The nature of Aristotle's logic

| 18th July 2019
Aristotle's logic provides the foundation for the dialectical theory of change.

The term contradiction is one we will return to in explaining dialectics, and also organisational science and systems theory.

Dialectics is the most powerful and comprehensive theory of change currently available. Yet it remains both largely unknown and widely misunderstood. It is a highly complex theory that almost defies definition: indeed, there is a heated debate in some quarters about whether to describe it as a “theory”, “a method”, a “form of logic” or simply “the dialectic”. It has also changed fundamentally over the centuries since it was first conceived, with almost as many versions of dialectics as there are people trying to describe it. 

Here I attempt to set out dialectics as conceived by Aristotle (382 BCE - 322 BCE), and later developed by GWF Hegel (1770 - 18310 and applied by Karl Marx (1818 - 1883). Dialectics is a universal theory which states that everything is related to everything else (although some things are more closely related than others) and that everything is always changing (so what is interesting is why some things appear to be static) which can be usefully applied in every intellectual discipline (from quantum mechanics to psychology). But it’s much more than that. 

The dialectic can facilitate a more comprehensive analysis of the relationship between nature and human societies, between groups and the individuals that populate them, and indeed between the individual and others. It is a universal theory which remains at odds with the predominant, mainstream thinking about how the universe functions. What follows is my attempt to present dialectics in a way that is accessible to someone who has never heard the term before, as briefly as possible, and which contains within it all that is useful in an attempt to effect change at the level of the self, the group and the whole of human society.

System

The dialectic as presented here begins with Aristotelian logic, but is different to what is often described as Aristotle’s dialectic. Aristotle claimed that knowledge could be developed through discussion. The first person would present a thesis, the second its opposite - an antithesis, and through rational debate the claims of each would be tested, falsity would be removed, and a more perfected synthesis would emerge. This may well be the case. But it is much more limited and much less useful than the dialectic I want to develop. 

It is well worth taking the time and effort to begin with Aristotle and his logic, not least because this is where many of the terms that will be used later in this exposition are used and defined (I have made the most important words bold). These same words exist in common language, but very often with different or less specific definitions. This, in my view, has led to people reading authors who rely on dialectical analysis without comprehending the full meaning of what is being said. When I say ‘people’, I definitely also mean ‘I’. Aristotle’s logic looks and feels like computer code, almost reducing all claims made in language down to binary form. Indeed, logic is the basis of some computer languages.

Aristotle, as natural scientist and as philosopher, endeavoured to classify almost everything, from animals to concepts. Indeed, he classified the different aspects of what can be known of any thing. He listed 10 types of information that can relate to the thing, and called these "the categories". The categories are classes of information which can relate to any 'thing'. He listed these as "substance, quantity, quality, relation, place, time, position, state, action or affection". Substance can be understood as the real, extant thing that is knowable through our senses. The other categories he described as 'accidents' which can describe the substantive thing. We can say that a human being is substantive. This human being may have the quantity of being 140lbs, the quality of being intelligent, the relation of being half as tall as another person, the place of being at the market, and the time of being yesterday. These classes of information are communicated through propositions, which have subject and a predicate where the subject is the thing about which the information relates and the predicate includes the information. 

What Aristotle succeeded in doing was to present an entire system of logic that organises the information about any thing, and all things. His is a highly complex and wide-ranging argument made up of a web of propositions where each conclusion can be traced back to its premises. The logic is set out through propositions, defined as “a statement or assertion that expresses a judgement or opinion”. Propositions contain a subject and a predicate. The subject-object relation is also the basis for grammar, the form through which language (at least Western language) organises meaning. We can organise everything we know about the world using just four forms of proposition, as set out below: all day is light, no night is light, some days are warm, some days are not warm. We can describe any one thing that we see in terms of what it is the same as, and what it is different to. The subject is the agent about which we have an opinion, and in grammar the predicate is the clause containing a verb and stating something about the subject.

Aristotle builds his system of logic from propositions, which are validated by the truth of the earlier propositions upon which they rest and the observation of the laws and rules of logic. This system of logic creates a series of consistent, internally coherent, arguments. If we begin with two premises based on observation of material reality, we can arrive by inference at a conclusion which will also correspond to material reality. Each conclusion in a series of arguments can be used as a hypothesis, which can then be tested against reality, often through empirical science. There is a significant risk here: if any one premise is false, a huge and intricate system of logical thought can in turn be completely false. If you rely entirely on logic, you must be sure of your premises.  

Aristotle begins his system of logic with the proposition: ‘all s is p’. This is an abstract statement, it does not at this stage describe concrete reality. The word abstract is really important here. I am going to use abstract and abstraction a lot in presenting dialectics. Here, I define abstract to mean “existing in thought or as an idea but not having a physical or concrete existence”. This comes from the Latin abstractus, ‘drawn away’. Here, the content has been drawn away from the symbols s and p. ‘Abstract’ is often used as a derogatory term, such has “that’s just an abstract argument”. However, the act of abstracting is incredibly useful. Mathematics is exactly this form of abstraction. We know 1 plus 1 equals 2. Here, 1 is abstract and can represent one anything. But we know from this proposition that if we have one cow, and we have another cow, we have two cows. This is obviously more useful when the terms used are more complex. The opposite of abstract is concrete, often defined as ‘real’, ‘material’, ‘extant’. It it quite difficult today not to think of the building material. Concrete comes from the Latin concrescere, or ‘to grow together’.

A, E, I, O

The proposition ‘all s is p’ is so abstract, and so simple, that it appears worthless. But it is the premise on which this logical system is predicated. This primary proposition sits as one in a set of four presented by Aristotle, each known by a letter:

A All s is p

E No s is p

I Some s is p

O Some s is not p

Here ‘s’ and ‘p’ are both the terms of the proposition. The letter ‘s’ represents subject, which in turn represents all subjects that can be used in this sentence. It is abstract in the sense that it is separate from any real examples. The p in our proposition represents the predicate. The ‘s’ is related to the ‘p’ and the proposition sets out the nature of the relationship. We can also say that ‘s’ is determined by ‘p’: in our statement, s stands by itself, and p is useful because it tells us something about s. What is really important here - in fact all we can know from this statement - is the relation between the ‘s’ and the ‘p’. One of the most important things about dialectics is that it is relations that matter most, even more than the nature of s or p. Dialectics is an alternative to the philosophy of René Descartes (1596-1650) who, in the simplest terms, proposed that there was a separation of mind (human consciousness) and matter (everything else) and that matter was mechanical. If you were to use the Cartesian scientific method, you would smash s apart and find out what it is made from. This, it is assumed, would help us discover the true nature of the ‘s’. With dialectics, the s is defined by its relation to p. We now want to define that relationship

In our primary statement, “all s is p” the “all” is called the qualifier, while the ‘is’ is known as the copula. Aristotle is concerned with the relations between these statements. The qualifiers “all” and “no” used here are described as being universal; the “some” and also the “some...not” as being particular. Philosophy has been deeply concerned about what is universal. If we know that a fact is true of all things this is considerably more useful than knowing what is true of just some things. These interest me because the challenges we face and the solutions we need impact all of us, and will take all of us to succeed. The difference between all and some has long been fundamental to politics: the universal declaration of human rights logically need apply to all humans (and not, for example, just men, or white men). The difference between universal and particular is described as a difference in quantity. All something minus some quantity of something becomes some: and in this case some is and some is not. 

The difference between the qualifiers “all” and “some...is” as opposed to the qualifiers “no" and “some...is not” is also described. The first (all) is named affirmative, and the second (no) is negative. The difference between affirmative and negative is described as a difference of quality. The definition of and use of the negative, and the relationship between quality and quantity, are an obsession of dialectics. We can here deduce that when there is a change in quantity between the universal (all) to the particular (some) there is also a change in quality, between affirmative and negative. Let us start with ‘all s is p’ where the quantity ‘all’ applies to s. We know that s has the quality of being affirmative in relation to p in all cases. When we change the proposition from “all s is p” to “some s is p” we must infer that “some s is not p” is at least possible. Here the quality of s in the first proposition remains affirmative but it follows from this that “some s is not p” is possible - and therefore s can also have the quality of being negative in some cases. This means when we change the quantity of s from all to some the quality also changes from affirmative to being possibly both affirmative and negative in relation to p.

Where all this starts to look useful here and now is when we can infer one proposition from one or two others in a process known as deduction. This is one method of extending our knowledge beyond what we can observe, beyond our sensations. Aristotle used the following example. Perhaps we know that “all dogs have four legs”. We are then told that “Fido is a dog”. We can then infer by deduction that Fido has four legs, without having to observe Fido. This is a very useful hypothesis, but it needs to be tested against reality. We could find out new information, such as “Fido had an operation to remove a leg”. However, it is statistically likely that it is true that Fido has four legs - useful information if we are buying him a gift of dog slippers.

The proposition with which we begin is known as a premise, the one with which we conclude as called a conclusion. The act of moving from the beginning to the conclusion is to infer. If we begin with one premise the inference is immediate. If we begin with two or more premises the inference is mediated: the first premise has been mediated by the second to infer the conclusion. (Note that here ‘immediately’ would not mean “without much time passing” but instead “with a single premise rather than two”. This is a good example of where the definitions used in logic are similar to but not the same as those commonly used). This is what happened with Fido: the premise that Fido is a dog is mediated by the premise that all dogs have four legs in order to infer that Fido has four legs.

As we see, when presented with an argument or a proposition, the first thing to do is to test whether you agree with the premises. If you do not agree that all dogs have four legs - some have had accidents - then you should also challenge the conclusion. We can see why universal claims are more useful than particular claims: if you substitute the all with some in the Fido premise, you end up with “some dogs have four legs” and can only infer that Fido may or may not have four legs. Logic is more useful when we can infer a series of propositions from the premise or premises, creating a body of knowledge from what can be a single fact. A long string of propositions, each resting on the former, is known as a sorites - sorites with two premises are known as a syllogism.

Atoms

Democratus (c460 BCE - c370 BCE), an ancient Greek philosopher and contemporary of Aristotle, observed that stone statues weathered. He noted a. hard, physical objects would wear down and lose their edges and b. the substance that must be being removed was invisible to the naked eye. Taking these two premises he deduced that even large, hard material objects were actually made up of parts which were so small that they could not be detected by human sight. He noted that in fact all objects of material reality transformed over time. From this he concluded that all material objects were made from objects too small to be seen. He described these as ‘atoms’, or the indivisible. Speculative philosophy, logical deduction, resulted in knowledge that was thousands of years in advance to the natural sciences.

Aristotle went to extraordinary lengths to find out what would happen when you changed each and every one of the different parts of a proposition, and what would happen if you piled one kind of proposition on top of the other. He started by holding ‘s’ and ‘p’ to be the same across all the propositions, like a scientist isolating the variables in an experiment. He proceeds by looking at what would happen if you just swapped the subject and the predicate; and what would happen if you changed the qualifiers, changing “all” to “some” (and vice versa). This may well be achingly dull now but becomes fascinating and useful later - and we know we have to get our premises in order if we want to deduce anything useful.

We start with immediate inference, because it’s simpler. When we start swapping the parts of the premise this is called eduction (and the various ways are called simple conversion; conversion by limitation and obversion). Aristotle established that in the sentence “all s is p” you could not assume that you can swap the s and the p. All s is p is not equivalent to all p is s (unless s is p). As we can simply show, when we swap s and p like this sometimes what we create is valid but sometimes it is invalid. For example, we know that “all humans are animals” is true but we also know that “all animals are humans” is false. Aristotle went through the postulates and worked out in every case when you could swap things around and still have propositions that were logically valid. This is why logic is useful - we can test whether the inference we have made from one truth to establish a later, inferred, truth is valid or invalid. To some degree, this can help us know who to believe and who to disbelieve without having to break out a chemistry lab during a late night pub conversation about whether “all philosophy is useless” or just “some philosophy is useless”.

Aristotle was particularly interested in how each and all of the forms of proposition A, E, I and O relate to the other forms when s is the same in every proposition and p is also the same. We know that each of the four forms of propositions are different, either in quality or quantity. He established that with some propositions you can swap the subject and predicate and your conclusion will remain true but some will be rendered false. But we can also see that some forms are in opposition to others - if one is true, the other is false. Most obviously, ‘all s is p’ is opposed to the proposition that ‘no s is p’. They cannot both be true. However, ‘some s is p’ is in agreement with ‘some s is not p’. The concept of opposition is perhaps the single most important in dialectics, as we will shortly see. 

The nature of oppositions between propositions is fascinating. There are four types of opposition: contradiction, contrariety, subcontrariety and subaternation. It is ‘contradiction’ that will be most useful to us. A proposition is in contradiction with a second proposition when the truth of the first immediately infers the falsity of the second, and further when the truth of the second immediately infers the falsity of the first. We can see from this that A contradicts O: if ‘all s is p’ then we know that ‘some s is not p’ cannot be true, but is false. We also know that if ‘some s is not p’ is true then ‘all s is p’ is false. This is the definition of contradiction in dialectical logic. In common language, contradicion can often be used for all oppositions. However, the definition of contrary is somewhat different. Here, we can infer from the truth of the first premise that the second premise is false; but we can only infer from the falsity of the first premise that the second premise is true or false. So if we know ‘all s is p’ is true, we can infer that ‘no s is p’ is false. But if we know ‘all s is p’ is false we can only know that ‘some s is p’ could either be true or false. Contradiction is therefore more thoroughgoing, and also symmetric.

There have been two particularly useful developments in traditional logic since Aristotle for our analysis: the emphasis on negation in the nineteenth century and also the modern emphasis on propositions where the terms ‘s’ and ‘p’ are ‘categories’, which I will explain more fully when discussing Hegel next. Aristotle’s logic has stood up to considerable scrutiny in the centuries since he first advanced them. More recent developments, including the concept of negation and the concept o have extended and advanced them. 

Negative

The introduction of negative values for s and p extends and improves Aristotle’s original logic. Professor James Wikenson Miller argued in The Structure of Aristotelian Logic in 1938, that “negative terms are of great advantage to traditional logic. On the one hand they increase the power and scope of the system. On the other hand they result in a genuine simplification of the system.” (p93). The second of Aristotle’s propositions (E), ‘no s is p’ includes the negative qualifier ‘no’, as noted above. However, this is extended by Hegel and others to include negative values for the terms themselves: ‘non-s’ and ‘non-p’. We can now have the proposition ‘all s is non-p’ and propositions with double negatives such as ‘no s is non-p’. This can be described as the negation of the negation, again a term which is essential for Hegel’s dialectic. This brings us to a particularly significant point. The proposition ‘all s is s’ is now not the same as, or equivalent to, ‘s is s’ as ‘all s’  includes the term with a negative value, ‘non-s’. The proposition ‘s is non-s’ is a contradiction, it is by definition false. The ‘law of identity’ which is foundational to dialectics describes ‘s is s’ but cannot be extended to ‘all s is s’. We will return to this later.

The second more recent addition to traditional logic is the introduction of the concept of ‘class’. A class, or classification, is like a basket of objects where those objects collected into the basket share one or more characteristic. Therefore “red” is a class of objects which present to the human eye with the colour red. The class red exists whether there are no objects that are red, and where all the objects in the class are red. Words can function in language as categories. The word ‘humans’ refers to those things that can be classified as ‘human’. They have whatever the properties are that would identify them as humans. The propositions we have presented so far contain the variables ‘s’ and ‘p’. These terms are propositional functions. When we replace these variable terms with a fixed value we then have a proposition. For example, ‘x is human’ is a propositional function. When the variable x is replaced with the value ‘Annie’ we have ‘Annie is human’, which is a true proposition. When we replace the variable x with Fido we then have the proposition Fido is human, which is false. Those values that replace the variable to create a true proposition are said to satisfy a propositional function.

The class ‘human’ are those entities that satisfy the propositional function ‘x is human’. (p85) The most general definition of class is contained in the descriptions, “a class which has no members” and “a class the membership of which coincides with the entire universe of discourse.” (p85) A class with no members is called a null class while the latter is called the universe class. Professor Miller presented a proof demonstrating that the inclusion of the modern definition and use of class into traditional logic is consistent. Further, he argued: “Traditional logic and modern logic are in perfect agreement with each other. The apparent disagreements between them reflect merely a difference of vocabulary...Modern logic is of course a much more extensive system than traditional logic. [Traditional logic] is genuinely a part of Logic - but only a part. It is a special case.”

The advantage of logic as presented here - a form of pure reason - is that as an idealised systematisation of concepts it does not depend on our interpretation of the world around us. Science has advanced in the last two thousand years, but Aristotelian logic has remained as robust and useful as when it was first advanced. However, the use of logic is severely limited when it is not related to, or grounded in, material reality. 

Reality

So how is logic useful to us today? I want to present an argument about the nature of change, at a personal, group and at a social scale. This argument relates to our material reality: what the world is really like, and what the humans who populate this world are really like. The first premise in this argument is simply this: the universe is made of differentiated matter. This claim is based on observation rather than a prior proposition. There are, for example, things in our world which we can sense through sight, sound, touch, smell and taste - and there are things which we cannot. For Democritus, the ancient Greek philosopher who introduced the concept of the atom, there is matter and void. This is different to Hegel, the idealist, whose system of logic does not rely on reference to material reality. The first premises are the concepts of pure being and nothingness. The relationship between these concepts and the nature we sense is a highly complex one, and we will return to this later. 

The second premise of this series is that the human mind is a differentiating machine. This is also based on observation. Through sight I can differentiate between different lengths of lightwaves, giving the sensation of colours. The brain is a differentiating machine, but it is equally a machine that identifies similarities. It can identify two apples as being the same, and an orange as being different. This is the duel movement of catagorisation. This is a materialist (where we begin with extant reality rather than logical concepts) rather than idealist foundation for much of what follows in our definition and discussion of identity and difference. There is a difference between the material universe and the human mind. However, there is also an identity: the material world can be differentiated and the human mind can differentiate it. It is the identity - the shared properties, the intersecting reality - between world and mind that interests us here. This, put in very simple terms, is what Hegel described as Geist, translated as both ‘spirit’ and ‘mind’. 

The objective of this book is to understand change at a personal, team, and society scale. Each of these clearly exists at the intersection of the human mind and the material universe. Therefore, the nature of the mind (without material reality) and material reality (without mind) is a philosophical realm which is out of scope. The argument about whether the dialectic exists within the natural world or is merely a product of the human mind is not addressed here. But I argue that we will benefit enormously from interpreting and organising information through the dialectic as we try and understand everything that can be known by the human mind about the entire natural universe.

This brief tour of Aristotelian logic is necessary but not sufficient for my presentation of the dialectic. It introduces the definitions of words such as subject and predicate, identity, negation, negation of the negation, immediate and mediate, opposition and contradiction, among others. These definitions will prove invaluable as we explore Hegel’s presentation of the dialectic. My own experience has been learning the meaning of the dialectic in reverse: starting with the crude and inaccurate version I learned from some academic Marxists, then from my own (mis)understandings of Marx’s writings, and then from Hegel and finally to Aristotle. If this series achieves nothing else, I hope it assists at least some people in avoiding this mistake. 

By beginning with Aristotle and his definition of these terms we can advance an understanding of the dialectic which is coherent, consistent and incredibly powerful in our interpretation of the world around us. Although, given the complexity of the concept, this understanding will never be complete, total or fixed. The relation between the subject and the predicate will be the basis of our understanding of the relation between humans and nature. We will be particularly interested in relations that are in opposition. The term contradiction is one we will return to in explaining dialectics, and also organisational science and systems theory. In the following text I am hoping to set out some universal claims about us human beings. I will argue that some behaviours can be found in all humans, and all human groups, and all interactions between humans and nature.

This Author

Brendan Montague is editor of The Ecologist. 

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