The word “predicate” is used differently in traditional and modern (i.e. predicate) logic. In traditional logic, this is the term that is affirmed or denied of the subject in a categorical sentence (see 2.1 Categorical sentences). For modern logic, “predicate” is synonymous with “general term” (for the difference between singular and general terms, see 3.1 General and singular terms). The second usage is broader than the first, because each predicate in terms of traditional logic is a predicate in terms of modern logic, but not vice versa. This is because the predicate (in terms of traditional logic) of a categorical sentence is always a general term (i.e. a predicate in the modern sense), but when it is a general term, the subject in a categorical sentence is never called “predicate” in traditional logic.
Another difference between traditional and modern logic is that for the latter the terms in a categorical sentence (the subject and the predicate) are always general terms, while for traditional logic the subject may be a singular term. This is due to the fact that sentences such as “Socrates is a philosopher” or “Socrates is not a politician” are analyzed in a completely different way by the two logics. Traditional logic considers them as a universal affirmative and a universal negative sentence, respectively (why this is so, see 2.1 Categorical sentences), while for modern logic these are not categorical, but atomic sentences, i.e. sentences obtained by connecting a general term with one or more singular terms (see 3.1 General and singular terms). The reason for the different analysis is that in traditional logic no distinction is made between general and singular terms (they are simply referred to as “terms”), while the distinction is key to predicate logic.
Another difference, which we discussed in detail in 2.6 Venn Diagrams, is that, unlike predicate logic, traditional logic tacitly assumes that the extensions of both terms in a categorical sentence (the subject and the predicate) are not empty. According to the contemporary approach, the extension of the subject in general affirmative and general negative sentences, and the extension of the predicate in general affirmative, general negative and particular negative sentences can be empty. This difference has significant consequences. One of them is that almost all logical relations in the square of opposition cease to be valid for modern logic – in particular the relation between contraries and the relation of subalternation. The only relation that is preserved is that along the diagonals of the square – between contradictories. Another consequence is that the immediate inference of conversion is no longer valid for universal affirmative sentences – they cannot be converted anymore. A third consequence is that the number of valid syllogisms decreases from 24 to 15, because all valid (for traditional logic) syllogisms with general premises and a particular conclusion are not valid anymore.
Categorical sentences are important and their discovery by Aristotle is an achievement of traditional logic. However, the logical resources of traditional logic do not allow the subject or the predicate of a categorical sentence to be analyzed when it has certain logical structure. Here is an example:
(1) | All predators that live in water are either fish or mammals. |
This is a universal affirmative sentence, whose subject is “predator that lives in water” and whose predicate is “fish or mammal”. In traditional logic, the sentence will be symbolized with “SaP”. This analysis of its logical form is not wrong but it is not complete enough, because the subject and the predicate are symbolized in the same way as a simple term would be symbolized. They are not simple, however, as each of them contains two other terms. The subject contains the terms “predator” and “lives in water”, and the predicate contains the terms “fish” and “mammal”. The problem with this is that the four simple terms may occur in other sentences that in combination with (1) have certain logical consequences. For example, (1) and the sentence “Some predators that live in water are not fish” logically entail the sentence “Some predators that live in water are mammals”:
All predators that live in water are either fish or mammals. |
Some predators that live in water are not fish. |
Some predators that live in water are mammals. |
In traditional logic, we cannot demonstrate that this inference is logically valid because there is no way to decompose the compound terms in it. We have to symbolize the first premise with “SaP”, where “S” corresponds to “predator that lives in water” and “P” to “fish or mammal”. The second premise and the conclusion have the same subjects, so they are symbolized with “S”, too. However, their predicates, “fish” and “mammal” respectively, are different from the predicate of the first premise, which is “fish or mammal”, so we have to symbolize them with different letters – say “P_{1}” and “P_{2}”. Thus, the whole inference is symbolically represented as follows:
SaP |
SoP_{1} |
SiP_{2} |
Of course, the above inference scheme cannot be logically valid since the only connection between the premises and the conclusion is S; P, P_{1} and P_{2} could be any terms. The crux of the problem is that the analysis of traditional does not reach the simple terms in the compound terms “predator that lives in water” and “fish or mammal”. The logical resources of predicate logic enable it to reach them. To see how, let us symbolize (1). Since this is a universal affirmative sentence, we use the standard way of symbolizing such sentences in predicate logic (see 3.2 Variables and quantifiers), and as a first step we transform (1) into the following semi-symbolic expression:
(2) | ∀x[(x is a predator that lives in water) → (x is a fish or a mammal)] |
(2) tells us that if a thing is a predator that lives in water, it is a fish or a mammal. As a next step, we turn to the open sentence “x is a predator that lives in water”, which is not jet symbolized. This is a conjunction of two atomic open sentences – “x is a predator” and “x lives in water”. If we symbolize the simple general terms “…is a predator” and “…lives in water” with “F” and “G”, the conjunction will be symbolized with “Fx ∧ Gx”. Then (2) becomes
(3) | ∀x[(Fx ∧ Gx) → (x is a fish or a mammal)] |
We now turn to the open sentence “x is a fish or a mammal”, which is not yet symbolized. It is a disjunction of two atomic open sentences – "x is a fish” and “x is a mammal”. Representing the simple general terms “…is a fish” and “…is a mammal” with “H” and “I”, we get the following final, entirely symbolic representation of the initial sentence, in which only simple terms are represented with letters:
∀x[(Fx ∧ Gx) → (Hx ∨ Ix)] |
Generally, if the sentence we are trying to symbolize is more complex or we are not quite sure how to symbolize it correctly, we should use the step-by-step procedure just demonstrated. Gradually moving from the whole to its parts allows us to divide our task into simpler subtasks and thus facilitates the process of symbolization to а great extent.
As a further example, let us consider the following logically valid inference:
All circles are figures. |
Everyone who draws a circle draws a figure. |
In the 17th century, the German mathematician, logician and philosopher Joachim Jungius (Jungius 1957) pointed out that despite the obvious logical validity of the inference, traditional logic is unable to demonstrate that it is valid. The premise and the conclusion are universal affirmative sentences. The premise’s subject and predicate are simple terms and those of the conclusion are compound. The problem is that in the analysis of the conclusion, traditional logic cannot reach to the premise’s terms (“circle” and “figure”), which are contained in the conclusion’s terms (“draws a circle” and “draws a figure”). As a result, the logical connection between the two sentences is lost. In traditional logic the inference will be symbolized as follows:
S_{1}aP_{1} |
S_{2}aP_{2} |
In this inference scheme there are four different terms (S_{1}, S_{2}, P_{1}, P_{2}) between which there is no connection that would justify a relation of logical inference between the sentences containing them.
The logical validity of the inference does not represent a problem for predicate logic. The premise “All circles are figures” is a simple A-sentence, which is symbolized in the standard way (“F” corresponds to “circle” and “G” to “figure”):
∀x(Fx → Gx) |
The conclusion is a more complex sentence, which is why we will use the step-by-step transformation of a natural language sentence into a predicate logic’s formula demonstrated in the previous example. “Everyone who draws a circle draws a figure” is a universal affirmative sentence (this becomes obvious if we paraphrase it with “All things that draw a circle are things that draw a figure”). As a first step, using the standard way of symbolizing A-sentences, we get:
(4) | ∀x (x draws a circle → x draws a figure) |
As a next step, we turn to the open sentence “x draws a circle”. In 3.2 Variables and quantifiers we saw that the indefinite article (“a”) in such sentences indicates an existential quantifier. Since this quantifier will be in the scope of the universal quantifier “∀x”, we will use another variable – “y”. “x draws a circle” is represented (still semi-symbolically) as follows:
∃y (y is a circle ∧ x draws y) |
Literally: “there is something that is a circle and that x draws”. “Circle” was represented with “F” and let us use “H” for the two-place predicate “...draws...”. Thus, the open sentence “x draws a circle” is represented (now completely symbolically) with
(5) | ∃y(Fy ∧ Hxy) |
In a completely similar way (by replacing “F” for “circle” with “G” for “figure”), the other open sentence “x draws (one, some) figure” is represented symbolically with
(6) | ∃y(Gy ∧ Hxy) |
Finally, replacing in (4) “x draws a circle” and “x draws a figure” with their symbolic representations (5) and (6), we obtain the following symbolic representation of “Everyone who draws a circle draws a figure”:
(7) | ∀x[∃y(Fy ∧ Hxy) → ∃y(Gy ∧ Hxy)] |
The final formulation of the inference
All circles are figures. |
Everyone who draws a circle draws a figure. |
in the predicate logic’s language is the following
∀x (Fx → Gx) |
∀x[∃y(Fy ∧ Hxy) → ∃y(Gy ∧ Hxy)] |
Here, in contrast to the representation of the inference in the notation of traditional logic, we have a connection between the premise and the conclusion, which is provided by the repeating letters “F” and “G” corresponding to “circle” and “figure”. Later, we will introduce a proof procedure by which we will be able to prove that the above inference scheme (and therefore the specific inference about drawing circles and figures) are logically valid.
In the previous subsection we examined categorical sentences (“All predators that live in water are either fish or mammals” and “Everyone who draws a circle draws a figure”) that cannot be symbolized well enough by traditional logic, as its logical apparatus does not allow it to analyze the logical structure of (general or singular) compound terms. There are other sentences that traditional logic not only cannot symbolize well enough – it cannot symbolize them at all. Such are the sentences about relations, which are very common in the natural sciences and mathematics (sentences taking of something being greater of less than something, of attraction or friction between objects, etc.). Consider, for example, the following logically valid inference:
All bodies attract each other. |
а and b are bodies. |
b attracts a. |
Traditional logic is completely helpless here, because it is unable to symbolize (i.e. to analyze the logical form of) any of the sentences in this inference. None of them is categorical. The first premise and the conclusion talk about certain relations, and the second premise has the form of a conjunction. Symbolizing and showing the logical validity of the inference is not a problem for predicate logic. “All physical objects attract each other” may be paraphrased with “For any two things, if they are physical objects, the first attracts the second and the second attracts the first”. If “F” symbolizes the two-place predicate “…attracts…” and “G” the one-place predicate “…is a physical object”, the translation of the last sentence in the predicate logic’s language is
∀x∀y[(Gx ∧ Gy) → (Fxy ∧ Fyx)] |
(“For any x and for any y, if x and y are physical objects, then x attracts y and y attracts x”). Symbolizing the second premise and the conclusion is quite easy. We get the following symbolic representation for the whole inference:
∀x∀y[(Gx ∧ Gy) → (Fxy ∧ Fyx)] |
Ga ∧ Gb |
Fba |
Soon we are going to introduce a proof procedure, by which we can easily prove that this inference scheme is logically valid.
A type of sentences that traditional logic is unable to logically analyze (and thus to symbolize) are those in which there is a multiple quantification, i.e. in which a quantifier is in the scope of another quantifier. Consider the following example. If we limit the universe of discourse to humans and put in front of the open sentence “y loves x” the existential quantifier “∃y”, we will get the open sentence “∃y(y loves x)”, which has the meaning of “x is loved by someone” (literally “There is a human who loves x”). Then, putting in front of that expression the universal quantifier “∀x”, we get a sentence in which there are no free variables, i.e. which is a statement:
∀х∃у(y loves x) |
This is the statement “Everyone is loved by someone” (literally “For every human x there is a human y who loves x”). Symbolizing “…loves…” with “F”, the purely symbolic representation of the statement is
(8) | ∀х∃уFyx |
Let us see what happens if we swap the places of the quantifiers:
(9) | ∃у∀хFуx |
The meaning of (9) is no longer that everyone is loved by someone (a different person for different people), but that someone loves everyone. The formula “∀xFyx” tells us that y loves everyone (literally “For every human x, y loves x”), so by putting “∃y” in front of it, we get “There is a human that loves everyone”, in short “Someone loves everyone”.
(8) and (9) represent completely different sentences. It could be true that everyone is loved by someone (we can imagine, for example, that everyone is loved by his or her mother), while it could hardly be true that someone loves everyone. The difference between them is better seen if we change the universe of discourse from the set of humans to the set of numbers and interpret “F” as “…is greater than…” instead of “…loves…”. Then (8) becomes the true sentence “For every number there is a greater number” (literally “For every number x, there is a number y that is greater than x”) and (9) becomes the false sentence “There is a number greater than every number” (literally “There is a number y such that for every number x, y is greater than x”). It can be shown that (9) logically entails (8). That (8) does not entails (9) is clear, as we have just found an interpretation for which the former is true and the latter is false.
Although (8) and (9) are different formulas with different meanings, under the humans and love interpretation used above, the statements corresponding to them may be conveyed in the everyday language with the same sentence – “Everyone is loved by someone”. In the case of (8), the someone in question is meant to be different for different people, while in the case of (9) it is meant to be the same for all people. This is an example of the imperfection of natural languages, which easily allow for ambiguities. An additional benefit of predicate logic (other than the purely logical benefit concerning the study of logical validity) is the elimination of ambiguities once a natural language sentence is translated into its symbolic language. The use of modern logic to clarify the meanings of terms and sentences is especially important for philosophy.
(1) Symbolize the following sentences using the given notations. |
1) | Any mineral with a hardness above 8 is precious. (F – …is a mineral, G – …has a hardness above 8, H – …is precious) |
2) | Not all poor people are unhappy. (F – …is poor, G – …is human, H – …is unhappy) |
3) | Bob borrowed a book from Alice but has not returned it to her. (F – …is a book, G – …borrowed…from…, H – …has returned…to…, b – Bob, a – Alice) |
4) | Expensive stones either have a high hardness or are rare. (F – …is expensive, G – …is a stone, H – …has a high hardness, I – …is rare) |
5) | Someone has hit or insulted Mary’s brother. (F – …has hit…, G – …has insulted…, a – Mary’s brother; D – the set of humans^{1}) |
6) | Bob is the author of a book that sells well. (F – …is the author of…, G – …is a book, H – …sells well, b – Bob) |
7) | Bob has a beautiful wife, but she hates him. (F – …is married to…, G – …is beautiful, H – …hates…, b – Bob; D – the set of humans) |
8) | Alice did not introduce Bob to all of her friends. (F – …introduced…to…, G – …is a friend of…, a – Alice, b – Bob) |
9) | Alice did not introduce Bob to any of her friends. (as above) |
10) | Animals that prowl at night always love to gaze at the moon.^{2} (F – …is an animal, G – …prowls at night, H – …always love to gaze at…, a – the moon) |
11) | Every animal is suitable for a pet, that loves to gaze at the moon. (F – …is an animal, G – …is suitable for a pet, H – …loves to gaze at…, a – the moon) |
12) | I detest animals that that do not take to me. (F – …is an animal, G – …detests…, H – …takes to…, a – I) |
13) | The only animals in this house are cats. (F – …is an animal, G – …is a cat, H – …is in…, a – this house) |
14) | No animals ever take to me, except what are in this house. (F – …is an animal, G – …takes to me, H – …is in…, a – this house) |
15) | No animals are carnivorous, unless they prowl at night. (F – …is an animal, G – …is carnivorous, H – …prowls at night) |
16) | Bob gave a book to each of his friends. (F – …gave…to…, G – …is a book, H – …is a friend of…, b – Bob) |
17) | All human heads are animal heads.^{3} (F – …is the head of…, G – …is human, H – …is an animal) |
18) | Everything is greater than something. (F – …is greater than…) |
19) | There is something than which everything is greater. (as above) |
20) | Everyone likes someone who likes Socrates. (F – …likes…, a – Socrates; D – the set of humans) |
21) | There is someone who likes everyone who does not like themselves. (as above) |
22) | Nobody likes someone who doesn’t like anyone. (as above) |
23) | There are books that all readers like. (F – …is a book, G – …is a reader, H – …likes…) |
(2) In the previous exercise, the following examples had to be symbolized by limiting the universe of discourse D. Symbolize them now without limiting D. |
1) | Someone has hit or insulted Mary’s brother. (F – …has hit…, G – …has insulted…, H – …is human, a – Mary’s brother) |
2) | Bob has a beautiful wife, but she hates him. (F – …is married to…, G – …is beautiful, H – …hates…, I – …is human, b – Bob) |
3) | Everyone likes someone who likes Socrates. (F – …likes…, G – …is human, a – Socrates) |
4) | There is someone who likes everyone who does not like themselves. (as above) |
5) | Nobody likes someone who doesn’t like anyone. (as above) |