Traditional logic originates with Aristotle, who is the father of the field of logic. It was the only logic the Western world had from the Middle Ages until the end of the 19th century when it was succeeded by the modern symbolic logic. (In Antiquity, apart from the logic of Aristotle, there was also the Stoic logic, which contained elements of the contemporary propositional symbolic logic but this logical tradition has not survived after Antiquity.) The main object of interest for traditional logic are certain kinds of inferences called syllogisms. Here is an example:
|Everything that is completely perfect exists.|
|God is completely perfect.|
Aristotle’s logical writings were called Organon, which means “instrument”, by his followers Peripatetics. The Organon consists of six works, of which the most important are three: Categories, On Interpretation, and Prior Analytics.
Aristotle’s logical views had a tremendous influence on Western thought. His logic has been further developed and supplemented (but only in the details rather than in essence) in the Middle Ages and in the Modern period.
After 17th century traditional logic has been psychologized. This means that instead of dealing with logical forms of language, it was considered dealing with forms of mind’s activity. Accordingly, instead of sentences (or statements) logician began to talk of judgements; instead of terms – of concepts or ideas, and so on. However, this psychologizing of logic was only external to its subject and did not affect its content substantially. Psychologized or not, from the point of view of logical analysis almost everything in traditional logic was already available in Aristotle. Aristotle’s approach itself was not psychological – he considered logic as related to reality and language rather than to mind. The emphasis on the connection between logic and language is even more characteristic of modern logic than of Aristotle.
From contemporary point of view, traditional logic is a fragment of predicate logic, which we will be dealing with in the next part, and is rather of historical significance.
Traditional logic takes into account only sentences called categorical. Roughly speaking, these are sentences affirming or denying something of something. Here are some examples:
|(1)||Tigers are mammals.|
|Socrates is not handsome.|
|Some black cats scratch.|
The word or phrase referring to what is being talked about in a categorical sentence is called its subject, and the word or phrase that is being affirmed or denied of what is being talked about is called its predicate. In the above three sentences, the subjects are “tigers”, “Socrates” and “black cats” respectively, and the predicates are “mammals”, “handsome” and “scratch”.
Sentences that are not categorical, i.e. which have no subject-predicate form and are therefore ignored by traditional logic, are for example “For every number there is a larger number” or “If today is Tuesday, tomorrow is Wednesday”.
The subject and the predicate are collectively called “terms” and are thought of as words or phrases under which things in the world fall or do not fall. For example, under the term “black cat” falls every single black cat and does not fall anything else. The man Socrates and nothing else falls under the term “Socrates”, etc.
Terms can be proper names, such as “Socrates”, or other denoting phrases, such “the discoverer of gravity”. They can be nouns, adjectives or verbs, as well as more complex, compound phrases – noun phrases, adjective phrases and verb phrases. For example, among the terms in (1) there are two nouns (“tiger” and “mammal”), one noun phrase (“black cat”), one proper name (“Socrates”), one adjective (“greedy”) and one verb (“scratch”). The terms can be simple expressions, such as “tiger” or “Socrates”, or quite complex expressions, such as “a friend of Alice’s employer’s wife”.
An important concept having to do with terms is their extension. The extension of a term is the class of things thats fall under it. For example, the extension of the term “human” is the class of all people; the extension of “scratch” is the class of all beings that scratch; the extension of the term “Socrates” is the class whose only member is Socrates; the extension of the term “number larger than any number”, under which nothing falls, is the empty class. (Instead of “class”, we may also use the word “set”.)
Categorical sentences are classified into four types according to their quality and quantity. The author of the classification is Aristotle.
In terms of its quality, a categorical sentence can be affirmative or negative. Affirmative are the sentences in which the predicate is affirmed of the subject; negative are the sentences in which the predicate is denied of the subject. For example, “Cats are predators” is an affirmative sentence and “Socrates is not handsome” is a negative sentence.
In terms of their quantity, categorical sentences are classified into universal and particular. A sentence is universal if the predicate is affirmed or denied of the entire extension of the subject. For example, in addition of being affirmative, the sentence “All cats are predators” is universal because the predicate “predator” is affirmed of all cats, that is it is affirmed of all members of the extension of the subject (the class of cats). Correspondingly, a sentence is particular if the predicate is affirmed or denied only of a part the extension of the subject. For example, the sentence “Some black cats scratch” is particular since the predicate “scratch” is affirmed only of some members of the class of cats, that is only of a part of the extension of the subject. In addition to being particular, that sentence is also affirmative.
Joining the classifications in terms of quality and quantity, we get a unified classification, according to which a categorical sentence falls into one of four possible types. It can be universal affirmative (e.g. “All cats are intelligent”), universal negative (“No cats are intelligent”), particular affirmative (“Some cats are intelligent”), particular negative (“Some cats are not intelligent”). Each of the four types is traditionally represented by a letter: “a” for universal affirmative, “e” for universal negative, “i” for particular affirmative, and “o” for particular negative. Thus, if we represent the term “cat” with “S”1 and the term “intelligent” with “P”, the sentence “All cats are intelligent” will be symbolized with “SaP”, “No cats are intelligent” with “SeP”, “Some cats are intelligent” with “SiP”, and “Some cats are not intelligent” with “SoP”.
The table below summarizes the classification of categorical sentences:
|Universal affirmative||All S are P.||All cats are intelligent.||SaP|
|Universal negative||No S are P.||No cats are intelligent.||SeP|
|Particular affirmative||Some S are P.||Some cats are intelligent.||SiP|
|Particular negative||Some S are not P.||Some cats are not intelligent.||SoP|
Categorical sentences whose subjects are proper names or other singular terms (like “Socrates” or “the discoverer of gravity”) are universal sentences. Though this may seem counterintuitive at first as these sentences look even less general then the particular ones beginning with “some”, it is in line with the definition of a universal sentence. The extension of a proper name is a class that has only one member. Therefore, when the predicate is affirmed or denied of such a term, it is affirmed or denied of its entire extension, not of part of it. For example, “Socrates is not handsome” is a universal negative sentence, as the predicate “handsome” is denied of the entire extension of the subject “Socrates”, which consists only of the man Socrates.
For brevity, we may refer to the universal affirmative sentences as “A-sentences”, to the universal negative as “E-sentences” and similarly for the other two types (I-sentences and O-sentences).
Whether a noun or verb is in singular or plural is irrelevant for the logical form of a categorical sentence containing it. This is obvious when the sentences are universal. “Every S is a P”, “Each S is a P” or “Any S is a P” have the meaning of “All S are P”. For example, “Every cat is intelligent” means the same as “All cats are intelligent”; similarly for “No cat is intelligent” and “No cats are intelligent”. The irrelevance of using singular or plural is not so obvious when we have “some”-sentences. In the everyday language use, we sometimes differentiate between “Some S is P” and “Some S are P” meaning that at least one S is P with the first but that at least a few (two, three or more) S are P with the second. In logic, we do not make this differentiation because we assume that the logical word “some” always means “at least one”. For example, we will not distinguish between “Some cat is intelligent” and “Some cats are intelligent”. Like the first sentence, the second will be considered true 1) if all cats are intelligent (then it is trivially true that some of them are intelligent), 2) if some are intelligent and some are not, and 3) if only one cat is intelligent and the rest are not.
Somewhat arbitrarily, we will regard the grammatical forms of the sentences in the above table as the standard forms of the four types of categorical sentences – “All S are P” for the universal affirmative, “No S are P” for the universal negative, “Some S are P” for the particular affirmative, and “Some S are not P” for the particular negative. There are many other ways to express the same what a sentence in standard form expresses. We will look at some of them in the next subsection.
When a categorical sentence is not in standard form, it is not always easy to analyze it – to determine which its subject and predicate are and which its logical type (A, E, I, O) is. To do this properly, it is often useful to try paraphrasing it into an equivalent sentence in standard form relying on our language intuition.
Let us look at some of the alternative ways to make a categorical statement.
Instead of “All S are P”, we often use the shorter “S are P”. Similarly, instead of “No S are P”, we sometimes say “S are not P”. For example, we may say “Cats are intelligent” instead of the more explicit “All cats are intelligent”. In the same way, instead of the standard form sentence “No cats are intelligent”, we may say “Cats are not intelligent”.
In spite of their grammatical form, sentences like “The dolphin is not a fish” usually do not refer to a particular thing (a particular dolphin in the example) but are general assertions. Taking into account what is meant, it may be paraphrased into the standard form universal negative sentence “No dolphins are fish”. So, sentences of the form “The S is (not) P” often have the meaning of “All S are P” (“No S are P”).
An important way to form universal affirmative and universal negative sentences is through an if-then-construction. “If anything is a cat, it is intelligent” has the meaning of “All cats are intelligent” and “If anything is a cat, it is not intelligent” the meaning of “No cats are intelligent”. Generally, “All S are P” can be rephrased as “If anything is an S, it is a P” and “No S are P” as “If anything is an S, it is not a P”. Paraphrasing a sentence into these forms is often even more useful for determining its logical form than paraphrasing it into standard form. Actually, for modern logic these are rather the standard forms of the A- and E-sentences.
Inserting “only” at the beginning of an universal affirmative sentence of the form “S are P” has the effect of switching the subject and the predicate – the subject becoming the predicate of the sentence and the predicate the subject. “Only cats are intelligent” does not necessarily mean that all cats are intelligent. It allows some cats not to be intelligent. What it does not allow is something that is not a cat to be intelligent. Thus, the sentence simply affirms that the class of intelligent beings is a subclass of the class of cats. It is equivalent to the sentence “All intelligent beings are cats”. So, while “S are P” has the meaning of “All S are P”, “Only S are P” has the meaning of “All P are S”.
The concepts of necessary condition and sufficient condition are closely connected with universal affirmative sentences.
Necessary condition. If only cats are intelligent, then being a cat is a necessary condition for being intelligent. Conversely, if being a cat is a necessary condition for being intelligent, then only cats are intelligent. We get that the sentence “Only cats are intelligent” is equivalent to the sentence “Being a cat is a necessary condition for being intelligent”. Generally, “Only P are S” is equivalent to “being a P is a necessary condition for being an S”. However, we saw above that “Only P are S” has the meaning of “All S are P” (“only” switches the subject and the predicate). Therefore, “to be a P is a necessary condition for to be a S” has the meaning of “All S are P” – every universal affirmative sentence is equivalent to the assertion that falling under its predicate is a necessary condition for falling under its subject.
Sufficient condition. If being a cat is a sufficient condition for being intelligent, then all cats are intelligent. Conversely, if all cats are intelligent, then being a cat is a sufficient condition for being intelligent. Generally, “All S are P” is equivalent to “being an S is a sufficient condition for being a P” – every universal affirmative sentence is equivalent to the assertion that falling under its subject is a sufficient condition for falling under its predicate.
Combining the two notions, we get that S is a necessary and sufficient condition for P when these two things are met: 1) all S are P (being an S is a sufficient condition for being a P) and 2) all P are S (being an S is a necessary condition for being a P). It follows that S is a necessary and sufficient condition for P if and only if the class of things that are S is the same as the class of things that are P, in other words S and P have the same extension. The table below summarizes the connection between the concepts of necessary condition and sufficient condition and A-sentences.
|All S are P.||P is a necessary condition for S.|
|All S are P.||S is a sufficient condition for P.|
|All S are P, and all P are S.||S is a necessary and sufficient condition for P.|
If I say “Nothing is a cat, unless it is intelligent” or “Nothing is a cat, except what is intelligent”, I am saying that to be intelligent is a necessary condition for to be a cat. Since, as we have seen, a necessary condition corresponds to the predicate of a universal affirmative sentence, the two sentences are equivalent to the sentence “All cats are intelligent”. Thus, “Nothing is an S, unless it is a P” and “Nothing is an S, except what is a P” have simply the meaning of “All S are P”.
A universal negative sentence in standard form “No S are P” can be paraphrased as “Nothing that is an S is a P” or as “Nothing is both an S and a P”. “Nothing that is a cat is intelligent” is equivalent to “No cats are intelligent” as is “Nothing is both a cat and an intelligent being”.
A particular affirmative sentence in a standard form “Some S are P” can be rephrased as “There is an S that is a P”, or as “Something is both an S and a P”, or as “There exists a PS”. For example, “Some cats are intelligent” is equivalent to “There is a cat that is intelligent”, as well as to “Something is both a cat and an intelligent being”, as well as to “There exists an intelligent cat”. The case with the particular negative sentences is similar. “Some cats are not intelligent” is equivalent, for example, to “There is (exists) a cat that is not intelligent”.
The above alternative ways to formulate categorical sentences do not exhaust, of course, all the ways – they cannot be exhausted. A general rule to follow when determining the logical structure of a natural language sentence is to consider the context and reflect upon what is meant by it. We should not relay too much on the grammatical structure, as it can be misleading. If there is a possibility that the sentence is categorical, we should try to paraphrase it in standard form.
|(1) Paraphrase each of the following categorical sentences into standard form and determine its type. What is its subject and its predicate? Represent it symbolically using “S” for the subject and “P” for the predicate.|
|1)||No metal is an isolator.|
|2)||Some amphibians are not vertebrates.|
|3)||All pearl hunters are good swimmers.|
|4)||Some particles move at a speed close to the speed of light.|
|5)||All active opponents of the corporative tax raise are members of the Chamber of Commerce.|
|6)||No medicine that can be bought without a prescription causes dependence.|
|7)||Crabs are the only animals on this island.|
|8)||All satellites that are currently in orbit are delicate devices that are very hard to manufacture.|
|9)||Blessed are the believers.|
|10)||There are insects with eight legs.|
|11)||The kangaroo is not suitable for a pat.|
|12)||There are no tigers that do not scratch.|
|13)||I hate snakes.|
|14)||Everyone likes Socrates.|
|15)||Only humans laugh.|
|16)||Nothing interests John, except his dogs.|
|17)||Only those who have known misfortune are capable of loving.|
|18)||A necessary condition for being an animal is to move.|
|19)||A sufficient condition for a number to be even is to be divisible by eight.|
|20)||John avoids everything he does not like.|
|21)||Bob likes everything Alice likes.|
|22)||John cannot outrun every man on the team.|
|23)||John cannot outrun any man on the team.|
|24)||Black swans do not exist.|
|25)||Everyone in the room speaks English.|
|26)||I want to go where you go.|
|27)||A lady is present.|
|28)||The rule applies to everyone.|