologies. Quite generally the empirical information content of logical truths is literally zero. They say nothing about how the world is. Consider the tautological truth if no horse is inorganic then anyone who rides a horse, rides something organic: -[-(H+(-O))] + [-(R + H ) +(R + O )].
Being logibraically equal to zero, it is empirically uninformative.
7. Having educed the +/- character of the logical constants of natural language, I soon came to believe that we instinctively reason with them by unconsciously regarding them as being positively or negatively charged. For example, we read some as some{+}, all as all{-}, if as if{-}, then as then{+}, and as and,{+} is as is{+} and reckon with them accordingly.
Admittedly, it is more than a little odd that we should constantly be reasoning by exploiting the +/- character of familiar logical words like all, some, is, not, and if without becoming consciously aware of their +/- character. That we instinctively exploit the +/- character of the natural constants of natural language in our ratiocinations is an empirical hypothesis. I expect that cognitive science, and more specifically, cognitive psychology, will eventually find ways to test this hypothesis and to confirm or disconfirm it. I am convinced that the hypothesis will not be falsified; it is the most reasonable explanation of our rational celerity, of why we are mentally as adept at reasoning with the logical constants of natural language as we are adept at reckoning with the plus and minus operators of elementary algebra and arithmetic.
Ratiocination is something that takes place in real time (in this case milliseconds) in real minds (including childrens) that reason competently with sentences of natural language. We may innately only have some rudiments of algebraic know-how but what innate knowledge we possess, enables us to reason logibraically even with relational sentences of natural language. Originally an inspired conjecture of Thomas Hobbes, the hypothesis that intuitive reasoning is an unconscious process of logibraic reckoning will, I predict, eventually come to be accepted as a psychological reality.
Of course cognitive scientists wont enter the picture to arrive at an empirical judgment, before they learn that the logical constants of natural language have a +/- character. They wont be learning that from philosophers of language or academic logicians. Despite repeated efforts on my part to attract attention to the +/- character of the natural language constants, I must report that I have failed to interest philosophers and logicians in the minus-like behavior of natural logical constants such as if, every, and not, or the plus-like behavior of then, some, and, and is.
Impressed by Freges concept-script, modern logicians and some linguists assume that a child who reasons that every pony is a horse so every rider of a pony is a rider of a horse or a teenager who says If some boy is taller than every girl, then every girl is shorter than some boy, must somehow be reasoning with sentences that have the mechanisms of the made-for-logical-grammar of MPL. (As if human beings have evolved to think and reason in the Procrustean bed of predicate logic to reason with in formulas of a constructionist notation devised by a brilliant 19th century mathematical logician.)
I have shown that the logical words of natural language have a +/- character. That
we actually reason by exploiting the +/- character of the natural formatives has not (yet) been shown. I expect that cognitive will find that to be so. In any case it is an intriguingly simple hypothesis and friends of MPL should have taken the +/- theses seriously and presented arguments against them. So far they have evinced very little interest in my formal thesis that the natural constants are oppositively charged and no interest at all in the empirical psychological hypothesis that we exploit their charged +/- character in actual reasoning.
Both theses have inappropriately been met, not with critical appraisal but with an obdurate silence. I published the first article on the formal thesis more than forty years ago and subsequently wrote a book called The Logic of Natural Language. That the logical formatives can be reckoned with logibraically is a surprising feature of natural language. That we actually reason logibraically is, if true, an important facet of our rational nature. Neither thesis has generated a single serious critical article in the literature.
No acceptable account of human reasoning can afford to make light of the fact that we think and reason in sentences of natural language. One of the main objectives of a course in logic is to clarify formally what is intuitively obvious. Most students find it obvious that some boy loves every girl entails every girl is loved by some boy. A course in logic should explain why the student finds that obvious. Ideally, the teacher should present a formal proof of the inference that elicits an Aha! reaction. Aha, so thats how we actually reckon with these sentences! Thats why I intuitively arrived so quickly and confidently to this very conclusion!
No student of logic presented with an MPL proof that Not: every archer had hit some target logically entails Some archer had missed every target will ever react to the conventional proof by saying: So thats why I find it so obvious that Some archer had missed every target follows from Not: every archer had hit some target. No student in my logic class reacting to my MPL proof that every pony is a horse entails every owner of a pony is an owner of a horse ever said Aha! So thats why I find this inference so obvious. When I changed my conventional MPL way of teaching logic to the +/- way, I was rewarded by many an Aha! reaction to the /- accounts of these and other obvious inferences.
On how the modern orthodoxy overrides scientific common sense
8. The orthodox view that quantifiers binding variables facilitates actual inference has even been embraced by linguists who are expressly in the business of explaining linguistic and logical competence. Noam Chomsky is rightly famous for his insistence on empirical explanations for the extraordinary linguistic competence of native speakers, including their competent use of natural language to reason deductively. However, Chomsky is all too aware that traditional term logic (TTL), which adhered closely to the quantifier-free and variable-free syntax of natural language, found it difficult to justify the validity of inferences involving multiple generality. Augustus De Morgans example Every horse is an animal, so every tail of a horse is a tail of an animal raised serious doubts about the ability of TTL to formally derive conclusions of form Every R to an X is R to a Y from premises of form Every X is Y. By contrast, modern predicate logic (MPL) with its quantifier/variable notation, could elegantly explain why Every R to an X is R to a Y follows from Every X is Y and many similar inferences whose validity traditional logic could not explain. Impressed by the superior inference power of modern predicate logic, Chomsky seems ready to abandon the search for an empirical explanation that would shed light on our linguistic competence in reasoning with the variable-free sentences of natural languages and to accept the orthodox logicians view that the sentences we use in such deductive inferences have a quantifier/variable syntax. We find him suggesting that the sentences that figure in our everyday reasoning may not be what they appear to be: The familiar quantifier/variable notation should in some sense be more natural for humans than a variable-free notation for logicAt one point he even says that [t]here is some empirical evidence that it [the brain] uses quantifier/variable rather than quantifier-free notation.
In the present, early, stage of its development , brain research is nowhere near the point of being able to specifying the syntax of sentences that figure in our deductive judgments. There can be no empirical evidence from brain science that we are using quantifier/variable notation. Nor is there any empirical basis for Chomskys astonishing suggestion that quantifier/bound variable syntax is more natural for humans than a variable-free syntax. That suggestion, like the one about the brain using a quantifier/variable notation, merely attests to Chomskys uncharacteristically unquestioning acceptance of a widely accepted conventional modern doctrine, in this instance, of the doctrine that many of sentences we reason with, are essentially quantifier/variable in forman all too typical example of doctrinal scientific orthodoxy dogmatically overriding ones scientific common sense.
Uncritical faith in formal technical advances made by Frege and codified in Principia Mathematica and in textbooks of logic such as Quines Methods of Logic, has led many contemporary philosophers and is sorely tempting some linguists to abandon the fundamental research program into the question that goes back to Platos Meno and that today is properly beginning to engage empirically-minded linguists and cognitive psychologists: How do people reason intuitively in their native languages?
Aristotles Logic of Natural Language
9. The +/- logic is a Logic of Natural Language because it focuses our attention on the logical formatives of natural language and shows how we can be reckon with them as we reckon with the plus/minus operators in elementary algebra. It is an Aristotelian logic because, the Dictum de Omni --- the governing principle of inference in Aristotelian Logic--- sanctions the +/- way of making inferences in natural language.
According to the Dictum de Omni:
Whats true of every{-} M is true of whatever is{+} an M.
By the D.O., when ( is said to be true every{-} M in one premise and is{+} an M is said to be true of something in a second premise, the middle term, M occurs negatively in the first premise, where it is said to distributed, and positively in the second premise, where it is said to be undistributed. When we add the first premise to the second, the negative middle term logibraically cancels the positive middle term of the second premise, replacing it by ( in a conclusion in which no middle terms appear:
{(}(-M)
+M
( (
For example: i. ii.
All{-}Mammals are{+} Warm-blooded All{-}Mammals are{+} Warm-blooded
All{-}Dolphins are{+} Mammals Some{+} Sea Creatures are{+}Mammals
(All{-}Dolphins are{+}Warm-blooded Some{+}Sea Creatures are{+}Warm-blooded
-M+W -M+W -D+M +S+M . ( -D+W ( +S+W
10. The logibraic form of categorical sentences
In today s standard logic, sentences of form every/some A is B are regimented as (x(Ax"Bx) and (x(Ax(Bx).
In traditional logic, a categorical sentence affirms or denies that some/every A/non-A is/isnt B/non-B:
Yes/ Not: Some/Every A/non-A is/isnt B/non-B
+/- ( +/- A/-A +/- B/-B)
Every categorical sentence has a positive or negative valence.
The valence of a sentence is positive or postively existential when its sign of judgment, Yes{+}/Not{}, is the same as its sign of quantity [Some{+}/Every{-}] i.e., when both signs are pluses [Yes{+}(Some{+}] or when both are minuses. [Not{-}(Every{-}]. Sentences whose valence is positive are existentially positive and are said to be particular in quantity.
The valence of a sentence is negative or negatively existential when its judgment sign and quantity sign differ as in Yes{+} (Every{-}- or No(t){-}(Some{+}. A sentence that is negative in valence, is said to be existentially negative and to have universal quantity.
Two sentences of the same valence are covalent. Sentences that differ in valence are divalent. Divalent sentences differ in quantity, one being particular and existentially positive, the other being universal and existentially negative. Covalence is a necessary condition of logical equivalence. Divalent sentences that are algebraically equal are not logibraically equal. For example, being divalent, some mammals arent whales [+M-W] and all whales are mammals[-W+M] are not logibraically equal.
THE LOGIBRAIC CONDITIONS FOR EQUIVALENCE:
Categorical propositions are logically equivalent if
and only if
they are both algebraically equal and covalent.
Some conventions of logibraic notation
10.1 Here are several examples of categorical sentences transcribed in +/- notation.
(1) Not a creature was stirring ! -(+C+S)
(2) Not every dog is friendly ! -(-D+F)
(3) Some bats weren t moving ! +(+B-M )
(4) Every creature was moving ! +(-C+M)
Since an affirmative statement such as (3) or (4) is only rarely prefixed by an explicit phrase such as it is the case that, or Yes, we do not normally transcribe them with a sign of affirmative judgment. Thus (3) may simply be transcribed as +B-M and (4) as -C+M. However in determining their valence we assume they are tacitly prefixed by plus- signs of affirmation.
The valence of (2) and (3) is positive. The valence of (1) and (4) is negative.
(1) is logically equivalent to
-C-S, [every creature wasnt stirring]
-C+(-S) [every creature was not-stirring]
-(C+S) [no creature was stirring]
(3) +B-M [some bats werent moving] is logically equivalent to
-(-B +M) [not every bat was moving]
+B+(-M) [some bats were motionless]
-(M+B) [nothing moving was a bat]
11. The logibraic characteristics of valid syllogisms
A syllogism or sorities is an argument that has as many terms as it has sentences. Standard syllogisms have three terms and three sentences. A1 and A2 are examples:
A1
every dolphin is a mammal. -D+M
every mammal is warm-blooded -M+W
( (every dolphin is warm-blooded ( -D+W
A2
not a creature was stirring -(+C+S)
some of the creatures were giants +C+G
so some giants werent stirring ( +G-S
Both syllogisms are valid. In each the conclusion is equal to the sum of the premises. But equality is not in itself sufficient to guarantee the validity of a syllogistism or a sorites. A1 and A2 also satisfy the Mood condition that a valid syllogistic argument must satisfy.
Only two moods are valid:
(1) Syllogisms like A1, that have only universal sentences. Call such syllogisms U-regular.
(2) Syllogims like A2, that have a particular conclusion and a single particular premise. Call such syllogisms P-regular. All other moods are irregular and invalid.
Syllogisms that have more than one particular premise, are irregular and invalid. For example, A3 is invalid.
A3
(1) Some A is B +A+B [e.g., some apes are black]
(2) Some C isnt B +C-B [e.g.,some cats arent black]
((3) Some A is C +A+C some ape are cats
A3s premises add up to its conclusion but its mood is irregular so it is invalid.
The following principle, called the The REGAL Principle, states the two necessary and sufficient conditions that any valid syllogistic argument must satisfy.
THE REGAL PRINCIPLE:
A syllogism or sorites is valid if and only if
(1) the sum of its premises is equal to its conclusion, and
(2) its mood is U-regular or P-regular.
In the 19th century, when logic was still natural language, it attracted a large an intelligent lay public. Syllogistic was taught in the secondary schools and most educated people knew a lot about them. Writers like Lewis Carroll made up many droll logical problems for the general public to solve. (His books on logic are still in print and are still selling well.) In solving them, I use the +/- calculus. We are given premises and asked to draw a conclusion.
(1) all puddings are nice -P+N
(2) some deserts are puddings +D+P
(3) no nice things are wholesome -(N+W)
((4) ? ? ?
Here we see a particular statement among the premises so we know that the conclusion must be a particular statement. We also know that a conclusion may be drawn by adding up the premises and canceling the middle terms.
Adding up we have -P+N+D+P-N-W = -W+D or +D-W.
Of the two algebraic alternatives for a conclusion, only the second will give us a syllogism that is valid in mood. For we need a particular conclusion. So our answer to Lewis Carroll's problem is
/(4) +D-W: some deserts aren't wholesome
The next Carroll example has six premises. But the same methods apply:
(1) Everything not absolutely ugly , may be kept in a drawing-room.
(2) Nothing that is encrusted with salt is ever quite dry.
(3) Nothing should be kept in a drawing room unless it is free from damp.
(4)Bathing-machines are always kept near the sea.
(5) Nothing that is made of mother-of-pearl can be absolutely ugly.
(6) Whatever is kept near the sea gets encrusted with salt.
We may transcribe it thus:
1. -(-U)+K
2.-(+E+D)
3.-(+K-D)
4. -B+S
5.-(M+U)
6.-S+E
All six premises are universal. So the conclusion must be universal. Adding up the premises we get -B-M or -(B+M) which is the transcription of
7. No bathing machine is made of mother of pearl.
Quine accepts the Procrustean bed of predicate logic because it facilitates logical inference. How facile is an MPL derivation of ((x(Bx&Mx) from the six premises of this sorites?
12. How the Dictum de Omni facilitates Relational Arguments
The D.O. sanctions the +/- application to relational reasoning. Take de Morgans Tail of a Horse inference:
(() Every horse is an animal, so every tail of a horse is a tail of an animal.
Applying the D.O., we may prove (() valid by an indirect argument showing that affirming its premise but denying its conclusion entails a self-contradiction. For suppose its true of every horse that it is an animal but also true of some horse that its tail isnt a tail of an animal. Since by the first premise, is an animal is true of every horse, it must, by the D.O., also be true of whatever is a horse. So, given the second premise, it would be true of a horse whose tail is not a tail of an animal that it is an animal whose tail is not a tail of an animal. This self-contradictory consequence of applying the D.O. comes out clearly if we logibraically add the premise of (() to the denial of its conclusion. For by applying the D.O. we cancel the middle term, horse and arrive at a blatant absurdity of form some X is not an X, viz., that some tail of an animal is not a tail of an animal.
(1) H+A: Every horse is an animal; Is an animal is
true of every horse.
+(2) +(t+H) (t+A); Some tail of a horse isnt a tail of an animal; Its true of some
thing that is a horse
that its tail is not a
tail of an animal.
((3) +(t+A) - (t+ A); Some tail of an animal horse isnt a tail of an animal; So, its true of some
animal that its tail is
not a tail of an animal.
This indirect reductio reasoning, which validates all arguments of form Every X is Y, so every R to an X is R to a Y, is an example of how an Aristotelian term logic of natural language expeditiously accounts for inference involving multiply general propositions.
Aristotelian Term Logic (ATL) is the classical logic of natural language, the variable free language of thought that we routinely use in everyday deductive reasoning. By contrast modern predicate logic (MPL) is a rational reconstruction of actual reasoning that uses a symbolic language of quantifiers and bound variables, that is not the language of actual ratiocination.
Contrast what a practitioner of MPL must go through to show that
(3) Every- boy envies some+ owner of a+ canine follows from
(1) Every- dog is+ a canine and (2) Every- boy envies some+ owner of a+ dog
He must find a way to derive
(3*) (x(Boyx " (y(Enviesxy & (z(Caninez & Ownsyz)))
from the premises,
(1*) (x(Dogx " Caninex) and (2*) (x(Boyx " (y(Enviesxy & (z(Dogz&dOwnsyz))),
By contrast any teenager, can apply the D.O. and instantly derive (3) from (1) and (2):
(1) Every- dog is+ a canine; -D3+C3
(2) Every- boy envies some+ owner of a+ dog; and+ -B1+E12 +(O23 + D3)
((3 Every- boy envies some+ owner of a+ canine -B1+E12 +(O23+ C3).
How we might ss show that we intuitively reason logibraically
13 That we normally reason by taking advantage of the +/- character of the natural formatives, needs to be empirically demonstrated. If we could show that we reason logibraically, that would explain our ability to reason as rapidly as we do.
I am not a cognitive psychologist. But I have a suggestion about one possible way we might experimentally support the hypothesis that +/- reasoning is actually transpiring in everyday mental reasoning.
Even microsecond differences in speed of reasoning are now experimentally discernible.
Consider the two valid inferences:
1. No noncitizens are voters 4" all voters are citizens ! -((-C)+V) 4"-V+C
2. Only citizens are voters 4" all voters are citizens & & & . 4"-V+C
To reckon (2) logibraically valid, we must first normalize only citizens are voters, by paraphrasing it as no non-citizens are voters: (2) may then be algebraically transcribed and reckoned in the manner of (1). Since normalizing (2) requires an extra step, one who reasons logibraically will find that reckoning (2) takes slightly more time than reckoning (1). That finding would count in favor of the hypothesis that +/- reckoning is the default way we reason deductively.
Quite generally, if we reason logibraically, then whenever we find two valid inferences that have the same conclusion but whose premises differ only in that one is already in logibraically normal form while the other needs to be paraphrased into normal form, we should expect the one needing normalization to take more reckoning time than the first. If we dont reason logibraically, there is no reason to expect a difference in speed of inference
We may, for example, find that people take less time to infer No Saint is unkind [( -(S+(-K))] from Every saint is kind [-S+K] than from Saints are always kind, which is not in normal +/- categorical form. That would suggest that the direct +/- route is the default way we actually reason. More generally, it would support the view that people routinely transform much of what they hear and read into plus/minus normal form for efficient logibraic reckoning.
Cognitive psychology should be able to test the +/- hypothesis by comparing the speed of reasoning with sentences that are in logibraic normal form to the speed of reasoning with sentences whose natural constants cannot be directly transcribed logibraically.
Frege and Aristotle
14. Frege had an ingenious idea of how we could be reasoning. Though he believed that we could and should be reasoning the quantifier- bound-variable way, he probably knew better than to claim outright that we actually reason that way. There is no good argument for the view that we are actually reasoning the QV or should be reasoning the QV way. Its true that natural language lacks the quantifier-binding mechanisms of modern predicate logic but false that this renders it deductively weaker than MPL as a vehicle for reasoning, .
Because Aristotles Physics has no place in a scientific Physics Department; Newtons physics has legitimately and irrevocably supplanted Aristotles Physics. By contrast, Aristotles term logic is not unscientific and its almost universal replacement by predicate logic as the new logic of the schools is not sanctioned by austere science.. Pace Quine, reasoning with the quantifiers and bound variables of predicate logic is in no way more scientific than reasoning logibraically with the formatives of ordinary language. Quite the contrary: reasoning the +/- way is simpler, faster, as well as more natural (non-Procrustean) than reasoning the QV way..
15. Logical theory and the teaching of logic now face a future that will increasingly be shaped by the empirical findings of cognitive science of how we actually reason mentally Once cognitive psychologists become aware of the formal, +/-, powers of the logical constants of natural language, they will look for and find ways to empirically test the hypothesis that we reason intuitively, by unconsciously exploiting the +/- character of the natural logical constants. They will then be in a position to confirm or disconfirm the +/- hypothesis. I, of course, believe it will be confirmed.
The hypothesis that we normally reason by instinctively reckoning with the +/- character of the logical particles of natural language, will, I believe, be found to be the best explanation of our naturally fluent rationality, confirming the view of ratiocination that Thomas Hobbes had peremptorily announced four hundred years ago.
That finding should spark a successful back-to-Aristotle counter-revolution that will set aside the way logic has been taught for the last hundred years. The logic of natural language will again be standard logic. Quantifiers and bound variables will be things of the past. But this time around, teachers of logic and reasoning will be like teachers of ballet or tennis coaches who know what sinews and tendons are engaged in the movements of ballet or tennis, and who train their subjects to get the best out of their natural abilities by deliberately focusing on doing artfully what they unconsciously do artlessly.
The goal of logic teachers will be similarly modest: to improve the students natural ability to reason by doing what comes naturally. They will first inform the students that they already reason quite well by instinctively exploiting the +/- character of the logical words and particles of the natural language in which they think and reason. They will then show them how the logibraic method applies to a variety of arguments, and assign exercises that literally provide exercises in natural reasoning. In my experience, students become more adept, versatile and stronger reasoners by doing knowingly and consciously what they naturally do unknowingly and unconsciously.
References
Thomas Hobbes
The English Works of Thomas Hobbes, vol. I, W. Molesworth (ed), London, Kessinger. 1839
N.Chomsky
Rules and Representations, New York: Columbia University Press,1980,
Lectures on Government and Binding, Dordrecht: Foris, 1981
M.Dummett,
Frege Philosophy of Language , Harvard University Press (2nd ed). 1982)
William and Martha Kneale,
The Development of Logic, Oxford University Press, 1960,
Leibniz:
Logical Papers, G.H.R. Parkinson (ed, transl), Oxford: Clarendon Press, 1966
W.V. Quine,
Philosophy of Logic, Prentice Hall (1970
Quiddities, Cambridge, Harvard University Press, (1987),
Fred Sommers
The Calculus of Terms, Mind, 79. January 1970
The Logic of Natural Language (Oxford, Clarendon Press, 1982),
An Invitation to Formal Reasoning (Aldershot, Ashgate, 2000), (with G. Englebretsen.)
On a Fregean Dogma,Problems in the Philosophy of Mathematics, I. Lakatos (ed), Amsterdam, North-Holland Publ. 1967
Do We Need Identity? The Journal of Philosophy, 66, pp. 499-504. 1969
Predication in the Logic of Terms; Notre Dame Journal of Formal Logic, 31, pp. 106-126. 1990
The World, the Facts, and Primary Logic, Notre Dame Journal of Formal Logic, 34, pp.169-182. 1993
Ratiocination: An Empirical Account; Ratio, xxi 2, June 2008 pp.116-13
Thomas (TiP) ONiell,
See The Calculus of Terms, Mind, 79, pp. 1-39.(1970)
Cf. W.V. Quine, Philosophy of Logic ,Prentice Hall (1970), p35-36.
W.V. Quine, Quiddities, Cambridge: Harvard University Press, (1987), p. 158.
M.Dummett, Frege Philosophy of Language (Prefece to 2nd Edition) p. xxxii.
Harvard University Press (1982).
7>DEQR
{
Pt
QabĽ~hcydh(B*phhcydh{dB*phhcydhtB*phhcydh^*B*phh_h:zxhcydh(he/hcydh^*hzh$5:h^*5:h$5:hzh^*5:CJhzh^*5:hzh^*5 h=51ERSVW
5d "%
&((d]gde/d`gde/
$da$gde/dgde/pq)Rt~38"', %ʿʷʷʪʢʚʚʁyrhe/h^*he/B*phhcydh^*PJhq!JB*phhB*phhB*phh_B*phhcydh^*:B*phhu[B*phhcydhtB*phhcydh^*B*phhi|h_hu[hcydhtjhcydh^*0JUhhcydh^*+%&'01;Zht5=ADMN3jtwy&5R[fks־ֶֶ֩֞ƞƾֶ||h9>Rhcydhuohh@hi|hhcydh^*hcydh{dB*phhcydh^*B*H*phhB*phhB*phhB*phh@B*phhcydh^*B*phhi|B*phhe/B*phhB*
phhe/h0Snwmr####+#.#2###;$M$$$4%I%%%
&&&&''''((((r(h7NhcydhXJhcydh^*:h;hcydh^*>*hrJh:zxh^*6hcydh:zxh:zxhcydh^*PJhcydh^*B*phh $Kh^*6jhcydh^*0JUhcydh^*h9>Rh $Khi|6r(|((())) *
***M*N*O*]*c*l**********+++++++++E+O+g+h+q+r+~+++++++++++ĽĽĵٖٖh=Th}hZC6hjxhcydhZChrJh}h}h6h}h}6h}hM<6hcydhhcydhM<hcydh}hcydhEAh0Zh;hcydhRjhcydh^*0JUhcydh^*hGU3()+y../0@1@577
::;2>4@|@d]gde/d]^`gde/d]^gde/d]gde/d^gde/d`gde/dgde/d]^gde/++++,,,.-/--...&.1.?.y.~...N/////0000=1>1?1@1F1g1h1q111111ᰣַ~z~zvrvnh"h@Skhrphjhx!jhcydh^*0JU\hcydh^*B*\ph h?\jhcydh^*0JUhcydh^*hcydh^*\
h"PJhcydh~PJhcydhrp\ hrp\
hjxPJhcydh^*PJhcydhZC
hrJPJhcydhrJPJ)1111111111112
2222'202d2e2y2222222222222
333O3S333333333
44444s4}4444hh^*6h=ThZC6 h&6hVhTh&hTh&6hTh^*6hcydh"hcydhZChtuhh?hcydh^hcydh^*h"hm9Ch@SkhrphjhT!84444444444445 5$5%5.5?5J5`5a5p55555596;6H6t666~77777777888"8{88O9P9h9
:ͬhcydh^*B*phhhcydh^*PJhcydh^*B*ph3fjhcydh^*0JUhhA6hrJhA6hVhcydhAhrphm9ChAhcydhZChcydh^*hcydh"hcydh~h&2
::
:$:&::::;;;;
<b<c<.=0=3=4=5=6=7=q=s=x=->.>1>:>ܿtmiiei^iei^QMh0jhcydh^*0JUhcydhahT!hOihcydh18jhcydh^*0J56B*CJOJQJUaJph+hcydh^*56B*CJOJQJaJph/hcydh^*56B*CJOJQJ^JaJph)hcydh^*56:CJOJQJ^JaJhcydh^*H*hcydh^*hcydh^*B*ph"jhcydh^*0JB*Uph:>E>N>>>>>??0@3@4@Y@|@@@@@@@@@@A A
AA*A7AVAAAAAAAAA3BgBqBBBBBBDDDDDE-E.E9EME_E`EzEEխh1hcydhD0Dhcydh^*B*ph
h\.^PJhcydh^*H*hcydhIhOihcydh1h;Ch\.^hcydh^*PJhe/h5hh^*hcydh^*h0hcydh0;|@ADFHLh35hcydhp5hcydhIhcydh[j!hcydh^*:h1hcydhD0Dh
hcydh^*FL2LAL]LLLLLLLLLLLMMMMNNNNN"N(NXDXHXTXXXדּhcydhuD!H*hcydhYH*hcydhYhoh&hcydh4hcydhuD!hcydhuD!PJ
huD!PJhcydhuD!:huD!h{KChcydhhcydh^* jhcydh^*:XX^XdXrXzX|XXXXXXXXXXXXXXXXXXXXXXYYY Y"Y.YLYNY^Y`YYYYYYYYYŶ͕ͩ͆xxxxph&B*ph jhcydh^*B*phhcydh^*B*H*PJphh&B*PJphhB/B*PJphhcydh^*B*PJphhcydh^*B*PJo(phhB/B*phhcydhB/H* hB/H* hkH*hcydh^*B*H*phhcydhkH*hcydh^*B*ph,YYYYYYYZZZZZZ#Z$Z&Z'Z3Z4Z:Z@ZNZPZ[Z_ZgZlZqZuZ{Z}ZZZZZZZZZ[#[j[k[m[n[[[[[)\+\9\D\\\\\]];]ǿǵǮǧǠǘǧǧhcydhuD!H*hcydhhcydhYhcydh4hoh^*6:hcydh^*]hcydh^*huD!hB/hB/B*phhcydh^*B*phh&B*ph jhcydh^*B*ph;;]:^@^e^g^m^r^^^^___E_I_J_U_\___`_a_d_n_z_{______
`$`W`_`c```````````ǽzpzhB/hsw6PJ
h`'PJ h\ h:H\ hsw\ huD!\hcydhsw\
hswPJhcydh^*hB/hJ:PJhB/h^*:PJhB/h{KC:PJ
hB/PJhcydh[j!PJhcydhPJhcydhOFPJ
h:HPJhcydhYPJhcydh^*PJ,;]Y__xabbXcccc-dddef/fuffd]gde/`d]`gde/$d]a$gde/d]^gde/d]gde/d]gde/d]`gde/d]gde/`
aaaa a!a+a/a6a9aGaJaQaRa]a`ahakatauavawaxaaaaaaaaab+b2b3b5b9b=bGbKbRbUbcbfbmbnbobbbbbȚؚؚؚؚȔhcydh^*H*PJ
h:3VPJhcydh`'PJ
huPJhcydhPJ
hPJ
hUPJhcydhOFH*PJ
h`'PJhcydhcH*PJhcydh`'H*PJhcydhcPJhcydh^*PJ
hcPJ
h:HPJ3bbbbbbbbbbbbbbbbbbbbbbbbbccccc c!c'c*c.c/c8c;cPJ
h>PJ
hkTPJ
hnPJhcydhOFPJhcydh^*PJhcydhOFPJhcydhOFH*PJhcydh^*:PJh>h^*:PJhcydh^*H*PJhcydh^*PJhcydh^*>*PJ7ccccccccccccccccccccccccccddd!d$dGdJdKdMdRdUd_dbdkdnd|dddddddddddddee^eceeŹhuD!h^*6PJ
h^*PJhcydh^*>*H*PJhcydh^*>*PJhcydh^*:H*PJhcydh^*:PJhcydh^*H*PJhcydh^*PJhcydhJPJ
h>PJhcydh^*PJ
hkTPJ:eeeeeeeeeeeeeeeeeff:f=fIfLfSfXf[fiflfmfuff gg g!g6g7g;g*PJhuh^*>*PJS*hcydh^*>*PJhuh^*PJS*
huPJh:Hhu6H*PJh:Hh^*6:PJh:Hh^*6PJhcydh^*H*PJhcydh^*H*PJhcydh^*PJhcydhuH*PJ.f$gGgggMhxhhKiCjllnopusFulwd^`gde/d^`gde/dgde/$d]`a$gde/d]`gde/$d]a$gde/d]gde/.h/hlhqhxh~hhhhhhhhhhhGiJiXicidiiiliriiiiiij j'j(j@jBjCjkkkk(l,l-l1lllllӯhh&:PJ
h^*PJhcydh
;|PJ
hPJ
hZ]1PJ
hBWPJ
hswPJ
hQPJhcydhc=)PJhcydh1PJ
h5PJ
h1PJ jahcydha)PJhcydha)PJhcydh^*PJhcydhJPJ/lnnooooo;pDpqusys{ssssssss>tAtItJtUtXt`tattttttttttt!u$u(u+uƹzkkkkkkh0h0B*H*PJphhvCkB*PJphhd`B*PJphh]B*PJphh0B*PJphh@B*PJphh0h0B*PJphh0h^*B*PJphh0h&PJhcydh^*B*phhcydh^*>*hcydh^*hcydh^*PJhcydh^*B*PJph*+u6u9u=u@uBuCuuuiwjwlwswwwwwwwwwSx^xxxxxxx#yB*PJphhcydh^*PJh>hD:B*PJphh>h^*:B*PJphh.:B*PJphhcydh^*B*PJphhd`B*PJphhvCkB*PJph&jh0h00JB*PJUphh0h0B*H*PJphh0h0B*PJphlwsww!y$yYyyyy'zzzzz
{|'}]}}~Q~~~~68d`gde/
$da$gde/dgde/JyKy_y`ygyhymy}y~yyyyyyyyzzzzz!z&z'zEzIzzzzzzzzzzzz{{{{t{u{N|ȾȾⱾ⧾}hcydhmB*PJphhcydh
;|B*PJph jhcydh^*B*PJphh]*B*PJphhQhQB*PJphhQB*PJphh>B*PJph jchcydh^*B*PJphhcydh^*B*PJph jhcydh^*B*PJph+N|Q|X|[|l|o|t|x|||||||}}} }}}}}"}%}*}+}2}3}7}8}>}?}M}N}[}\}r}s}~~~"~0~3~9~<~C~F~L~O~Q~~~~~~~~䫞hcydhpB*PJphhcydh
;|B*PJphhcydhLB*PJph jhcydh^*B*PJphhcydhDB*PJphhcydhDB*H*PJphhcydh^*B*PJphhcydh^*B*H*PJph9~~~~~~~~~~~~~~~~~~~
"$&(,678T`廱 jhcydh^*B*PJphhcydhloB*PJphhloB*PJphhxXB*PJphhcydh^*B*PJph jhcydhG~B*PJphhcydhG~B*PJphhcydhLB*PJphhcydhpB*PJph2 Zg̀.:<IJNde+/=>RSfgr⒃⒃ttttdd j\hcydh^*B*PJphhcydh^*>*B*PJphhloh^*6B*PJphhloh^*6B*PJ]phhcydhloB*PJph jhcydh^*B*PJphhcydh^*B*PJ]phhloB*PJphhxXB*PJphhcydh^*B*PJph jhcydh^*B*PJph'ہ>{|}҂pHa߇ՈLԌl gde/]gde/dgde/dgde/rs{|Ђ҂6B̓Ճփ߃9:;MXƷ~sks`skXsksPHPsXhloB*phh~B*phh%B*phhcydhloB*phh-&B*phhcydh^*B*phhcydh^*PJh%h~6:B*PJphh%h#>6:B*PJphh%h^*6:B*PJphh%h%:B*PJphh%h^*:B*PJphhcydh#>B*PJphhcydh^*B*PJph j\hcydh^*B*PJphXZcmnpƄȄɄۄ,HMPSZq{~"#ݿշ~~~~qqc jhcydh^*B*phhcydh^*B*H*phhFt'hFt'B*H*phhcydh]*B*phhcydh\TfB*phhcydh^*B*phhcydh^*B*phhFt'hFt':h^*"jhcydh^*0JB*UphhFt'h-&hcydh^*hcydh^*B*phh%B*phh-&B*ph&#$(,/59&23=@GWŇއ߇ӈԈՈrxz~obhcydh^*B*^Jphhcydh^*B*H*^Jphh#B*phhcydh^*^Jhcydh^*PJhcydhPJhcydh:B*phhcydh^*:B*phhFt'B*phhcydhmB*ph jhcydh^*B*phhcydh^*B*H*phhcydhB*phhcydh^*B*ph&ʊЊҊԊ.0vx8np~ΌЌԌ܍ލ -:=BE̽̽xxhcydhB*phhcydhGB*ph&jhcydh^*0JB*U^Jph jhcydh^*B*phhcydh^*B*^Jphhcydh^*B*H*^Jphhcydh^*B*H*phhcydh^*:B*ph jhcydh^*B*phhcydh^*B*ph/ )~GLfʐːa͒ld]`gde/d]gde/d]gde/d]gde/d]gde/dgde/
gde/EHKNRV[_orw{ݎ'):=@BCELc
yęhQB*phh~B*phhcydhGB*H*phhcydhGB*phhcydhB*phh?B*ph jhcydh^*B*phhcydh^*B*H*phh%h^*:B*phhcydh^*B*ph9ɐʐːΐאېabő͒/69@ac"#`chkrҪvvvvhcydh^*B*H*phhcydh^*:B*ph2hcydh^*5:B*CJOJQJ\^JaJph/hcydh^*5B*CJOJQJ\^JaJphhcydhfPJhcydh^*hcydh^*PJ
h?PJhcydh^*B*
phh?B*phhcydh^*B*ph.rst|}єؔ!%+/15*Lpw{ߴߝߝߝ않zhB*phhQB*phhcydh#B*phh^*B*phhcydh^*B*H*phh%:B*ph jh%h%:B*phh%h^*6:B*phh%h^*6B*phhcydh^*:B*phhcydh^*B*phhDB*ph0 !%()klǛțۛ$(-䯥ݛݛݓݛ݄zhcydh^*:]hcydh^*:hcydh3g5hcydh^*]hG>
h
PJhcydh^*H*PJhcydh^*PJ jhcydh^*B*phhcydh^*B*H*phhQB*H*phhcydh^*hcydh^*B*phhQB*phh
B*ph0lEԚ+~ܧ1˱ù^gde/
X d]gde/d`gde/dgde/d]`gde/
d]gde/d]gde/$d]a$gde/-017;<@CDEJNQSTW]`afijlrwz{~Hdhmpӧӧۣۚ|hcydhiPJ]hcydh^*PJ]hcydh^*:PJhcydh^*PJh.hcydh3g5]hcydh^*:H*hcydh3g5:hcydh^*H*hcydh^*H*]hcydh^*]hcydh^*hcydh^*:hcydh^*:]hcydh^*:H*]0Ýɞ̞kn*+2lǠˠؠ٠"FJU\`xԜhcydh#>PJ
h?PJ
h=PJ
h=PJh=h^*6PJhcydh}PJ
h.PJhcydh^*B*PJph
hG>PJhcydhiPJhG>hi6PJh"h^*6PJhcydh^*PJ8EGHJT`hƢ=CadimnsxyCW38?@Bjky~ŦϦ٦XZ붰߶h%hH*PJ
hjT*PJhhhhPJ
h"PJ
hhPJ
hl+PJ
hPJh%hPJhcydh^*B*
PJphhcydh3NPJhX!h^*6PJ
h.PJ
hG>PJhcydh^*PJ
h=PJ
hX!PJ4Z{|ۧܧ)1256abߨ
$,Ūƪ̪٪@ANOSU`hѿѿѹѿѿѪѪјяѹяцhcydhPJhcydhePJjh0JPJU
hjT*PJhcydh1PJ
hGPJ
hj^PJhcydh}PJhcydh;0PJhcydh^*PJh%hhhh6PJ
hPJ
hhPJh%hPJ2hk{ëǫʫ2367ìŬƬ֬+3;Aѭխ֭߭%+0Ϲ{{ hy] hxYV] h,
]h,
hC~6] hj^]hcydhC]hcydhC~]hcydh]hcydh^*]hcydhC~h=h^*6hcydh}hcydh^*
hj^PJhcydhWPJhcydh1PJhcydh^*PJhcydh%\)PJ0018:;ͮӮ!CnǯC]kmnrv|Ұװ);dgy|Ǳȱ˱ٴǴǴǬǬǥǥǥǥҥhyh&hUw6hcydhUwhz[DhQ6hz[DhyhND6hcydhNDhcydhQh&hcydhhcydhChcydh^*hcydh^*]hcydhC]hcydh];˱ֱر[b%)!)ƶ*+¹ù*0I^ٽߨs]s+hcydh^*>*B*OJPJQJ]^Jph(hcydh^*>*B*OJPJQJ^Jphh!=vB*OJPJQJ^Jphh_HB*OJPJQJ^Jph%hcydh^*B*OJPJQJ^Jphjhcydh^*0JPJU
h,
PJ
hyPJhcydh;0PJhcydh^*PJhcydh^*:PJhcydhUw: hy:ùI?8bm5Pdd\$]Pgde/`d]`gde/d]gde/$d]a$gde/
^gde/^gde/d1$]^`gde/)>?ǿ -.2DܷܷzldzWzh_Hh^*6B*phhcydh^*:hcydh^*:B*]phhcydh^*B*phhz[Dh^*6hcydhUwhcydh^*PJhcydh^*%h>h^*B*OJPJQJ^Jph(hcydh^*B*OJPJQJ]^Jphh,
B*OJPJQJ^Jph%hcydh^*B*OJPJQJ^Jphh>B*OJPJQJ^Jph
!69@Rabvy|}
qthcydh^*PJ h/ j\hcydh^*h h>*hcydh^*>*hcydh^*H* jYhcydh^*hcydh^*hcydh^*:H*]hcydh^*:]hcydh^*B*]phhcydh^*B*phhcydhUwB*ph3 "4569<HKfiz}$:ƼθδμμέΜhcydhX<hcydh3Nhcydh3N:hcydh^*:hcydh^*PJhh^*hhz[D j\hcydh^*hcydh^*H*hcydh^*hcydh^*>*hcydh^*>*H*hcydh/H*hcydh/h/
h/PJ 1$z: Ub$
) da$gde/
) 6d`6gde/
) dgde/
$da$gde/$Pd]P^a$gde/P]Pgde/dd\$gde/
(Saghjrxzbjkﱸﱸwhcydh^*:^J
h/:^J
h/^Jhcydh^*^Jhcydh?BYPJhcydh^*PJhcydh^*:hcydh?BYhcydh^*hcydh3N>* j&hcydhX< j$hcydhX<hcydhX<PJQJo(hcydhX<H*hcydhX< j"hcydhX<* ADJMFIPSTVbejn8T^ Abk~߫ߒ}hcydh^*PJhR>B*]^Jphhcydh^*B*]^JphhR>B*^Jphh/B*^Jphh/:B*^Jphhcydh^*:B*^Jphhcydh^*B*H*^Jphhcydh^*B*^Jphhcydh^*^Jhcydh^*H*^J-JL*FGPd]P^`gde/
Pd]P^`gde/Pd]P^gde/P d]P^ gde/dgde/$
) da$gde/ln,.JLTYFGKSHV?C}s}s}chcydh^*5CJOJQJaJ j\hcydh^*hcydh^*hcydh^*5CJaJhcydh^*OJQJaJhcydh^*:OJPJQJaJhcydh^*OJPJQJaJh%RB*PJphhiUB*PJphhR>B*phhcydh^*B*PJo(phhcydh^*B*phhcydh^*B*PJph&GH>?CCd]gde/
@d]^@gde/
@d]^@gde/$
@d]^@a$gde/
@d]^gde/
@d]^@gde/dgde/ :;PR~Wz{*3JjY!Tjk$9=>?@{hcydh^*B*
PJphhcydh^*B*
PJphhcydh^*B*
PJ
phhcydh^*PJ
j$hcydh^* j~hcydh^*hcydh^*H*hcydh^*5CJaJhcydh^*>*hcydh^*]hcydh^*: j\hcydh^*h%Rhcydh^*0{,
Yi!T@d]^@gde/@d]^@gde/d]gde/
d]gde/
Pd]^Pgde/
@d]^@gde/
@d]^@gde/;j{|de_`@]^@gde/
0@d*$]^@gde/
0d*$]^`gde/
0d*$]^`gde/@d]^@gde/ZXYtuABt9gde/@]^@gde/
0@d*$]^@gde/@d]^@gde/9??51CaPPd]Pgde/p`d]p^`gde/0d]^0gde/0d]^0gde/@0d]@^0gde/P0d]P^0gde/ d] gde/dgde/d]gde/
@AEFGRS{ade=PUVjnt盼Ð~u~uhcydhIKPJ
hcydh^*PJ
hcydh^*PJhcydh?BYB*phhcydh?BYhcydh^*\hcydh^*S* j\hcydh^*hcydhIKhcydh^*B*ph jDhcydh^*hcydhhcydh^*>*hcydh^*hcydh^*PJhcydh^*PJ.P=QVWNvW$d]a$gde/d]gde/d]gde/$
!d]a$gde/d]gde/"#34?@`ahi "26BFҗҡhcydh^*H*PJ j$hcydh^*PJ
hcydh^*OJPJQJ^Jhcydh^*H*PJ
j"hcydh^*PJ
hcydh^*PJhcydhIKPJ
hcydh^*H*PJ
hcydh^*PJ
hcydhR
PJ
;
*DHJRx*+PQSVWbcst̤̏wmhcydh-uCH*PJ
j\hcydh^*PJ
hcydhR
>*PJ
S*hcydh-uCPJ
hcydhR
>*H*PJ
hcydhR
>*PJ
hcydh^*>*PJ
hcydh^*PJ
S*hcydhR
H*PJ
hcydhR
PJ
hcydh]mPJ
hcydh?BYPJ
hPJ
hcydh^*H*PJ
hcydh^*PJ
*1?@BF)09dù{reXeNDhcydh]mPJ]hcydh^*PJ]hcydhq^B*PJphhcydhB*PJphhcydhPJ hq!J:
hq!JPJhcydh^*:B*PJphhcydh]mB*PJphh;B*PJphhcydh^*B*PJphhB*PJphhcydh^*B*
PJph3hcydh^*PJhcydh^*PJ
hcydh-uC>*PJ
hcydh-uCPJ
hcydh-uCH*PJ
@<pZrA+;Oz(
{od]gde/
!d]`gde/$d]a$gde/0d]0gde/d`gde/dgde/d]gde/46LPX'(ABXY@:;OPQRνǟzhcydh^*hcydh^*:hcydh^*B*
PJphhcydh^*PJ jahPJ]hcydhf>PJ]hcydh]mPJ]hcydh]mPJ]
hPJ]hcydh^*PJ]hcydh^*PJ]o(
h_HPJ]hcydhq^PJ]hcydh^*PJ].RS'1OTu:>Yvxyz6<BCENXar
h8SNPJ
hmKxPJ
hPJ
hcydhDyPJ
hcydhisPJ
hiUPJ
hcydh-PJ
hcydh^*PJ
h_HPJ
jhcydh^*0JUhhUV-h-6h,Jhcydhhcydh-hcydh^*]hcydh^*hq!J2!+13=>]lmuv # ( + , 0 b
k
l
v
)*2r
žž沬۔hcydhO2hThcydhHhOhcydh^*PJ
hq!JPJhcydh^*\h,Jhcydh-hcydhishhcydh&h8SNhiUhcydhDyhcydh^*hiUh^*6hiUh^*6PJ
5
[i
iu\_ijq);<no}thz5CJaJh(#hzCJaJh(#hz:CJaJhT:CJaJh
j:CJaJ hT:hcydh^*:hz;hcydhHhcydh(hz;h8SN6 h06hz;hW6hz;h^*6h8SNh0hOhcydh^*h75hcydhW)omnxQRn"$
+T;Kd1$]^`a$gde/dgde/$
T;Kd]a$gde/ $
T;Kd1$]^`a$gde/
@dgde/dgde/8mnpxyϿhYG8,hzCJPJ^JaJh(#hzCJPJ^JaJ"hV
hz>*CJPJ]^JaJh>*CJPJ]^JaJ/hV
hz>*B*CJOJQJ]^JaJph,hV
hz>*B*CJOJQJ^JaJph)h(#hzB*CJOJQJ^JaJph%h(#hzB*CJOJQJaJphhV
B*CJOJQJaJphh(#hzCJaJhV
hz>*CJaJhV
hz5:>*CJaJhz5:CJaJ
$%'CEQRmno殟lYN@Nh(#hzCJ]^JaJhV
CJ]^JaJ$h(#hzCJOJPJQJ^JaJ*hV
hz>*CJOJPJQJ]^JaJhV
hz>*CJOJQJaJhz5CJOJQJaJh(#hzCJOJQJaJhV
CJOJQJaJh(#hzCJH*aJhV
CJaJhV
hz>*CJaJh(#hz:CJaJh(#hzCJaJh(#hV
CJPJ^JaJ@sZ'x}gd^*$0d]^`0a$gde/$d]a$gde/
$da$gde/dgde/d]gde/dgde/'(@AKLMm=stuѽїƏƃƃxjƏbj^Wh(#hzhhCJaJh(#hz>*CJ]aJh>*CJ]aJh(#hz:CJaJh
jCJaJhV
hz>*CJaJh(#hzCJ]aJhV
hz>*CJ]aJhV
CJ]aJh(#hzCJaJhV
CJaJhV
CJ^JaJh(#hzCJ^JaJhV
hz>*CJ]^JaJ uv
'(,-CZ]u&')+>?@ALQ˸ˬø˸˔ˌø{peeehzhCJaJhzh
jCJaJhzCJaJh
j>*CJaJhiCJaJhihz>*CJaJhihz6CJaJh
jh
j6CJaJhzhzCJaJhCJaJh
jCJaJhihzCJaJ h5 h
j5hzhz5hzhzh(#hz:'QTUZacwxy|}~*+8Byzøwii][Uhe/hT!CJH*aJhe/hT!6CJ]aJhe/hT!6:CJaJhe/hT!:CJaJhe/hT!CJaJ!jhe/hT!0JCJUaJhh-,CJaJh(#hzCJaJhzhzCJaJhiCJaJhzh
jCJaJhih
j>*CJaJh
jCJaJh(#hCJaJ#*y'enoqr"<Pd]Pgd?$0]^`0a$gd@]gd^*$a$gd^*$a$gd^*$
+T;K1$]`a$gd^*]gd^*gd^*Dummett, ibid. p.20
See below, 3.2
The English Works of Thomas Hobbes, vol. I, W. Molesworth (ed), London, Kessinger. 1839, p.3.
William and Martha Kneale, The Development of Logic, Oxford University Press, 1960, p. 511.
Leibniz: Logical Papers, G.H.R. Parkinson (ed, transl), Oxford: Clarendon Press, 1966, p. 3. iii
Cf. Fred Sommers The Logic of Natural Language (Oxford, Clarendon Press, 1982), An Invitation to Formal Reasoning (Aldershot, Ashgate, 2000), (with G. Englebretsen.) and Predication in the Logic of Terms, Notre Dame Journal of Formal Logic 31 (1990): 10626
However Leibnizs idea of regarding propositions as a kind of term could properly ascribe a semantic primacy to the logic of terms if we understand propositions to be about the world as well as about things in the world, p saying of the world that is a p-world (i.e., a world in which p), not-p saying of the world that it isnt a p-world, p&q saying that it is both p-world and a q-world, if p then q; saying that isnt a p world and a not-p world cf. The World, the Facts, and Primary Logic, Notre Dame Journal of Formal Logic, 34, (1993), pp.178ff. and Invitation to Formal Reasoning, (2008), pp. 201-209.
The approach here used to determine the +/- character of some and every is similar to the one taken in propositional logic when one starts with & and ( as primitive sentential operators and defines other sentential operators such as ", v, and a".
The equivalence of No S is P and not: some S is P is logibraic: -(S+P) = def. -(+S+P).
The Calculus of Terms, Mind, 79. January 1970
N.Chomsky, Rules and Representations, New York: Columbia University Press,1980, p. 165.
7N. Chomsky, Lectures on Government and Binding, Dordrecht: Foris, 1981, pp. 34f.
See above, 1.2, where Quine and Dummett are cited saying that predicate logic facilitates inferences that a traditional logic of natural language cannot facilitate.
PAGE
PAGE 20
PAGE
PAGE 20
!&'(efz9[pq}~±zll`The/hT!:CJaJhe/hT!CJ]aJhe/hT!>*CJ]aJhe/hT!6CJ]aJ$he/hT!CJOJPJQJ^JaJ*he/hT!6CJOJPJQJ]^JaJ he/hT!CJOJQJ^JaJ)jhe/hT!0JCJOJQJUaJhe/hT!CJPJaJ!jhe/hT!0JCJUaJhe/hT!CJaJ "$:<>@DNt誝ˎssYD)he/hT!B*CJOJQJ^JaJph2jhe/hT!0JB*CJOJQJUaJphhe/hT!:CJaJhe/hT!B*CJaJphhe/hT!B*CJaJphhe/hT!CJPJaJ$he/hT!CJOJPJQJ^JaJ j~he/hT!CJaJ!jhe/hT!0JCJUaJhe/hT!>*CJaJhe/hT!CJaJhe/hT!6CJaJ<>/,-/0235689B&`#$gdTC$
+T;K 6]^ `6a$gd^*$
+T;K 61$]^ `6a$gd^*gdgd^*/0<^+,-.0134679:@ABDEKLNOķ|xpxpxpxpxf`f`\f`fQfhe/0JmHnHuhT!
hT!0JjhT!0JUjhyUhy!jhe/hT!0JCJUaJhe/hT!CJaJhe/hT!B*CJaJphhe/hT!6CJPJ]aJhe/hT!CJPJaJhe/hT!CJH*PJaJ)he/hT!B*CJOJQJ^JaJph/he/hT!6B*CJOJQJ]^JaJphBCDPQR[\]ijklmdgde/h]hgdTC&`#$gdTC&`#$gdTCh]hgdTC
OPRSYZ[]^deghiklmhh-,CJaJhyhe/0JmHnHujhT!0JUhT!
hT!0J,1h/ =!"#$%^ 666666666vvvvvvvvv666666>6666666666666666666666666666666666666666666666666hH6666666666666666666666666666666666666666666666666666666666666666662 0@P`p2( 0@P`p 0@P`p 0@P`p 0@P`p 0@P`p 0@P`p8XV~_HmH nH sH tH @`@uD!NormalCJ_HaJmH sH tH Z@Z^* Heading 1$<@&5CJ KH OJQJ\^JaJ f@f^* Heading 2$<@&)56B*CJOJQJ\]^JaJph333`@`^* Heading 3$<@B*CJOJ
QJ
\^J
aJph333J@J^* Heading 4$<@&5CJ\aJH@H^* Heading 6
<@&5CJ\aJDA`DDefault Paragraph FontRiRTable Normal4
l4a(k (No Listhh^*Heading 3 Char35B*CJOJ
QJ
\^J
_HaJmH ph333sH tH ^R@^^*Body Text Indent 2hdx^hB*\ph333jj^*Body Text Indent 2 Char$B*CJ\_HaJmH ph333sH tH 4 @"4^*Footer
!F1F^*Footer CharCJ_HaJmH sH tH B@BB^*
Footnote TextB*\ph333`Q`^*Footnote Text Char$B*CJ\_HaJmH ph333sH tH Z@b^*-Plain Text,Char Char Char Char,Char Char Char&Z1$7$8$H$]^`ZB*OJQJ\^Jph333q^*=Plain Text Char,Char Char Char Char Char,Char Char Char Char10B*CJOJQJ\^J_HaJmH ph333sH tH @@@^*Comment TextB*\ph333^^^*Comment Text Char$B*CJ\_HaJmH ph333sH tH D&@D^*Footnote ReferenceH*^JHC@H^*Body Text Indenthx^h<P@<^*Body Text 2dxH2@H^*List 2^`OJQJ\aJ.)@.^*Page Number4@4^*Header
!PK![Content_Types].xmlN0EH-J@%ǎǢ|ș$زULTB l,3;rØJB+$G]7O٭V$!)O^rC$y@/yH*)UDb`}"qۋJחX^)I`nEp)liV[]1M<OP6r=zgbIguSebORD۫qu gZo~ٺlAplxpT0+[}`jzAV2Fi@qv֬5\|ʜ̭NleXdsjcs7f
W+Ն7`gȘJj|h(KD-
dXiJ؇(x$(:;˹!I_TS1?E??ZBΪmU/?~xY'y5g&/ɋ>GMGeD3Vq%'#q$8K)fw9:ĵ
x}rxwr:\TZaG*y8IjbRc|XŻǿI
u3KGnD1NIBs
RuK>V.EL+M2#'fi~Vvl{u8zH
*:(W☕
~JTe\O*tHGHY}KNP*ݾ˦TѼ9/#A7qZ$*c?qUnwN%Oi4=3N)cbJ
uV4(Tn
7_?m-ٛ{UBwznʜ"ZxJZp;{/<P;,)''KQk5qpN8KGbe
Sd̛\17 pa>SR!
3K4'+rzQ
TTIIvt]Kc⫲K#v5+|D~O@%\w_nN[L9KqgVhn
R!y+Un;*&/HrT >>\
t=.Tġ
S; Z~!P9giCڧ!# B,;X=ۻ,I2UWV9$lk=Aj;{AP79|s*Y;̠[MCۿhf]o{oY=1kyVV5E8Vk+֜\80X4D)!!?*|fv
u"xA@T_q64)kڬuV7t'%;i9s9x,ڎ-45xd8?ǘd/Y|t&LILJ`& -Gt/PK!
ѐ'theme/theme/_rels/themeManager.xml.relsM
0wooӺ&݈Э5
6?$Q
,.aic21h:qm@RN;d`o7gK(M&$R(.1r'JЊT8V"AȻHu}|$b{P8g/]QAsم(#L[PK-![Content_Types].xmlPK-!֧60_rels/.relsPK-!kytheme/theme/themeManager.xmlPK-!0C)theme/theme/theme1.xmlPK-!
ѐ' theme/theme/_rels/themeManager.xml.relsPK]
p!'=)/O1
23-6gj$zOp@
fnWho3 @ %0>>>A%r(+14
::>ELOSXXY;]`bce.hl+uJyN|~rX#Er-Zh0˱@R
uQOm(|@6R;]flw lù9Po<Bm!%,07:A!!!!G
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
NNNNNNNNNN N
NNN
NE++>>'?D3DlYl|ӄ`FǊYboԳYiy9MM'YYeem7SA
!"#$%&'()*+-,./0213546789:<;=>?@ABCDEFI//>?*? D6DoYl|ׄcIʊ\erس\l|FSS&11bblww@VA
!"#$%&'()*+-,./0213546789:<;=>?@ABCDEF9*urn:schemas-microsoft-com:office:smarttagsStateBE*urn:schemas-microsoft-com:office:smarttagscountry-region9F*urn:schemas-microsoft-com:office:smarttagsplace8!*urn:schemas-microsoft-com:office:smarttagsCity?G*urn:schemas-microsoft-com:office:smarttagsstockticker=*urn:schemas-microsoft-com:office:smarttags PlaceName=*urn:schemas-microsoft-com:office:smarttags PlaceTypeMGFEGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGF!FF!FFF!FFF!F!FGFFF!GGGGST
u v
IJ~z{))//C >AST/~ E
R
bk=BZg6E,
(ch#5& 9 !! ""^"i"""##N$X$$$_&j&&&''(()))*++q-v--...//W/c/00g1p1J2d2d3f3E6T6667788:2:;;<$<< =>>rB{BCDvD|DEEEEFFFFFF.G:GaGeGGGHHCJFJJJJ KHKPKzO~OOOPPiRvRiStSATITTTTTXW_WXXXXXX6Y@mq/1#:@ px0;/n .@Gbl6EHJ~bli ,`luIKakM P +
6
68v >A33333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333WWGGLL00r#~#$$C_C_tuu W
.
/
>
''(>aprXYabgh "%-0;AWWGGLL00tu >A<V>R
V
ql+M5?am1==Tu[]./XJQ! uD![j!x!#-&C&Ft'c=)%\)a)@*jT*UV-B/[0;01Z]1O2BX435753g5ZC7wh7R8Z#9M>:g:{;<M<X<j<=e>?@@EAkaACm9C{KCZC-uCDD0Dz[DOF:H=H_HJq!J&J,J $Ko0KIK@rM8SNqyOP3Q%R9>RSTSkTqUV:3VxYVXxX?BY0Z7]q^\.^j^cyd{d\TfiOi9mikvCk@Skm]mo$ouorpis!=v*wswmKx:zxuzz
;|i|{|~C~/#>vbtQG~(sz;Lf>C3NT)G$lo0,
T?A^*>%G_""=}(jD>yu
1}v-,G>Uw-iTCJL&I;jx~p^;Ch4j]*
}Wd`;BWp5Y|NDrJ`O&0c&xVn.e=O}n
Z&9R>iUzHy`'
tu
jT!Eq'Dy
&e/0X!"5a:7NWQfURGU+puw@%E%F%N%O%%%%%]]EE
@@N@X@@@@@@@@UnknownG*Ax Times New Roman5Symbol3.*Cx ArialC (PMingLiUe0}fԚ;|i0Batang9MT Extra7
Sylfaen5..[`)Tahoma;Wingdings;= |8SimHeiўSOI.??Arial Unicode MSG=
jMS Mincho-3 fgA$BCambria Math7@Cambria?= *Cx Courier New"qho c Ǔ!,'I),'I)!243qHP ?+p2!xx> 9*16*2012 HOW WE NATURALLY REASON FredTamlerOh+'0, <H
ht
@ 9*16*2012 HOW WE NATURALLY REASON FredNormalTamler147Microsoft Office Word@@qϘ@~@@5W),'I՜.+,00hp
Microsoft? 9*16*2012 HOW WE NATURALLY REASON Title
!"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~
"#$%&'(-Root Entry FUJ#W/1TableQWordDocument.SummaryInformation(DocumentSummaryInformation8!CompObjr
F Microsoft Word 97-2003 Document
MSWordDocWord.Document.89q**