Department of Philosophy (PHI)

Equivalences Among Categorical Statements

Two statements are equivalent if they have the same logical meaning, so that they cannot possibly differ in truth value. By the same token, each of two equivalent statements logically implies the other, and each can be substituted for the other in arguments without any change in validity. As a result, it is often possible to determine whether an unfamiliar form of argument is valid or invalid by transforming it into a familiar form through substitution of equivalent statements for its premises and/or conclusion. For instance, the confusing argument on the left is actually equivalent to old familiar BARBARA (on the right):

Only non-Fs are non-Gs	All Fs are Gs
All non-Hs are non-Gs	All Gs are Hs
No Fs are non-Hs	All Fs are Hs

Each categorical statement is actually equivalent to five others in "standard" form. For example, No Fs are Hs, No Hs are Fs, All Fs are non-Hs, All Hs are non-Fs, Only non-Fs are Hs, and Only non-Hs are Fs are all equivalent to one another. Each says, in essence, that Fs that are Hs do not exist.

Categorical statements can be checked for equivalence by diagramming, or by simply comparing their interpretations (keeping in mind the fact that Fs that are Gs are the same things as Gs that are Fs). But it is more practical to memorize a few simple rules or laws of equivalence rather than testing them repeatedly. The main laws of equivalence for simple categorical statements are summarized in the remainder of this document.

Conversion = swapping subject and predicate terms.

Conversion laws:

a NO statement = its converse

a SOME statement = its converse

the other four types are not equivalent to their converses

Note: in these laws the quantifiers are assumed to remain the same.

Examples:

No non-Fs are Gs = No Gs are non-Fs

Some Fs are non-Hs = Some non-Hs are Fs

Contraposition = swapping terms and negating both.

Contraposition laws:

an ALL statement = its contrapositive

an ONLY statement = its contrapositive

a NOT ALL statement = its contrapositive

a NOT ONLY statement = its contrapositive

the other two types are not equivalent to their contrapositives

Note: in these laws the quantifiers are assumed to remain the same.

Examples:

All Fs are Gs = All non-Gs are non-Fs

Not only Gs are Hs = Not only non-Hs are non-Gs

Only non-Fs are Hs = Only non-Hs are Fs (double negation cancelled)

Not all non-Fs are non-Gs = Not all Gs are Fs (double negations cancelled)

All-Only (Not all-Not only) laws

an ALL statement = an ONLY statement with the terms swapped

a NOT ALL statement = a NOT ONLY statement with the terms swapped

an ALL statement = an ONLY statement with both terms negated

a NOT ALL statement = a NOT ONLY statement with the terms negated

Examples:

Only Gs are Fs = All Fs are Gs

Not only Fs are Gs = Not all Gs are Fs

Only Fs are Gs = All non-Fs are non-Gs

Not only non-Fs are non-Gs = Not all Fs are Gs

ALL-NO / NOT ALL-SOME Laws:

an ALL statement = a NO statement with the predicate negated

a NO statement = an ALL statement with the predicate negated

a NOT ALL statement = a SOME statement with the predicate negated

a SOME statement = a NOT ALL statement with the predicate negated

Note: in these four laws, the subjects are assumed to be the same

Examples:

No Fs are Gs = All Fs are non-Gs

Not all Fs are Gs = Some Fs are non-Gs

No Fs are non-Gs = All Fs are Gs

Not all non-Fs are non-Gs = Some non-Fs are Gs

ONLY-NO / NOT ONLY-SOME Laws:

an ONLY statement = a NO statement with the subject negated

a NO statement = an ONLY statement with the subject negated

a NOT ONLY statement = a SOME statement with the subject negated

a SOME statement = a NOT ONLY statement with the subject negated

Note: in these four laws, the predicates are assumed to be the same

Examples:

Only Fs are Gs = no non-Fs are Gs

Not only non-Fs are Gs = Some Fs are Gs

Only non-Fs are Gs = No Fs are Gs

Not only non-Fs are non-Gs = Some Fs are non-Gs

General Summary. All of these equivalences are summed up in the following charts, one for universal statements and one for existentials. To transform a statement into an equivalent statement with the same quantifier, follow the rule in that quantifier's corner (contraposition or conversion). To transform a statement into an equivalent statement with a different quantifier, follow the directions along the line between the two quantifiers.