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Immediate inference
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An immediate inference is an inference which can be made from only one statement or proposition.[1] For instance, from the statement "All toads are green", the immediate inference can be made that "no toads are not green" or "no toads are non-green" (Obverse). There are a number of immediate inferences which can validly be made using logical operations. There are also invalid immediate inferences which are syllogistic fallacies.
Valid immediate inferences
[edit]Converse
[edit]- Given a type E statement, "No S are P.", one can make the immediate inference that "No P are S" which is the converse of the given statement.
- Given a type I statement, "Some S are P.", one can make the immediate inference that "Some P are S" which is the converse of the given statement.
Obverse
[edit]- Given a type A statement, "All S are P.", one can make the immediate inference that "No S are non-P" which is the obverse of the given statement.
- Given a type E statement, "No S are P.", one can make the immediate inference that "All S are non-P" which is the obverse of the given statement.
- Given a type I statement, "Some S are P.", one can make the immediate inference that "Some S are not non-P" which is the obverse of the given statement.
- Given a type O statement, "Some S are not P.", one can make the immediate inference that "Some S are non-P" which is the obverse of the given statement.
Contrapositive
[edit]- Given a type A statement, "All S are P.", one can make the immediate inference that "All non-P are non-S" which is the contrapositive of the given statement.
- Given a type O statement, "Some S are not P.", one can make the immediate inference that "Some non-P are not non-S" which is the contrapositive of the given statement.
Invalid immediate inferences
[edit]Cases of the incorrect application of the contrary, subcontrary and subalternation relations (these hold in the traditional square of opposition, not the modern square of opposition) are syllogistic fallacies called illicit contrary, illicit subcontrary, and illicit subalternation, respectively. Cases of incorrect application of the contradictory relation (this relation holds in both the traditional and modern squares of opposition) are so infrequent, that an "illicit contradictory" fallacy is usually not recognized. The below shows examples of these cases.
Illicit contrary
[edit]- It is false that all A are B, therefore no A are B.
- It is false that no A are B, therefore all A are B.
Illicit subcontrary
[edit]- Some A are B, therefore it is false that some A are not B.
- Some A are not B, therefore some A are B.
Illicit subalternation and illicit superalternation
[edit]- Some A are not B, therefore no A are B.
- It is false that all A are B, therefore it is false that some A are B.
See also
[edit]References
[edit]- ^ Churchill, Robert Paul (1990). Logic: An Introduction (2nd ed.). New York: St. Martin's Press. p. 162. ISBN 0-312-02353-7. OCLC 21216829.
Immediate inference is the assumption, without intervening—or 'mediating'—premises, that because one categorical statement is true (or false), a logically equivalent categorical statement must also be true (or false).
Immediate inference
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Immediate inference is a fundamental concept in categorical logic, referring to the process of deriving a conclusion directly from a single premise by rearranging or transforming its terms, without requiring additional premises.[1] This form of deductive reasoning applies to the four standard categorical propositions—A (All S are P), E (No S are P), I (Some S are P), and O (Some S are not P)—and relies on logical equivalences that preserve the truth value of the original statement.[2] Unlike mediate inferences, such as syllogisms, which involve multiple premises and three terms, immediate inferences focus solely on the relations between a subject and predicate or their complements.[1]
The main techniques of immediate inference include conversion, obversion, contraposition, and subalternation, each with specific rules of validity depending on the type of proposition. Conversion involves transposing the subject and predicate terms while preserving the quality (affirmative or negative); it is fully valid for E propositions (e.g., "No S are P" implies "No P are S") and I propositions (e.g., "Some S are P" implies "Some P are S"), but limited for A propositions (e.g., "All S are P" implies only "Some P are S").[3] Obversion changes the quality of the proposition and replaces the predicate with its complement, making it valid for all four types (e.g., "All S are P" implies "No S are non-P").[2] Contraposition switches the subject and predicate to their complements, which is valid for A and O propositions but not for E or I.[1] Subalternation, meanwhile, draws a particular conclusion from a universal premise of the same quality (e.g., from A to I, or E to O), though the reverse does not hold.[3]
These inferences are rooted in the traditional square of opposition, which maps the logical relationships—contradiction, contrariety, subcontrariety, and subalternation—among the categorical propositions, enabling the testing of validity through methods like Venn diagrams or existential assumptions.[1] In modern interpretations, validity can be affected by the existential import of universal propositions (e.g., empty classes like "unicorns" render some subalternations invalid), contrasting with the traditional view that assumes existence.[1] Immediate inferences serve as building blocks for more complex arguments in syllogistic logic, providing a straightforward way to expand or rephrase premises while maintaining deductive rigor.[2]
In these forms, "some" denotes at least one instance, avoiding existential import assumptions in modern interpretations unless specified.[5][6][4]
Examples illustrate each form clearly: the A proposition "All dogs are mammals" asserts universal inclusion; the E proposition "No cats are dogs" asserts universal exclusion; the I proposition "Some birds are flightless" asserts partial inclusion; and the O proposition "Some fruits are not apples" asserts partial exclusion.[4][6]
Symbolic notation uses the letters A, E, I, and O to represent these forms concisely in logical analysis. Venn diagrams provide a visual notation, using two overlapping circles for S and P: A shades the part of S outside P; E shades the overlap between S and P; I places an "X" in the overlap; and O places an "X" in the part of S outside P. These representations aid in understanding the class relations without implying further inferences.[5][6]
This table illustrates how assuming existential import preserves the interdependencies, such as the flow of truth downward (from universal to particular) and falsity upward in subalternation.[8]
Fundamentals of Categorical Logic
Standard Forms of Categorical Propositions
Categorical propositions are statements in traditional logic that relate two classes or categories, typically denoted as subject (S) and predicate (P), through a copula (such as "are" or "are not") and a quantifier that specifies the extent of the relation. These propositions assert either inclusion or exclusion between the classes, either completely (universal) or partially (particular), and either affirmatively or negatively. This structure forms the basis of categorical logic, as developed from Aristotelian principles and formalized in medieval and modern treatments.[4][5] The four standard forms of categorical propositions are distinguished by their quantity (universal or particular) and quality (affirmative or negative), labeled with the vowels A, E, I, and O from the Latin phrases used in scholastic logic. These forms are:| Form | Statement | Quantity | Quality | Description |
|---|---|---|---|---|
| A | All S are P | Universal | Affirmative | Every member of the subject class S is included in the predicate class P. |
| E | No S are P | Universal | Negative | No member of the subject class S is included in the predicate class P. |
| I | Some S are P | Particular | Affirmative | At least one member of the subject class S is included in the predicate class P. |
| O | Some S are not P | Particular | Negative | At least one member of the subject class S is excluded from the predicate class P. |
Relations in the Square of Opposition
The square of opposition is a diagrammatic representation in classical categorical logic that depicts the pairwise logical relations among the four standard forms of categorical propositions: universal affirmative (A: "All S are P"), universal negative (E: "No S are P"), particular affirmative (I: "Some S are P"), and particular negative (O: "Some S are not P").[7] These relations—contradiction, contrariety, subcontrariety, and subalternation—enable immediate inferences by establishing how the truth or falsity of one proposition determines the status of others.[7] The diagram arranges the propositions at the corners of a square, with A at the upper left, E at the upper right, I at the lower left, and O at the lower right; horizontal lines connect contraries and subcontraries, vertical lines link subalterns, and diagonals indicate contradictories.[7] This framework originated in Aristotelian logic, where Aristotle distinguished basic oppositions like contradiction and contrariety in his Prior Analytics (circa 350 BCE), but the full square diagram emerged later with the Roman philosopher Apuleius in the 2nd century CE, who introduced subcontrariety, and was further formalized by Boethius in the 6th century CE during the medieval scholastic period.[7] In traditional Aristotelian logic, the relations hold under the assumption of existential import, whereby universal propositions (A and E) presuppose the existence of at least one member of the subject class (S), ensuring that the propositions refer to actual entities rather than merely possible ones.[8] Particular propositions (I and O) inherently possess existential import due to their quantifier "some," which asserts existence, while this assumption for universals supports the full set of oppositional relations; without it, as in modern Boolean logic, only contradictories remain valid.[8] The relations are defined as follows: contradictories (A–O and E–I) cannot both be true or both false, so they always have opposite truth values—for instance, if A is true, O must be false, and vice versa.[7] Contraries (A–E) cannot both be true but can both be false; thus, if A is true, E is false, but if A is false, E may be true or false.[7] Subcontraries (I–O) cannot both be false but can both be true; accordingly, if I is false, O must be true, but if I is true, O may be true or false.[7] Subalterns (A–I and E–O) involve one-way implication: if the universal (A or E) is true, the corresponding particular (I or O) is true, and if the particular is false, the universal is false; however, the converse does not hold—for example, if A ("All humans are mortal") is true, then I ("Some humans are mortal") is true, but I true does not entail A true.[7] These truth-value implications can be summarized textually for clarity:| Proposition | If True | If False |
|---|---|---|
| A | E false (contrariety); O false (contradiction); I true (subalternation) | E undetermined; O undetermined; I undetermined |
| E | A false (contrariety); I false (subalternation); O true (contradiction) | A undetermined; I undetermined; O undetermined |
| I | A undetermined (subalternation); E undetermined; O undetermined (subcontrariety) | A false (subalternation); E undetermined; O true (subcontrariety) |
| O | A undetermined; E undetermined (subcontrariety); I undetermined (contradiction) | A true (contradiction); E undetermined; I false (subcontrariety) |
Valid Forms of Immediate Inference
Conversion
Conversion is a form of immediate inference in categorical logic whereby a new proposition, called the converse, is derived by interchanging the subject term (S) and the predicate term (P) of the original proposition, known as the convertend, while preserving the quantity and quality of the proposition.[9] This operation relies on the standard forms of categorical propositions (A: All S are P; E: No S are P; I: Some S are P; O: Some S are not P) as its foundation.[10] The validity of conversion depends on the type of proposition. For universal negative (E) propositions, conversion is fully valid: "No S are P" implies "No P are S," as the complete exclusion of one class from another is symmetric. For instance, "No reptiles are mammals" converts to "No mammals are reptiles," maintaining the truth value because the relationship of mutual exclusion holds regardless of term order.[10][11] Similarly, for particular affirmative (I) propositions, conversion is fully valid: "Some S are P" implies "Some P are S," reflecting the symmetry in partial overlap between classes. An example is "Some artists are musicians," which converts to "Some musicians are artists," as the existence of shared members in both directions is equivalent.[9][10] In contrast, conversion is invalid for universal affirmative (A) and particular negative (O) propositions. For A propositions, "All S are P" does not imply "All P are S," because the inclusion of S within P does not guarantee the reverse; the predicate class may encompass additional members beyond the subject. For example, "All dogs are mammals" is true, but its converse "All mammals are dogs" is false, as mammals include non-dogs like cats.[11][10] For O propositions, "Some S are not P" does not imply "Some P are not S," since the original asserts only that not all S belong to P, without symmetrically excluding parts of P from S; the converse may fail if all P are included in S. Consider "Some cats are not black," which is true, but "Some black things are not cats" does not necessarily follow from it alone, as it depends on external facts about black things.[9][11] Attempting full conversion for A or O propositions results in the fallacy of illicit conversion.[10] The convertibility of E and I propositions aligns with their positions in the square of opposition, where E and I are subcontraries (in the modern interpretation without existential import), allowing symmetric inferences due to their particular or fully negative nature, whereas A and O's universal or partially negative structures break this symmetry.[11]Obversion
Obversion is a form of immediate inference in categorical logic that transforms a proposition by reversing its quality—changing an affirmative statement to negative or vice versa—while replacing the predicate term with its complement, denoted as "non-P," which refers to the class of all things that are not P.[12][11] This operation preserves the truth value of the original proposition, making the obverse logically equivalent to the obvertend across all four standard categorical forms (A, E, I, O).[9][13] The specific transformations for each form are as follows:- For an A proposition ("All S are P"), obversion yields an E proposition: "No S are non-P."
- For an E proposition ("No S are P"), obversion yields an A proposition: "All S are non-P."
- For an I proposition ("Some S are P"), obversion yields an O proposition: "Some S are not non-P."
- For an O proposition ("Some S are not P"), obversion yields an I proposition: "Some S are non-P."
- The A proposition "All humans are mortal" obverts to "No humans are immortal" (non-mortal).[13]
- The E proposition "No dogs are cats" obverts to "All dogs are non-cats."[9]
- The I proposition "Some birds are penguins" obverts to "Some birds are not non-penguins."[11]
- The O proposition "Some birds are not penguins" obverts to "Some birds are non-penguins."[12]