Introduction – logic and logical form

What does logic do?

To be logical means, when arguing or thinking, to use only valid arguments. Logic is the scientific study of valid arguments.

Each argument consists of premises and a conclusion. Both the premises and the conclusion are sentences. Here are two examples:

Bob is not sick with the flu because if he was, he would have a fever, and he doesn’t.
If Bob was sick with the flu, he would have a fever. He has a fever. Therefore, he is sick with the flu.

Both arguments have two premises. When analyzing an argument we will often write its premises one below the other, and below them, separated by a line, we will write the conclusion:

If Bob is sick with the flu, he has a fever.
Bob does not have a fever.
Bob is not sick with the flu.
If Bob is sick with the flu, he has a fever.
Bob has a fever.
Bob is sick with the flu.

An argument is valid when it is impossible for the conclusion to be false in case all the premises are true. Otherwise, it is invalid. The first of the above arguments is valid because if its premises are true, its conclusion cannot be false. On the contrary, the second one is invalid because it is possible for Bob to have a fever and not be sick with the flu. He may be ill with something else, for example. In this case, the premises will be true and the conclusion false. The first argument is an example for a logically valid inference and the second for a logical fallacy.

Logical inference and logical form

An inference is logically valid if it is (logically) impossible for the conclusion to be false when the premises are true. In that case we say that the premises logically entail the conclusion, or that the conclusion logically follows or that it can be logically inferred from the premises, etc. The relation of logical inference between sentences is of fundamental interest to logic, so an important question is whether there are universal procedures through which we can always determine whether a certain sentence can be logically inferred from certain premises. Another important question is this: is there a universal procedure by which, if a sentence follows logically from certain premises, we can prove that this is the case? Without being one and the same question, the questions are related and are not at all trivial. Modern logic gives a negative answer to the first one – there is no a universal recipe through which it can be always determined whether a sentence follows logically from certain premises. However, the second question has an affirmative answer – there are proof procedures through which, if it is a fact that a sentence is logically entailed by certain premises, this can be demonstrated.

The premises and the conclusion of each inference (argument) are not any sort of sentences. They are sentences that can be (and are) true or false. We will call such sentences “propositions”. From the sentences below, only the last one is a proposition.

Is it Tuesday tomorrow?

Say what you think without worrying.

If only I were ten years younger!

All mushrooms are plants.

Questions, orders, requests, exclamations, etc. are sentences but they are not propositions as they are not true or false.

It is important to distinguish between whether or not an inference is logical valid and whether or not its constituent sentences are true. Whether a certain conclusion follows logically from certain premises has nothing to do with the truth or falsity of these premises and conclusion. Let us look at the following two inferences:

All winged creatures are gods.
All humans have wings.
All humans are gods.
All predators are animals.
All tigers are animals.
All tigers are predators.

All three sentences in the first inference are false and all three in the second are true but nevertheless the first is logically valid and the second is not. Anyone giving the first argument would be more logical than anyone giving the second, although the world of the former person would be much farther from reality than the world of the second, less logical person. The reason for the logical validity of the first inference is that if the premises were true, the conclusion would be (necessarily) also true, i.e. it is impossible for the conclusion to be false if the premises are true, which, as we noted above, is the criterion for logical validity. On the contrary, while all sentences in the second inference are true, it is not impossible for the conclusion to be false when both premises are true, because it is not logically impossible for tigers and predators to be animals without tigers being predators. The diagram below shows why such a state of affairs (though not real) is possible.

The circles in the diagram are related to the corresponding sets of things. For example, that the circle of predators is fully included in the circle of animals corresponds to the state of affairs in which the set of predators is a subset of the set of animals, i.e. in which all predators are animals (the first premise). That the circle of predators and the circle of tigers are included in the circle of animals corresponds to the assumption that the premises (“All predators are animals” and “All tigers are animals”) are true. Despite this assumption, the circle of tigers could have three different positions relative to the predator circle – it could be fully included in it (the actual state of affairs), only partially included, or completely excluded. The second and the third cases would make the conclusion (“All tigers are predators”) false, while preserving the truth of the premises. This shows that it is (logically) possible for the conclusion to be false when the premises are true, which means that the inference is invalid – it does not guarantee the truth of the conclusion if the premises are true.

On the contrary, the diagram below shows that the conclusion of the second inference could not be false, if the premises were true.

If the premises (“All winged creatures are gods” and “All humans have wings”) were true, the circle of winged creatures would be fully included in the circle of gods and the circle of humans would be fully included in the circle of winged creatures (as is in the diagram). But then the circle of humans must be fully included in the circle of the gods – i.e. there is no way for the conclusion to be false if the premises are true. This shows that the inference is logically valid, even though the sentences of which it is composed are in fact false. Logic is beyond the actual state of affairs – it takes into account every possible world.

What determines the logical validity or invalidity of an inference is its structure, not the words it consists of – important is the form, not the content. No matter how we change the content of the above two inferences, as long as we preserve their structure (i.e. preserve their logical form), the first will continue to be logically valid and the second logically invalid. For example, the inferences below are derived from the inferences above by changing the substantive words in them leaving everything else unchanged. As a result, the subject has been completely changed but the form of the new inferences is the same:

Everyone who has ever forgotten their umbrella at my place, is a man.
All my friends born in March have forgotten their umbrellas at my place.
All my friends born in March are men.
Every squirrel is a mammal.
Every elephant is a mammal.
Every elephant is a squirrel.

The first inference is obtained from the first above by replacing “winged creature” with “who has ever forgotten his umbrella at my place”, “god” with “man”, and “human” with “my friend”, and the second is obtained from the second above by replacing “predator” with “squirrel”, “animal” with “mammal”, and “tiger” with “elephant”. As a result of the substitution, each of the three false sentences in the first inference above has turned into a sentence that could be true or false (it depends who says it); the premises of the second inference continue to be true but the conclusion has become false. The important thing is that all these changes do not affect the validity or invalidity of the inferences – the first continues to be valid and the second invalid, and the reason is the same as indicated in the diagrams above.

Because logical validity or invalidity of inferences is determined by their form, not by their content, in logic we use symbols to replace the “contentful” expressions so that the logical form of inferences stands out. For example, if we replace some of the expressions in the four inferences discussed above by the symbols “S”, “M” and “P”, we will make their logical form explicit. The form of the two valid inferences is

All M are P.
All S are М.
All S are P.

and of the two invalid is

All P are М.
All S are М.
All S are P.

Accordingly, that every inference having the first form is logically valid and that every having the second form is logically invalid can be seen from the following two diagrams.

Let me emphasize again: whether or not an argument is logically valid is determined by its logical form, not by the content of the propositions involved in it, or by whether or not they are true. So (although there is a connection between the two) being logical and telling the truth are two different things. One may only use logically valid arguments but proceeding from false premises it is very likely to come to a false conclusion; and vice versa – although using invalid arguments, one could start with true premises and (by chance) come to a true conclusion. What logical validity guarantees is that if the premises are true, the conclusion will be also true. So the ideal case is to always start with true premises and use only logically valid arguments – then we will have the strongest possible guarantee – the logical – that our conclusions will be true.

Logical words

If what makes an argument logically valid or invalid is its logical form, then what exactly is it?

The logical form of an argument depends on the logical form of the sentences it consists of (premises and a conclusion), and, in turn, their logical form depends on words or expressions that we might call logical. Such are, for example, the words “all” and “are”. Above we made explicit the logical form of certain inferences containing them by replacing the other, non-logical words or expressions with symbols (“S”, “M” and “P”). Thus, we obtained two inference schemes:

All M are P. All P are М.
All S are М. All S are М.
All S are P. All S are P.

No matter by what words or expressions we replace the symbols “S”, “M” and “P” (the same symbol – by the same expression), the first scheme will always produce a logically valid inference and the second – a logically invalid one.

The same is true of the two examples I gave at the very beginning:

If Bob is sick with the flu, he has a fever.
Bob does not have a fever.
Bob is not sick with the flu.
If Bob is sick with the flu, he has a fever.
Bob has a fever.
Bob is sick with the flu.

The logical form of these two inferences is determined by the logical expressions “if... then...” and “not”. Replacing the other (non-logical) expressions, which in this case are whole sentences, with symbols, we get the following inference schemes:

If A, then B. If A, then B.
not-B. B
not-А. A

Every argument that takes the form of the first inference scheme (i.e. that is derived from it by replacing “A” and “B” with arbitrary sentences) will be logically valid, and every one that takes the form of the second will be logically invalid.

Here are some examples of logical words and expressions:

“every”   “any”   “each”   “some”   “a” (indefinite article)   “the” (definite article)   “not”   “no”   “is”   “are”   “exist”   “there is”   “and”   “but”   “or”   “if... then...”   “only if”   “neither... nor...”   “unless”   “identical”   “other”   “necessary”   “possible” …

Unlike non-logical words or expressions, the logical ones occur in all areas and contexts. This corresponds to the fact that, irrespective of area and context, when spoken rationally, arguments are usually given. The validity or invalidity of arguments depends on their logical form, and the latter depends on those words or expressions – not on the content of the sentences. If a person is logical, he or she will be such regardless of whether they are engaged in politics, philosophy, biology, etc.; in the same way if one is not logical.

A brief history of logic

Aristotle is the founder of logic as a science. The main subject of his logical analysis are the so-called syllogisms – certain kinds of inferences that involve the logical words “all”, “some”, “none”, etc. Above we used syllogisms as examples. The following logically valid inference is, for example, a syllogism:

All winged creatures are gods.
All humans have wings.
All humans are gods.

Aristotle’s logical writings are called by his followers, the peripatetics, The Organon, which means “instrument”. The Organon includes the works Categories, Topics, On Interpretation, Prior Analytics, and Posterior Analytics. The main part of Aristotle’s logic, which deals with the syllogisms and is therefore sometimes referred to as syllogistics, is set out in Prior Analytics.

Laying the foundations of logic, Aristotle demonstrates two of its key characteristics:
1) abstraction from the content of inferences and interest in their form
2) use of symbols to explicate the form

Aristotle formulates two principles that underpin any theoretical activity: the law of non-contradiction and the law of excluded middle. The first law states that one proposition and its negation cannot be both true, and the second states that at least one of them is true. For example, we do not know whether there is extraterrestrial life, or not, but the law of non-contradiction guarantees that both cannot be true (it is not possible that the sentences “There is extraterrestrial life” and “There is no extraterrestrial life” are both true). In turn, the law of excluded middle guarantees that at least one of them is true.

Aristotle’s logical views have a huge influence on Western thought. In essence, they exhaust almost everything that the Western world has as logic from the end of Antiquity until the late 19th century, when modern logic emerged.

Megarian-Stoic logic dates back to Euclid of Megara (5-4 c. BC), a pupil of Socrates and a contemporary of Plato. His students were called “Megarians” and later “Dialecticians”. By “dialectic” the ancient Greeks understood the exchange of arguments between the defenders of a thesis and its antithesis, which could result in the rejection of one of the two positions or some kind of combination (synthesis) between them. Notable figures among the Megarians were Diodorus Cronus (4-3 c. BC) and his student Philo (4-3 c. BC).

Later (3 c. BC), the logical views of the Megarians were adopted and systematized by the Stoics, hence the name “Megarian-Stoic logic”. The most notable logician among the Stoics was Chrysippus (278-206 BC), who wrote over 300 logical works, none of which survived.

One of the things Megarian-Stoic logic dealt with was the questions of necessity and possibility. For example, Diodorus insisted that “the impossible cannot be inferred from the possible”, and Chrysippus denied it. (Nowadays, modal logic deals with necessity and possibility.)

They were also interested in when a conditional proposition (“If A, then B”) is true. Philo thought that in order for it to be true, it is enough that things do not stand so that A is true and B is false, while Diodorus argued that it should not be possible A to be true and B false. The first view corresponds to the modern understanding of material implication (or conditional) and the second to the modern understanding of strict implication. On the whole, a striking difference between the Megarian-Stoic logic and Aristotle’s logic is that the representatives of the former were interested in the logical connections through which propositions are linked in more complex propositions, such as “if... then...”, while Aristotle’s logic completely ignored them. Nowadays, propositional logic deals with these connections.

Another thing that interested the representatives of Megarian-Stoic logic was the notion of meaning. The Stoics, for example, thought that what is expressed by a sentence (not the things it refers to, but its meaning) is real. Nowadays, such a real abstract entity, which corresponds to the meaning of a sentence, is often called “proposition”1. It is another question that many philosophers do not thing that such an entity exists.

For centuries, Megarian-Stoic (not Aristotle’s) logic has been the dominant logical doctrine in the ancient world. After the end of the Roman Empire (5 c.), the pursuit of logic ceased for a long time, and no original works of Megarian-Stoic logic survived. This is the reason for the disappearance of this logical tradition. Aristotle’s logic was more fortunate.

During the so-called “Dark Ages” (5-10 c.), one of the few connections with the logic of the ancient Greeks was the Christian philosopher Boethius (480-524), who was familiar with some of Aristotle’s logical writings. Until the 12th century, the only available Aristotelian works in the Western world were Categories and On Interpretation. In the early 13th century, the rest of The Organon (Prior and Posterior Analytics and Topics) was discovered, resulting in the so-called scholastic logic, a refined form of Aristotelian logic that further developed it in the details but that was not new in substance.

In the broader sense, the term “traditional logic” refers to Aristotle’s logic as opposed to modern logic. In the narrower sense, the term refers to the characteristic of the early modern period psychologized version of Aristotle’s logic, the beginning of which is a textbook written in the late 17th century – Logic or the Art of Thinking (1662) by Antoine Arnauld and Pierre Nicole, which became known as the Port-Royal Logic. Port Royal refers to a monastery in France, the center of theological movement of Jansenism. The two authors are prominent representatives of the Jasenism and are influenced by Descartes’ philosophical views. Some of the textbook is probably written by Blaise Pascal. The textbook gained immense popularity and became the standard textbook on logic in France and England in the 17th and 18th centuries. As logical content, traditional logic follows Aristotle’s logic, but psychologizes it. While Aristotle and the subsequent scholastic tradition relates logic to arguments and sentences – i.e. to linguistic expressions, traditional logic relates it to ideas and thoughts as actions of the mind. For traditional logic, logical form is a form of thinking, not a form of language. For example, instead of the word “man”, one would talk in traditional logic of the concept of man; instead of the sentence “All humans are mortal” – of the judgement that all humans are mortal; instead of arguments expressed by language – of reasoning as an action of the mind, etc. In its extreme, such an approach views logic as subordinate to psychology and is called “psychologism in logic” (a term with a negative connotation).

In the late 19th and early 20th centuries, with the emergence of modern logic, psychologism in logic was subjected to a devastating criticism. Modern understanding of logic, even more so than Aristotle’s, links it to language – arguments are sequences of sentences (i.e. linguistic entities), and their form (what makes them logically valid or not) depends on certain logical words and expressions.

Modern logic

In the late19th century predicate logic, also known as first-order predicate calculus, emerged and in 20th century established itself as the modern logic. Its resources of logical analysis are much larger than the resources of traditional (Aristotelian) logic. Part of it, propositional logic, deals with the inferences whose logical validity or invalidity depends on the so-called logical connectives – logical words that connect sentences into more complex, compound sentences (such as “and”, “or”, “if... then...”, etc.). Within the scope of propositional logic are, for example, the aforementioned inference schemes

If A, then B. If A, then B.
not-B. B
not-А. A

Beyond the scope of propositional logic, but within that of predicate logic, which contains propositional logic as a part, are inferences whose logical validity or invalidity depends on logical words such as “some”, “all”, “none”, “exist”, etc. Traditional (Aristotelian) logic deals with such inferences, but syllogisms are only a small (though important) part of them. There are many other such inferences (including some syllogisms) that cannot be analyzed adequately by the means of traditional logic. The logical resources of predicate logic allow it to adequately analyze each such inference.

Predicate logic, along with its part – propositional logic, is what we might call classical (modern) logic. In addition, there are extensions or alternatives to it, sometimes referred to as non-classical logics. First of all, here comes the group of modal logics, which are extensions of classical logic derived from the latter by adding to its logical symbols the so-called modal operators. Modal operators are the symbolic equivalent of words such as “necessary”, “possible”, “permitted”, “obligatory”, etc. Modal logics differ from each other in the words to which the modal operators correspond. In alethic modal logic, these are “necessary” and “possible”; in deontic modal logic – “obligatory” and “permitted”; in temporal modal logic – expressions related to the past and the future, such as “always was”, “once was”, “will always be”, “will once be”; in the epistemic modal logic – words like “knows”, “believes”, and more.

Modal logic is an attempt to extend classical logic because it accepts its principles as it seeks to extend its scope. In addition, there are alternatives to classical logic that reject its basic principles, replacing them with others. Such is, for example, the three-valued logic, which rejects the principle of bivalence. This principle states that every proposition is true or false. The three-valued logic assumes that some propositions are neither true nor false, but have some third truth value, such as undetermined or meaningless.

Another alternative to classical logic is intuitionistic logic, which denies the law of excluded middle as well as other principles of classical logic.

1. The sense in which we use this word is completely different – for us it means “a sentence”, not something extra-linguistic denoted by a sentence.