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QN. 1 LOGIC:
Present and discuss Categorical propositions: universal and particular propositions; the square of propositions; contradictories and contraries; the inconsistencies of a set of propositions. Finally, explain composed (hypothetical) propositions: copulative propositions; disjunctive propositions; conditional propositions.
Type of propositions
Form and structure of Categorical Propositions
Quality, Quantity, and Distribution
The Square of Oppositions
Analysis of the Square of Oppositions
Mediate inferences from the Square of Oppositions
Contradictory Relation inference
Contrary Relation Inference
Subcontrary Relation Inference
Subalteration Relation Inference
Immediate Inferences from the Square of Oppositions
Inconsistencies of a set of propositions
The Existential Import Fallacy
The Illicit Fallacies of contrary, Subcontrary, and Subalteration relations
The possible solutions to the inconsistencies
Composed (hypothetical) Propositions
The Conditional Propositions
The Disjunctive propositions
The Copulative Propositions
A proposition refers to a statement which has a truth value. The truth value here refers to a position of a statement as either being true or false. A proposition is a declarative statement as opposed to other such as interrogative (questions), imperatives (commands), performatives, suggestions and exclamation marks.
Example: The University of Nairobi is in Kenya.
A proposition either becomes a premise or a conclusion according to how it is used in an argument.
Types of Propositions
There are basically two classical types of propositions. That is, simple and compound propositions. A simple proposition is also referred to as categorical proposition whereas compound propositions have many formulations such as; hypothetical, disjunctive, conjunctive, conditional, copulative propositions.
A categorical proposition is one that relates two classes. The classes in question are denoted as the subject class and the predicate class respectively. This type of proposition asserts that either all or part of the class denoted by the subject class is included or excluded from the class denoted by the predicate class. In a nutshell, a categorical proposition is one that asserts that the subject class is either wholly or partly included or excluded from the predicate class.
Form and structure of categorical propositions
Forms of categorical propositions
Since any categorical proposition asserts that either all or part of the class denoted by the subject is included or excluded from the class denoted by the predicate terms or class, it follows that there are exactly four types of categorical propositions. These four types have long been named A, E, I, and O. Of the four types, there is one which asserts that the whole subject class is included in the predicate class (A); there is one which asserts that the hole subject class is exclude from the predicate class (E); another one hold that part of the subject class is included in the predicate class (I); and finally there is one which asserts that part of the subject class is excluded from the predicate class (O). The four types are based on the Latin connotations of Affimo referring to the affirmative propositions A and I and Nego referring to the negative propositions E and O. The A and E propositions are referred to as universal propositions because they make reference to all the members of the subject class whereas the I and O propositions are referred to as particular propositions because they make reference to only some members of the subject class.
All human beings are mortal – Universal affirmative
E. No human beings are mortal – Universal negative
I. Some human beings are mortal – Particular affirmative
O. Some human beings are not mortal. – Particular negative.
Structure of a Categorical Proposition
A standard form categorical proposition is composed of four basic elements. These include the quantifier, the subject, the copula and the predicate. The quantifier indicates whether reference is being made to all or just some members of the subject term or class. The subject term or class is that about which the proposition is talking about. The copula is the expression which joins the predicate to the subject class. It is usually some form of the verb "to be." The predicate term or class is that which is asserted or denied of the subject.
Example: All Men are Mortal.
In the above mentioned example; 'All' is the quantifier, 'Men' is the subject, 'are' is the copula and 'Mortal' is the predicate.
Quality, Quantity and Distribution
Quality of a categorical proposition refers to whether the proposition affirms or denies the inclusion of a subject class within a predicate. Thus the affirmative or negative character of a categorical proposition is said to be its quality. For instance, an A proposition All S are P is affirmative since it states that the subject is contained within the predicate. On the other hand, an O proposition I Some S are not P is negative since it excludes the subject from the predicate.
The quantity if a categorical proposition refers to the amount of members of the subject class that are used in the proposition. The universal or the particular character of a categorical proposition is said to be its quantity. Thus, if the proposition refers to all member of the class, then it is universal in quantity. However, if the proposition does not employ all the members of the subject class it is said to be of a particular quantity. For instance, an E proposition No S are P is universal because it refers to all the members of the subject class. On the other hand, the I proposition Some S are P is particular because it refers only to some of the members of the subject class.
Quantity and Quality in summary
All S are P.
No S are P.
Some S are P.
Some S are not P.
The two terms (subject and predicate) in a categorical proposition may each be classified as distributed or undistributed. If all members of the term's class are affected by the proposition, that class is distributed; otherwise if not it is undistributed. Every proposition therefore has one of four possible distributions of terms.
An A -proposition distributes the subject to the predicate, but not the reverse. Consider the following categorical proposition: "All dogs are mammals". All dogs are indeed mammals but it would be false to say all mammals are dogs. Since all dogs are included in the class of mammals, "dogs" is said to be distributed to "mammals". Since all mammals are not necessarily dogs, "mammals" is undistributed to "dogs".
An E -proposition distributes bi-directionally between the subject and predicate. From the categorical proposition "No beetles are mammals", we can infer that no mammals are beetles. Since all beetles are defined not to be mammals, and all mammals are defined not to be beetles, both classes are distributed.
Both terms in an I-proposition are undistributed. For example, "Some Americans are conservatives". Neither term can be entirely distributed to the other. From this proposition it is not possible to say that all Americans are conservatives or that all conservatives are Americans.
In an O-proposition only the predicate is distributed. Consider the following: "Some politicians are not corrupt". Since not all politicians are defined by this rule, the subject is undistributed. The predicate, though, is distributed because all the members of "corrupt people" will not match the group of people defined as "some politicians". Since the rule applies to every member of the corrupt people group, namely, "all corrupt people are not some politicians", the predicate is distributed.
The distribution of the predicate in an O-proposition is often confusing due to its ambiguity. When a statement like "Some politicians are not corrupt" is said to distribute the "corrupt people" group to "some politicians", the information seems of little value since the group "some politicians" is not defined. But if, as an example, this group of "some politicians" were defined to contain a single person, Albert, the relationship becomes clearer. The statement would then mean, of every entry listed in the corrupt people group, not one of them will be Albert: "all corrupt people are not Albert". This is a definition that applies to every member of the "corrupt people" group, and is therefore distributed.
In short, for the subject to be distributed, the statement must be universal (e.g., "all", "no"). For the predicate to be distributed, the statement must be negative (e.g., "no", "not")
All S are P.
No S are P.
Some S are P.
Some S are not P.
The Square of Propositions
In the Aristotelian logic, the square of proposition/opposition is a diagram representing the different ways in which the four propositions of the system are logically related (opposed) to each other. The logical opposition in the square of propositions/opposition occurs among the standard form categorical propositions if they have the same subject and predicate terms and if they differ in quality and quantity or in both. The four forms of categorical propositions admit four kinds of opposition among themselves. These oppositions with reference to the square of oppositions are; contradictory opposition, contrary opposition, sub - contrary opposition, and sub - alternate opposition.
Sub – alternate
Sub - alternate
O I ISub-contraries
Analysis of the square of Opposition/ Propositions
The four relations in the traditional square of opposition may be characterized as follows:
Contradictory;- which denotes opposite truth value
Contrary;- which denotes that at least one is false (not both can be true at the same time)
Sub-contrary;- which denotes that at least one is true (not both can be false at the same time)
Sub – alternate; - which denotes that truth flows downward and falsity flows upwards.
2.2. Mediate inferences from the Square of Propositions
This is a kind of relation found between the A and O propositions and separately between the E and I propositions. The statements in this case cannot both be true and both be false at the same time. They express opposite truth values to each other respectively. Thus if a certain A proposition is given as true, then the corresponding O proposition must be treated as false and vice versa as a result of logical consequence. The same relation holds between the E and I proposition in the sense that if any E proposition is given as true, then the corresponding I proposition will necessarily be false and vice versa as a result of logical consequence.
The proposition All Men are Mortal is contradicted by the proposition Some Men are not Mortal. The proposition No Men are Mortal on the other hand is contradicted by the proposition Some Men are Mortal.
The common feature with contradictories is that they both differ in quality and quantity. While the A proposition has a Universal quantity and an Affirmative quality, the corresponding O proposition has a Particular quantity and a Negative quality. The same analysis is applicable for the E and I propositions.
This is a kind of relationship between the universal categorical propositions A and E. Though they bear the same universal quantity, they differ in quality because the former has an affirmative quality while the latter has a negative quality. The contrary opposition is different from the contradictory opposition in that it expresses only a partial opposition. For instance, if a certain A proposition is given as true, the corresponding E proposition has to be false due to the fact that at least one is false and that they cannot both be true at the same time. However, if a certain A proposition is given as false, the truth value of the corresponding E proposition could either be true or false without violating the at least one is false rule. In such a case, the E proposition has a logically indeterminate truth value. Similarly, if an E proposition is given as false, the corresponding A proposition has too, a logically indeterminate truth value.
All Cats are Animals (A)
No Cats are Animals (E)
This is a kind of relation that exist between the particular propositions I and O. though they are of the same quantity, they differ in quality since the former has an affirmative quality while the latter has a negative quality. In this type of relation, if a certain I proposition is given as true, the corresponding O proposition is has an undetermined truth value. However, if the I proposition is given as false, the corresponding O proposition is true because of the at least one is true rule.
Subcontrary relation can be summarized as follows;
Some cats are dogs
Some cats are not dogs
The I and O propositions cannot both be false, but they can both be true. Example: "Some animals are cats," and "Some animals are not cats," are both true.
This is a kind of relation between the universal propositions (the superalterns) and the particular propositions (subalterns). That is, it is a kind of relation that is manifested by the 'A' and 'I' propositions and the 'E' and 'O' propositions on the square of oppositions. Though they bare different quantities, they have the same qualities respectively. That is, the 'A' and 'I' propositions have the same affirmative qualities while the E and the O propositions have the same negative qualities. This kind of relationship is always denoted by two opposing arrows on the square of oppositions. This is such that, the downward arrow is marked 'T' and the upward arrow is marked 'F.'
These arrows can be conceived as pipelines through which truth values flow. The arrow marked T only transmits truth while the arrow marked F only transmits falsity. In a nutshell, the truth is always transmitted downwards while falsity is always transmitted upwards in subalteration relation.
Therefore, if the 'A' and 'E' propositions are given as true, their corresponding 'I' and 'O' propositions are true because truth flows from the superalterns to the subalterns. However, if any of the superaltern is given as false, its corresponding subaltern will have an indeterminate truth value because falsity in not transmitted downwards.
On the other hand, if any of the subalterns is given as false, the corresponding superaltern will be false due to the fact that falsity is transmitted upwards as indicated by the square of oppositions. However, if any of the subalterns is given as true, the corresponding superaltern has an indeterminate truth value since truth is not transmitted upwards but downwards.
Subalteration relation can be summarized as follows;
All men are mortal
Some men are mortal
No men are mortal
Some men are not mortal
Some men are mortal
All men are mortal
Some men are not mortal
No men are mortal
Immediate inference from the Square of Propositions
What we have discussed so far concerns the four forms of categorical propositions and their relationship in the square of oppositions. However, there are some of the inferences that are of interest. These inferences are always generally referred to as inferences by eduction (from any proposition taken as true, we may derive other propositions implied by it. It implies deriving a second proposition which is consistent in meaning with the original proposition). They are also referred to as immediate inferences and they are basically of three type: Conversion, Obversion and Contraposition.
This is a type of an immediate inference which is arrived by reversing the positions of the subject and predicate terms of a categorical proposition. In this case, the original proposition is referred to as the convertend and the resulting proposition is referred to as the converse. When this is applied to all the four forms of the categorical propositions, we get;
All S are P All P are S
E. No S are P No P are S
I. Some S are P Some P are S
O. Some S are not P Some P are not S.
Thus the process of converting a proposition can be referred to as inference and the inference made can either be valid or invalid. Conversion applies validly to the E and I proposition where we have the subject and predicate terms as distributed and undistributed respectively.
Therefore with the conversion of A and O propositions, the respective converse propositions are certainly not valid. Example: The converse of the convertend, "All trees are plants'' which is, "All plants are trees," is certainly an invalid inference since the latter cannot be validly inferred from the later. The converse of the convertend, "Some trees are not plants" which is "Some plants are not trees," is also certainly an invalid inference.
The conversion of an E proposition; "no Kenyans are Americans," the converse, "no American are Kenyans" in a validly inferred proposition. This is due to the fact that the converse is implied by the convertend and vice versa. It is also favored by the law of distribution since no member of the predicate class is included in the subject class/term. That is, the subject and the predicate terms are distributed. Thus the propositions "No Kenyans are Americans" and the corresponding converse "No Americans are Kenyans" can be said to be logically equivalent because each is implied by the other.
In the conversion of an I proposition "Some Kenyans are Americans" the converse, "Some Americans are Kenyans" is certainly a valid inference because it is favored by the law of distribution in the sense that both the subject and the predicate terms are undistributed in all the instances of convertend and the converse.
This is a type of inference whereby there are two steps are observed for its success. The first step being the changing of the quality of the obvertend or the original proposition and the second being the replacement of the predicate term with its complement. It should be noted that the complement of a term consists of all things which fall outside that class or term. Thus the complement of Kenyans is non – Kenyans, that is, all things that are not Kenyans. The complement of a 'horse' is 'non – horse,' that is, all things that are not horses. Therefore, plants, human beings, books, electronics, and many more are in the class or term 'non – horse'
With obversion, we are not concerned with contraries but rather with the complement of the term given. Hence the term 'winner' does not become 'losser' rather its correct complement is 'non – winner' which implies all things that are not winners.
Applied to all the four forms of the categorical proposition, we get;
Changing the quality
All S are P
No S are P
No S are non – P
No S are P
All S are P
All S are non - P
Some S are P
Some S are not P
Some S are not non – P
Some S are not P
Some S are P
Some S are non – P
A little reflection on all the types of categorical propositions shows that obversion is always valid. This is because the obverse and the obvertend have the same meaning. For instance; in obverting the A proposition "All human beings are mortal," the obverse, "Non human beings are non mortal" is logically a valid inference due to the logical equivalence implied by the obverse and the obvertend. It implies that no human being can be non – mortal or immortal which is the same as saying that all human beings are mortal. This same interpretation is the same with the remaining three forms of the categorical propositions as attested to by the table above.
It should be noted that obversion is not only always valid, but the obverse and the obvertend are also logically equivalent. Obversion is always reversible since one can be inferred from the other without altering the meaning of either the obverse or the obvertend.
This is the third form of inference which involves conversion and obversion though at different levels and perspectives. It involves taking the complement of both the subject and the predicate terms and then reversing their positions. In a particular; contraposition proceeds as follows: first we begin by obverting the original proposition, then converting it, and then obverting it again. Thus in finding the contrapositive of the A proposition, "all S are P," we will have to proceed as follows;
Step one: Obversion: No S are non – P
Second step: Conversion: No non – P are S and finally
Third step: Obversion: All non – P are non – S
Since it has been noted above that obversion is always valid, the first and the third steps in contraposition are therefore true since they both involve obversion. The only test for the validity of the contrapositive lies in the second step which involves conversion. Since conversion is valid only for the E and I propositions, therefore whenever an E or an I proposition falls in the second step in the process of contraposition, the entire process and the resulting contrapositive will denote a valid inference. If either an A or an O proposition occurs in the second step in the process of contraposition, then the entire process and the resulting contrapositive will denote an invalid inference. This is so because conversion is invalid for the A and O propositions.
Since the E and I propositions are in the second steps of the A and O propositions in the process of contraposition, we can deduce right away that contraposition is therefore valid for the A and O propositions and invalid for the E and I propositions.
The contraposition of the E, I and O propositions are as follows:
E. No S are P
Step one: Obversion: All S are non – P
Step two: Conversion: All non – P are S
Step three: Obversion: No non – P are non – S and this is the contrapositive
I: Some S are P
Step one: Obversion: Some S are not non – P
Step two: Conversion: Some non – P are not S
Step three: Obversion: Some non – P are non – S and this is the contrapositive.
O: Some S are not P
Step one: Obversion: Some S are non – P
Step two: Conversion: Some non – P are S
Step three: Obversion: Some non – P are not none – S and this is the contrapositive.
One should therefore, not lose sight of the fact that contraposition is valid only if each of the three steps is valid. If any step is invalid, then the whole inference is invalid.
Inconsistencies of a set of propositions
While analyzing the square of oppositions, it should be noted that there is no much problem encountered with the contradictory relationship apart from the problem of existential import which will be discussed latter in this paper. However, the inconsistencies of a set of propositions in the square of oppositions are brought about by the fact of dubitability of the truth values of some propositions. This is with reference to their corresponding contrary, subcontrary, and subalteration relations. Thus we can infer from this that we can possibly have contrary, subcontrary, and subalteration inconsistencies.
The guiding question in arriving at the above conclusion is why the square of oppositions which is so much exalted, cannot be used to solve the puzzle of a double truth value. That is, the conclusion that a proposition or an inference is either true or false given the truth value of a corresponding proposition.
The Existential (import) Fallacy
The problem of the truth value as seen above amounts to a major inconsistency which has severally been termed as the existential fallacy. This fallacy is committed in any inference based on the relations of Aristotelian square of oppositions, which involves propositions that we cannot safely presuppose exist. This is a type of fallacy that is committed when and only when contrary, subcontrary, and subalteration relations are used incorrectly to draw a conclusion from a premise about things that do not exist
Existential fallacy is never committed with the contradictory relation, nor is it committed in connection with conversion, obversion and contraposition all of which hold regardless of existence.
Subalteration existential fallacy can be demonstrated as follows:
"All flying witches are fearless women"
Therefore, "Some flying witches are fearless women."
The interpretation is that is flying women actually existed, the argument would be valid. But since they do not exist, the argument is invalid and commits the fallacy of existential fallacy.
Contrary existential fallacy on the other hand can also be expressed as follows:
"No wizards with magical powers are malevolent beings"
Therefore, it is false to say that, "All wizards with magical powers are malevolent spirits."
Noting that in contrary opposition, if an E proposition is given as true, the corresponding A proposition is false. The inference above thus commits a fallacy of existential import because magical wizard do not actually exist hence the argument is invalid
In conclusion, the existential fallacy occurs when contrary, subcontrary and subalteration relations are used on premises whose subject terms refer to no-existent things such as; flying horse, fire-breathing goat, a talking pig, a walking snake and so forth.
The Illicit fallacies of Contrary, Subcontrary and Subalteration relations
Besides the existential import fallacy, there are three other formal inconsistencies which may occur when the square of oppositions is used to draw conclusions on contrary, subcontrary and subalteration relations. The various inconsistencies or fallacies manifested by these relations are; illicit contrary, illicit subcontrary, and illicit subalteration.
An Illicit contrary inconsistency or fallacy results from the incorrect use and application of the contrary relation. Example: If the premise; "No viruses are structures which attack T – cells is given as false, the conclusion (which is the contrary) "All viruses are structures which attack T – cells" has a logically undetermined truth value, and so the inference is invalid. This is so because of lack of knowledge of whether the contrary to the original proposition is true or false.
Analogously, inferences that depend on an incorrect application of subcontrary relation commit the formal fallacy of illicit subcontrary. In a similar way, inferences that depend on an incorrect application of subalteration relation commit a formal fallacy of illicit subalteration.
The Possible Solutions to the Inconsistencies
There is no express solution to the problem of the inconsistencies regarding the illicit use of contrary, subcontrary, and subalteration relations. However, the only way out of this dilemma is a presupposition. That is, if we want to proceed within a certain line of thought without encountering any obstacles; we have to assume something as true or false without taking recourse on both.
The same idea of presupposition can be applied in trying to solve the problem of existential import fallacy. For instance, one can presume that flying witches actually exist in order to proceed within a certain line of thought. On the other hand, one can opt for thing that actually have real existence in trying to create an argument such as; digs, cows, flying mammal or bat, man and so forth.
Composed (Hypothetical) Propositions
Composed or hypothetical propositions are the type of propositions which signify something to be true from the supposition indicated by the particulars such as; if, unless, whenever, when e.t.c. these propositions enunciate the relationship between a condition (the antecedent) and the conditioned (the consequent). All hypothetical propositions are affirmative in the sense that if the condition happens then the consequent follows. It expresses the dependence of one affirmation or denial on another affirmation or denial. There are three types of hypothetical propositions: conditional hypothetical propositions, disjunctive hypothetical propositions and conjunctive hypothetical propositions.
3.1 The Conditional Hypothetical Propositions
A conditional proposition is one that connects two propositions in such a way that one is a condition for the other. It is a hypothetical proposition which expresses a relation in virtue of which one proposition necessarily flows from the other because a definite condition is verified or not verified. Sometimes these are called the "if" propositions.
It asserts that the subject is true on the condition that the truth of the predicate inferred. The basis of these types of propositions is the plurality of things and their relations. Conditional propositions usually combine two simple propositions.
Example: "If Otieno passes examinations, he will become the first in his village." The first part of the conditional proposition above, "If Otieno passes examinations" is the antecedent and the second part "He will become the first in his village" is the consequent. This is the reason as to why these propositions are also known as implicative propositions because the antecedent implies the consequent. It should be noted that all conditional propositions are logically universal.
Examples: "If the barometer falls, there will be a storm." "If Peter is a good boy, he will be able to go on the trip." "If I make a lot of money, then I will be able to buy a mansion."
The part of the proposition containing the "if" is called the "condition" or the "antecedent." The other part is called the "conditioned" or the "consequent." Notice there is a strict relation expressed in a conditional hypothetical proposition. The "antecedent" must be true before the "consequent" can follow.
The truth of conditional hypothetical propositions does not depend on the truth of the statements taken by themselves or individually. The truth depends on the relation between the statements.
For example, take the proposition "If the barometer falls, there will be a storm." We are not asserting that the barometer is falling. We are not asserting that a storm is coming. We are simply saying that the coming of a storm is dependent on low atmospheric pressure which is indicated by the falling of the mercury in a barometer.
In a conditional hypothetical proposition, it is the dependence of one idea on the other that is affirmed or denied. The truth of the whole statement rests on the truth of the dependence.
3.2. The Disjunctive Hypothetical Proposition
This is a kind of hypothetical proposition whose constituent simpler propositions are connected by the form either… or… statement, indicating that the implied judgments cannot be true together nor false together, but one must be true and the other must be false.
Examples: "Either the sun or the earth moves in an orbit." "An automobile is either in motion or at rest."
This form, "either… or…," makes these alternatives to be also known as alternatives or disjuncts.
This is a type of a bi-conditional proposition where two or more simple or categorical propositions are joined by a connective such as and. It expresses or indicates pure sum, simultaneity, or succession of the component propositions.
Examples: Logic is a science and art (pure sum), Otieno sleeps and Ogaba runs (simultaneity), and finally turn and go (succession).
For the copulative proposition to be true, all the constituent propositions must be true. If one simple proposition is false, then the whole proposition will be false.
Logic being the branch of philosophy that reflects upon the nature of thinking in attempting to answer the question such as: What is reasoning? What distinguishes good reasoning from bad reasoning, a good argument from a bad one, a valid argument from an invalid one? These question among others thus exalts logic to a greater height since all the other branches of philosophy employ reasoning and the correct application of these laws in any field will depend on the laws of logic.
It is only to that the laws of logic and their proper application is what makes up a valid argument. It is in the light of the test for the validity of the various propositions such as the categorical proposition and the composed hypothetical propositions that the paper has been presented. Since the propositions which in turn are used as premises in an argument, are the basic foundations of any form of any reasoning, the paper has provided a foretaste in trying to avail a better understanding and interpretation of the major tool of logic which is referred to as the square of oppositions. This paper is not much exalted since it has only emphasized on some of the basic elements of an argument with a bit of new insights noted in the understanding of the inconsistencies of a set of propositions. This thus calls for a further research in the field since the principles of reasoning are too vast to be amicable discussed within the scope of this question directed paper.
However, the aim of this paper which has been achieved was to identify the logical interpretation of the propositions, the square of oppositions and the composed proposition in the light of an argument and application. Therefore, it should be noted that in our daily activities, reason is employed and caution should be taken not to fall prey of ambiguity.