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In Peter of Spain’s Summaries of Logic he describes different laws of logic, and how to interpret a variety of propositions. He illustrates these ideas using what is known as a square of opposition, in which a set of four statements is placed in the four corners of the square. These statements are placed in each of the four corners of the square based on their logical relation to one another. If the statements are directly above each other, for example the statement in the top left corner and the statement in the bottom left corner, then they are what are known as subalternates. If the statements are in the top right and left corners of the square then they are contraries, while if they are in the bottom right and left corners of the square they are subcontraries. If the statements are diagonally across from each other in the square then they are contradictories.
In the square of opposition, the statement that goes in the top left corner is that “every A is B”, the statement in the top right corner is that “no A is B”. The statement in the bottom left corner is that “a-certain A is B”, and in the bottom right corner it is “ a-certain A is not B”. There are three kinds of propositions: natural, contingent, and eliminated. Natural matter occurs when the predicate is of the being or its property, such as “a man is an animal”. Contingent matter is when the predicate can be present or absent from the subject, such as “man is white” or “man is not white”. Finally, eliminated matter is when the predicate cannot fit the subject, for example, “man is a donkey”, a man can never be a donkey because it is genetically impossible.
If we take the statement “ no donkey is a fish” and assume it to be true, then we can look to find its contrary, subalternate, and its contradictory. When we look at our truth, we see that it belongs in the top right corner of the square of opposition because it states that “no A is B”, by placing it there we can see that its contrary would be in the top left corner and would be in the form “every A is B” or in this case, every donkey is a fish. We can also see that the subalternate to this statement would be directly below it and would be in the form “a-certain A is not B” or in this case, a-certain donkey is not a fish. Lastly, when looking to see its contradictory we must look diagonally across the square to the bottom left corner, we can see that the form of the statement will be “a-certain A is B” and in this particular case, a-certain donkey is a fish.
The square of opposition allows us to conclude a few things about the truth and falsity of these propositions. We will start by looking to the law of contraries, where Peter states that “if one is true, the other is false, but not conversely; for it is possible for both to be false in contingent matter” (Peter of Spain, 1.14 p. 7). In this particular case, the statement about the donkey is eliminated matter because the predicate cannot fit the subject. According to the law, it is always the case that when one is true the other is false, so we know that it is always false to state that “every donkey is a fish”. Next, we can look to the law of subalternations, which states that “if the universal is true, the particular is true, but not conversely, and if the particular is false, the universal is false, but not conversely” (1.14, p. 7). In this case, the universal, every donkey is a fish, is false, and so we can see that a-certain donkey is not a fish is true. Lastly, we can look to the law of contradictories which states “that if one is true, the other is false, and conversely; for in no matter is it possible for both to be true, or both to be false (1.14, p. 7). The truth in this case is that no donkey is a fish, so its contradictory, a-certain donkey is a fish, has to be false.
In the square of opposition, the statement that goes in the top left corner is that “every A is B”, the statement in the top right corner is that “no A is B”. The statement in the bottom left corner is that “a-certain A is B”, and in the bottom right corner it is “ a-certain A is not B”. There are three kinds of propositions: natural, contingent, and eliminated. Natural matter occurs when the predicate is of the being or its property, such as “a man is an animal”. Contingent matter is when the predicate can be present or absent from the subject, such as “man is white” or “man is not white”. Finally, eliminated matter is when the predicate cannot fit the subject, for example, “man is a donkey”, a man can never be a donkey because it is genetically impossible.
If we take the statement “ no donkey is a fish” and assume it to be true, then we can look to find its contrary, subalternate, and its contradictory. When we look at our truth, we see that it belongs in the top right corner of the square of opposition because it states that “no A is B”, by placing it there we can see that its contrary would be in the top left corner and would be in the form “every A is B” or in this case, every donkey is a fish. We can also see that the subalternate to this statement would be directly below it and would be in the form “a-certain A is not B” or in this case, a-certain donkey is not a fish. Lastly, when looking to see its contradictory we must look diagonally across the square to the bottom left corner, we can see that the form of the statement will be “a-certain A is B” and in this particular case, a-certain donkey is a fish.
The square of opposition allows us to conclude a few things about the truth and falsity of these propositions. We will start by looking to the law of contraries, where Peter states that “if one is true, the other is false, but not conversely; for it is possible for both to be false in contingent matter” (Peter of Spain, 1.14 p. 7). In this particular case, the statement about the donkey is eliminated matter because the predicate cannot fit the subject. According to the law, it is always the case that when one is true the other is false, so we know that it is always false to state that “every donkey is a fish”. Next, we can look to the law of subalternations, which states that “if the universal is true, the particular is true, but not conversely, and if the particular is false, the universal is false, but not conversely” (1.14, p. 7). In this case, the universal, every donkey is a fish, is false, and so we can see that a-certain donkey is not a fish is true. Lastly, we can look to the law of contradictories which states “that if one is true, the other is false, and conversely; for in no matter is it possible for both to be true, or both to be false (1.14, p. 7). The truth in this case is that no donkey is a fish, so its contradictory, a-certain donkey is a fish, has to be false.
