Introduction The definition of logic



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Aristotle's Logic



Aristotle, marble copy of bronze by Lysippos. Louvre Museum, France




  1. Introduction

  2. The definition of logic

  3. Aristotle’s logic

    1. The oragon

      1. Categories (Latin: Categoriae)

      2. On Interpretation (Latin: De Interpretatione)

      3. Prior Analytics (Latin: Analytica Priora)

3.1.4 Posterior Analytics (Latin: Analytica Posteriora)

      1. Topics (Latin: Topica)

3.1.6 On Sophistical Refutations (Latin:De Sophisticis Elenchis)

    1. Syllogism

    2. Definition

    3. Demonstration

  1. Conclusions

  2. References



1. Introduction
Aristotle (Greek: Αριστοτέλης Aristotelēs) lived in 384 BC – March 7, 322 BC. He was an ancient Greek philosopher, student of Plato and teacher of Alexander the Great. He wrote many books about physics, poetry, zoology, biology, rhetoric, government, and logic. Aristotle is one of the most important figures of the western philosophy and science. He systemised philosophy and scientific way of thinking as a whole and thus is considered the father of many sciences.
When Aristotle was 18 years old he joined Plato’s Academy in Athens and worked there nearly twenty years of his life. After that he worked as the teacher of Alexander until in 336 BC he returned to Athens to start up his own school lykeion. The most significant of his writings are from the time after returning to Athens. Aristotle died in 332 BC in Euboia where he escaped the restless political times after the death of Alexander the Great.
Aristotle's logic, especially his theory of the syllogism, has had an unparalleled influence on the history of Western thought. His logical works contain the earliest formal study of logic that we have. Together they comprise a highly developed logical theory, one that was able to command immense respect for many centuries: Kant has even said that nothing significant had been added to Aristotle's views in the intervening two millennia. And Jonathan Lear has said, "Aristotle shares with modern logicians a fundamental interest in metatheory": his primary goal is not to offer a practical guide to argumentation but to study the properties of inferential systems themselves.

2. The definition of Logic

Logic, from Classical Greek λόγος (logos), means originally the word, or what is spoken, (but comes to mean thought or reason). The exact definition of logic is a matter of controversy among philosophers, but It is often said to be the study of arguments. However the subject is grounded, the task of the logician is the same: to advance an account of valid and fallacious inference to allow one to distinguish good from bad arguments.

Traditionally, logic is studied as a branch of philosophy. Since the mid-1800s logic has been commonly studied in mathematics, and, even more recently, in computer science. As a science, logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and through the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as study of fallacies and paradoxes, to specialist analyses of reasoning such as probably correct reasoning and arguments involving causality.
For Aristotle the name logic is unknown, his own name for this branch of knowledge, or at least the study of reasoning is ‘analytics’, which primary refers to to the analysis of reasoning into the figures of syllogism, but into it may be included the analysis of the syllogism into propositions and of the proposition into terms. The term logic he reserved to mean dialectics.

3. Aristotle’s logic
Aristotle divides sciences in three groups:


  1. theoretical

  2. practical

  3. productive

He sees that the primary purpose of all of them is to know, but knowledge , conduct and the making of beautiful or useful objects become the ultimate objects.

If one would try to enter logic into these groups, it would respectfully belong to the group of theoretical sciences, but according to Aristotle the only theoretical sciences are mathematics, physics and theology or metaphysics, and logic cannot belong to any of these. Thus logic is not a substantive science but a part of general culture which anyone should undergo before he studies any science, and which alone will enable him to know for what sorts of proposition he should demand proof and what sorts of proof he should demand for them.
3.1 The Organon
Aristotle failed to understand the importance of his written work for humanity. He thus never published his books, except from his dialogues. Most of Aristotle's work is probably not authentic, since students and later lecturers most likely edited it. Aristotle's works on logic, are the only significant works of Aristotle that were never "lost"; all his other books were "lost" from his death, until rediscovered in the 11th century.
The Organon was used in the school founded by Aristotle at the Lyceum, and some parts of the works seem to be a scheme of a lecture on logic. So much so that after Aristotle's death, his publishers (e.g. Andronicus of Rhodes in 50 BC) collected these works. In these works we can find the first ontological category theory (relevant in some branches of intensional logic), the first development of formal logic, the first known serious scientific inquisitions on the theory of (formal and informal) reasoning, the foundations of modal logic, and some antecedents of methodology of sciences.

The logical works of Aristotle were grouped into six books by he ancient commentators at about the time of Christ. They all go under the title Organon ("Instrument"):




  1. Categories

  2. On Interpretation

  3. Prior Analytics

  4. Posterior Analytics

  5. Topics

  6. On Sophistical Refutations

The order of the books (or the teachings from which they are composed) is not certain, but this list was derived from analysis of Aristotle's writings. There is one volume of Aristotle's concerning logic not found in the Organon, namely the fourth book of Metaphysics.
The title Organon reflects a much later controversy about whether logic is a part of philosophy (as the Stoics maintained) or just a tool used by philosophy (as the later Peripatetics thought); calling the logical works "The Instrument" is a way of taking sides on this point. Aristotle never uses this term himself.

3.1.1 Categories (Latin: Categoriae)


The Categories introduces Aristotle's 10-fold classification of that which exists. The book begins with by consideration of linguistic facts; it distinguishes ‘things said without combination’ from *things said in combination’. I.e. words and phrases such as ‘man’, ‘runs’, ‘in the Lyceum’ from propositions such as ‘man runs’. ‘words uncombined’ are said to mean one or other of the following things. These categories consist of:
Substance (e.g. man),

Quantity (e.g. three cubits long),

Quality (e.g.white),

Relation (e.g. double),

Place (e.g. in the lyceum),

Date (e.g. yesterday),

Posture(e.g. sits),

Possession (e.g. is shod),

Action (e.g. cuts),

Passion (e.g. is cut).


Subjects and predicates of assertions are terms. A term (horos) can be either individual, e.g. Socrates, Plato or universal, e.g. human, horse, animal, white. Subjects may be either individual or universal, but predicates can only be universals: Socrates is human, Plato is not a horse, horses are animals, humans are not horses.

The word universal (katholou) seems to be an Aristotelian coinage. Literally, it means "of a whole"; its opposite is therefore "of a particular" (kath’ hekaston). Universal terms can properly serve as predicates, while particular terms cannot.

This distinction is not simply a matter of grammatical function. We can readily enough construct a sentence with "Socrates" as its grammatical predicate: "The person sitting down is Socrates". Aristotle, however, does not consider this a genuine predication. He calls it instead a merely accidental, incidental (kata sumbebêkos) predication. Such sentences are, for him, dependent for their truth values on other genuine predications (in this case, "Socrates is sitting down").

Consequently, predication for Aristotle is as much a matter of metaphysics as a matter of grammar. The reason that the term Socrates is an individual term and not a universal is that the entity which it designates is an individual, not a universal. What makes white and human universal terms is that they designate universals.

These categories appear in almost all of Aristotel’s works, but he is not very consistent about the number of the categories, for example posture and possession only reappear a few times and in fact it can be said that Aristotle later come to the conclusion that posture and possession are not ultimate, unanalysable notions.

“Of things said without any combination, each signifies either substance or quantity or quality or a relative or where or when or being-in-a-position or having or doing or undergoing. To give a rough idea, examples of substance are man, horse; of quantity: four-foot, five-foot; of quality: white, literate; of a relative: double, half, larger; of where: in the Lyceum, in the market-place; of when: yesterday, last year; of being-in-a-position: is-lying, is-sitting; of having: has-shoes-on, has-armor-on; of doing: cutting, burning; of undergoing: being-cut, being-burned”. (Categories 4, 1b25-2a4, tr. Ackrill, slightly modified)



3.1.2 On Interpretation (Latin: De Interpretatione)


On Interpretation introduces Aristotle's conceptions of proposition and judgement, and treats contrarieties between them. It contains an account of simple sentences (what later come to be called "propositions." And deals with quantifiers ("all," "some," "none") and their logical relations. As such, it contains Aristotle's principal contribution to philosophy of language.
In On Interpretation Aristotel traces the possible oppositions between propositions. He takes the existential judgment as the primary kind and argues that to every affirmation there corresponds exactly one denial such that that denial denies exactly what that affirmation affirms. The pair consisting of an affirmation and its corresponding denial is a contradiction (antiphasis). In general, Aristotle holds, exactly one member of any contradiction is true and one false: they cannot both be true, and they cannot both be false. An example of possible varieties:
A (i.e. some) man exists.

A man does not exist.

A not-man exists.

A not-man does not exist.


3.1.3 Prior Analytics (Latin: Analytica Priora)


The Prior Analytics introduces his syllogistic method, argues for its correctness, and discusses inductive inference.
Aristotle's anaylsis of the simplest form of argument: the three-term Syllogism. The standard example in philosophy has always been:

      • All men are mortal. [Premise1 in the form: All B's are C's.]

      • Socrates is a man. [Premise 2 in the form: (All) A is B.]

      • Therefore, Socrates is mortal. [Conclusion in the form: All A's are C's.]

This example is somewhat misleading, despite the fact that it is the standard one, since it treats a proper name ("Socrates") as a term (or class name.) One of the fundamental departures of modern (19th & 20th Century C.E.) symbolic logic is that it treats sentences about individuals differently from the way it treats sentences about classes. But with this first figure form of the syllogism Aristotle arrives at a clear and explicit distinction between truth and validity, where the latter is a property of argument forms. (If the premises of a valid argument are true, the conclusion must be true.)

Syllogism will be discussed more in the later parts.


3.1.4 Posterior Analytics (Latin: Analytica Posteriora)


The Posterior Analytics discusses correct reasoning in general. The book is for the most part occupied with demonstration, which presupposes the knowledge of first premises not themselves known by demonstration.

Here Aristotle identifies the valid forms of the syllogism. He identifies the formal key to valid syllogistic forms in the middle term (identified in the form above by "B.") The middle term must be "distributed" (quantified) if an argument form is to be valid. (Of course this is a necessary but not sufficient condition. Not every argument form with a distributed middle term is valid.)


For a syllogism to achieve the status of a demonstration the argument form must be valid and the premises must be true, and must be known to be true unconditionally. The premises must, therefore, either be themselves derivable as conclusions of other demonstrations following necessarily from necessarily true premises or they must be known by "intuition".

At the end Aristotle talks about the question how these are known. What is the faculty by which we know them: and is the knowledge acquired, or is it latent in us from the beginning of our lives? It would be hard to imagine, that knowledge would be there from the beginning on without us knowing it but just as hard is to imagine, that it would be acquired without any pre-existing knowledge. Aristotle sees that these two problems can be passed by assuming that some humble faculty of knowledge is a starting point from which on we can further develop our knowledge. Such faculty is perception, the discriminative power that is inborn in all animals. From perception to knowledge the first stage is memory, the remaining of the percept when the perception is over. And the second stage is experience or framing, on the bases of repeated memories of the perceived things.


Later he raises the question whether defining and demonstrating can be alternative ways of acquiring the same knowledge. His reply is complex:

  1. Not everything demonstrable can be known by finding definitions, since all definitions are universal and affirmative whereas some demonstrable propositions are negative.

  2. If a thing is demonstrable, then to know it just is to possess its demonstration; therefore, it cannot be known just by definition.

  3. Nevertheless, some definitions can be understood as demonstrations differently arranged.

As an example of case 3, Aristotle considers the definition "Thunder is the extinction of fire in the clouds". He sees this as a compressed and rearranged form of this demonstration:

  • Sound accompanies the extinguishing of fire.

  • Fire is extinguished in the clouds.

  • Therefore, a sound occurs in the clouds.

We can see the connection by considering the answers to two questions: "What is thunder?" "The extinction of fire in the clouds" (definition). "Why does it thunder?" "Because fire is extinguished in the clouds" (demonstration).
Definition and demonstration will be discussed more later.

3.1.5 Topics (Latin: Topica)


The Topics treats issues in constructing valid arguments, and inference that is probable, rather than certain. It is in this treatise that Aristotle mentions the idea of the Predicables, which was further developed by Porphyry and the scholastic logicians.
The "Topics" identify strategies and techniques Aristotle identified for constructing valid arguments. The general name for this kind of reasoning is dialectic. Dialectic begins with a opinion or belief, examines, criticizes, and revises that opinion/belief in the light of reason and other things known or believed to be true, in order to establish scientifically known premises which can then be used in demonstrations to generate syllogistically the truth of conclusions derived. Aristotle's account of dialectic owes much to the "method of hypotheses" in Plato's Phaedo.
“We should distinguish the kinds of predication (ta genê tôn katêgoriôn) in which the four predications mentioned are found. These are ten in number: what-it-is, quantity, quality, relative, where, when, being-in-a-position, having, doing, undergoing. An accident, a genus, a peculiar property and a definition will always be in one of these categories”. (Topics I.9, 103b20-25)


3.1.6 On Sophistical Refutations (Latin:De Sophisticis Elenchis)


On Sophistical Refutations deals with a variety of bad or invalid argument forms: "fallacies" and gives a treatment of logical fallacies, and provides a key link to Aristotle's work on rhetoric.
3.2 Syllogism
The most famous achievement of Aristotle as logician is his theory of inference, traditionally called the syllogistic (though not by Aristotle). That theory is in fact the theory of inferences of a very specific sort: inferences with two premises, each of which is a categorical sentence, having exactly one term in common, and having as conclusion a categorical sentence the terms of which are just those two terms not shared by the premises. Aristotle calls the term shared by the premises the middle term (meson) and each of the other two terms in the premises an extreme (akron). The middle term must be either subject or predicate of each premise, and this can occur in three ways: the middle term can be the subject of one premise and the predicate of the other, the predicate of both premises, or the subject of both premises. Aristotle refers to these term arrangements as figures (schêmata): Syllogism is defined by Aristotle as a 'discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so'.
The propostions of a categorical syllogism must between them employ exactly three terms, each term appearing twice as, for example, in

1.) All men are mortal

2.) No gods are mortal

Therefore

3.) no men are gods.
or
1.) Everybody likes Fridays
2.) Today is Friday
Therefore
3.) Everybody likes today,
or
All B's are A's.

All C's are B's.

All C's are A's.
The syllogism has two premises and a conclusion. Each premise is a proposition with a subject term and a predicate term. In the conclusion, the subject term is C and the predicate term is A. There is also a "middle term" B, which is the term linking the C's and the A's. Hence Aristotle regards the middle term as what provides the explanation (i.e., B explains why all C's are A's.)
Aristotle recogniced four tipes of categorical sentences, which all include a subject (S) and a predicat (P):
A,B,C.... Terms

a universal affirmation: belongs to everybody

i particular afformation: belongs to some

e universal negation: doesn’t belong to anybody

o particular negation: doesn’t belong to some
These sentences can form syllogisms in many ways, both non-logical and logical. In the Middle age students of Aristotelian logic categorised all logical possibilities and named them:

(www.wikipedia.com)



3.3 Definition
For Aristotle, a definition is "an account which signifies what it is to be for something" (logos ho to ti ên einai sêmainei). The phrase "what it is to be" and its variants are crucial: giving a definition is saying, of some existent thing, what it is, not simply specifying the meaning of a word (Aristotle does recognize definitions of the latter sort, but he has little interest in them).
In the second book Aristotle considers demonstration as the instrument than with what definition is reached. The four great problem types, the ‘That’, the ‘Why’, the ‘If’, the ‘¨What’ are all concerned with the middle term. These terms are in all five objects of knowledge:

1. What a name means

2. That the corresponding thing is

3. What it is

4, That it has certain properties

5. Why it has certain properties


An example of using a syllogism in demonstrating something could be:


Question: What is an eclipse?

Answer: a blocking of the moon's light by the earth.

Let A=eclipse, B=blocking by the earth, and C=moon.

B is A.


C is B.

C is A.


In this example, asking whether the moon is eclipsed = asking whether B is or is not.

We have the "account" of eclipse (namely, B, the middle term), so we learn both the fact (that there is eclipse) and the reasoned fact (why) at the same time.

Alternatively we might only know the fact, not the reason.

Let A=eclipse, B=inability of moon to cast shadows, C=moon.

If it's clear that A belongs to C, then to inquire why it belongs is to inquire into what B is (blocking? rotating? extinguishing?). B is an "account" or explanation of one of the other two "extreme" terms, A (eclipse).
Another example: A=thunder, B=extinguishing of fire, C=cloud. Then we get an account of thunder as "extinguishing of fire in the cloud."
Since a thing's definition says what it is, definitions are essentially predicated.

en predicate X is an essential predicate of Y but also of other things, then X is a genus (genos) of Y. A definition of X must not only be essentially predicated of it but must also be predicated only of it: to use a term from Aristotle's Topics, a definition and what it defines must "counterpredicate" (antikatêgoreisthai) with one another. X counterpredicates with Y if X applies to what Y applies to and conversely. Though X's definition must counterpredicate with X, not everything that counterpredicates with X is its definition. "Capable to cry", for example, counterpredicates with "human" but fails to be its definition. Such a predicate (non-essential but counterpredicating) is a peculiar property or proprium (idion).

Finally, if X is predicated of Y but is neither essential nor counterpredicates, then X is an accident (sumbebêkos) of Y.



    1. Demonstration

1. Whatever is scientifically known must be demonstrated.

2. The premises of a demonstration must be scientifically known.


Aristotle says all teaching and all learning start from pre-existing knowledge. The knowledge thus presupposed is of two types of fact; it is knowledge ‘that so-and-so is’ (which is a question also in the centre of many of Plato’s dialogues), or knowledge of ‘what the word used mean’. With regard to some things, the meaning of the words being quite clear, all that needs to be explicitly assumed is that the thing is so; this is true, for example of the law that everything may with truth either be affirmed or be denied. With regard to others it is enough if we know explicitly the meaning of the name; it is then sufficiently obvious that the thing exists, and this need not be explicitly stated. With regard to other things we must explicitly know both what the name means and that the thing is.
Demonstration is scientific syllogism, i.e. a syllogism which is through and through knowledge and not opinion. A demonstration (apodeixis) is "a deduction that produces knowledge". Aristotle's Posterior Analytics contains his account of demonstrations and their role in knowledge. From a modern perspective, we might think that this subject moves outside of logic to epistemology. From Aristotle's perspective, however, the connection of the theory of sullogismoi with the theory of knowledge is especially close.

Aristotle says that a demonstration is a deduction in which the premises are:

1. true

2. primary (prota)

3. immediate (amesa, "without a middle")

4. better known or more familiar (gnôrimôtera) than the conclusion

5. prior to the conclusion

6. causes (aitia) of the conclusion

4. Conclusions
It is difficult to write a shortly on Aristotle, there is just too much ‘stuff’ in his writings. A Finnish philosopher Eero Ojanen writes about this in his book ‘Mitä Aristoteles on opettanut minulle’ (Eero Ojanen ja Kirjastudio, Helsinki 2005). He also talks about his experiences in reading Aristotle, how he feels like this is what he’s always been reading. I had very much the same experience with reading his logical writings and the articles people have written about them. Aristotle is so much inside our culture, that even when we are not directly reading him, we come in touch with his ideas everywhere. As Ojanen puts it (translation is by me, since the book is so far only published in Finnish): “Aristotle is not a target of knowledge, he is the tool, the method we have for knowing and sharing what we know”(p.19). This is of course very much true when one talks about his logical works, they certainly are a tool. His logical works have been criticised very harshly, for example by Bertrand Russel in the History of western philosophy, where he in the end of his introduction to Aristotle’s logic, states that he concludes that all the Aristotle’s teachings he presents in the introduction are completely wrong, except syllogism. But even he has to admit the talent of the writings and the role of Aristotle in history. “The historical importance of Aristotle is true”, as Ojanen puts it (p.115, syllabus mine).
References:

- Ojanen Eero, Mitä Aristoteles on minulle opettanut, Eero Ojanen ja Kirjastudio, Helsinki 2005

- Oliver, Martyn, Filosofian Historia (orig, The Hamlyn History of Philosophy), Gummerus kirjapaino oy, China 1997

- Russel, Bertrand, Länsimaisen filosofian historia (orig. The History of Western Philosophy), WSOY, Porvoo 1992

- Sir David Ross, Aristotle, London 1995

- Smith Robin, Aristotle’s logic, http://plato.stanford.edu



- www.wikipedia.com

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