Aristotle recognized the importance of observation. But his primary concern was still with what one can rationally infer. This generated a sharp interest in the processes and patterns of reason itself, and that motivated his systematic mapping of what we call logic. His ideas suggested a vision of a logical system where all knowledge is deducible from a set of indubitable axioms. Aristotle’s logic focused on what we can directly infer from individual assertions (by immediate inference) and on what we can figure out from pairs of assertion (by using categorical syllogisms).
Aristotle makes room for the important role that observation plays in thinking—providing data for our contemplation and raw material for our inferring.
A. His work on biological taxonomy illustrates this nicely.
B. Alexander the Great is said to have been so impressed with Aristotle’s interest in observing and understanding the natural world that he regularly sent samples to his former teacher.
But Aristotle’s primary concern was still with what one can rationally infer.
A. For Aristotle, the bases of our inferences are observations and generalizations of them, rather than our memories or intuitions of the Forms.
B. But he is still working with a vision (that started with Plato and culminates with Descartes) that all general knowledge is deducible from foundations that are (or nearly are) indubitable and directly apprehended in one way or another.
This generated his sharp interest in the processes and patterns of reasoning itself, and that motivated his systematic mapping of what we call logic.
A. Logic is not necessarily a map of how we, in fact, think. It is, rather, a rational reconstruction of what constitutes reliable thought.
B. The use of logic presupposes that every statement that we use in our reasoning is either true or false, never both and never neither, and that the denial of a true statement is false and the denial of a false statement is true. These “laws of thought” are often summarized as the Law of Contradiction, the Law of Identity, and the Law of Excluded Middle.
C. Although we are confining our attention to logic with two values (true or false), with no tertium quid available, contemporary logicians have discovered logical systems that can work with three values or more.
Aristotle’s logic focused on what we can directly infer from individual categorical propositions and on what we can infer from pairs of them in categorical syllogisms.
Categorical propositions declare what is or is not the case.
A. Every categorical statement has a subject term and a predicate term and asserts something about the relationship between the sets (categories) named by those terms.
B. A categorical proposition can be analytic or synthetic. The meaning of the predicate of an analytic categorical proposition is “contained” in its subject’s meaning. The meaning of the predicate of a synthetic categorical proposition “adds to” its subject’s meaning.
C. Categorical statements with the same subject and predicate terms can differ in quality (affirmative or negative) and quantity (universal or particular).
D. The standard forms for categorical propositions are:
Universal affirmative (called an A proposition): All S are P.
negative (called an E proposition): No S are P.
Universal Particular affirmative (called an I proposition): Some S are P.
Particular negative (called an O proposition): Some S are not P.
E. Why are the propositions labeled A, E, I, and O?
1. The A and O propositions are both affirmative, and the letters A and O are the vowels in the Latin (and English) word AFFIRM.
2. The E and O propositions are both negative, and E and O are the vowels in the Latin word NEGO (meaning “I deny”).
Immediate inferences from standard-form categorical propositions include obversion, conversion, and contraposition.
A. Note that the complement of a class name is the name of the class comprised of everything not in the class named by the term itself. For example, the complement of the term heroes is non-heroes (not cowards!). Note also that not is an indicator of a proposition’s quality, but non- is part of the class name.
B. Obversion is changing the quality of a categorical proposition (affirmative to negative or negative to affirmative) and replacing the predicate term with its complement. Obversion works reliably for A, E, I, and O propositions.
A: All S are P. Obverse: No S are non-P. (reliable)
E: No S are P. Obverse: All S are non-P. (reliable)
I: Some S are P. Obverse: Some S are not non-P. (reliable)
O: Some S are not P. Obverse: Some S are non-P. (reliable)
C. Conversion amounts to swapping the subject and predicate terms of a categorical proposition. Conversion works reliably only with E and I propositions. CONVERSION A: All S are P. Converse: All P are S. (unreliable) E: No S are P. Converse: No P are S. (reliable) I: Some S are P. Converse: Some P are S. (reliable) O: Some S are not P. Converse: Some P are not S. (unreliable)
D. Contraposition amounts to replacing the subject of a categorical proposition with the complement of its predicate and replacing the predicate with the complement of the subject. Contraposition works reliably only with A and O propositions.
A: All S are P. Contrapositive: All non-P are nonS. (reliable)
E: No S are P. Contrapositive: No non-P are nonS. (unreliable)
I: Some S are P. Contrapositive: Some non-P are non-S. (unreliable)
O: Some S are not P. Contrapositive: Some non-P are not non-S. (reliable)
[Courtesy: Professor James Hall]