Department of Philosophy (PHI)
Categorical Statements
Categorical statements were the subject of the earliest formal treatments of logic. Aristotle (384-322 BC) distinguished four types of categorical statement on the basis of quantity (universal or existential) and quality (affirmative or negative). The idea expressed is that that all or some members of one class (the subject) are members or non-members of another class (the predicate). Schematically:
| All Ss are Ps | universal affirmative |
| All Ss are non-Ps | universal negative |
| Some Ss are Ps | existential affirmative |
| Some Ss are non-Ps | existential negative |
The standard form for a categorical statement is Quantifier + Subject + Predicate. Some examples:
| Quantifier | Subject | Predicate | Aristotlean Classification |
| All | people | dream. | Universal Affirmative |
| No | insect | is harmful. | Universal Negative |
| Only | birds | fly South. | N/A |
| Some | planets | are gaseous. | Existential Affirmative |
| Not all | fish | taste good. | Existential Negative |
| Not only | politicians | lie. | N/A |
Universal Statements
All, only, no and their synonyms (each, every, none but, not one etc.) are the quantifiers used in universal statements.
Universal statements say that every member of one class belongs to another class. In the examples:
'All people dream' says that every person is a member of the class of dreamers;
'No insect is harmful' says that every insect belongs to the class of harmless things;
'Only birds fly South' says that everything that flies South belongs to the class of birds.
Existential Statements
Some, not all, not only and their synonyms are the quantifiers used in existential statements.
Existential statements say that at least one member of one class belongs to another class. In the examples:
'Some planets are gaseous' says that at least one planet belongs to the class of gaseous things;
'Not all fish taste good' says that at least one fish belongs to the class of things that do not taste good;
'Not only politicians lie' says that at least one liar is a non-politician.
Interpretation of Categorical Statements
Although universal statements say that every member of one class belongs to another, it is useful to paraphrase or interpret them as non-existence claims. Such an interpretation brings the logical relationship between universal statements and existential statements into sharper focus, and it also helps in the construction of their Venn diagrams. The equivalence of universal statements to non-existence claims is most easily seen with respect to 'no'-type statements. To say, for instance, that no dog is green, is simply to deny the existence of green dogs. In general, to say that no Ss are Ps is to say that Ss that are Ps do not exist.
The other two types of universal statement can also be seen to be equivalent to non-existence claims. To say that all men are mortal, for example, is to say that no man is immortal, i.e., that men that are immortal don't exist. In general, to say that all Ss are Ps is to say that Ss that are not Ps don't exist. And to say that only humans play guitar is to say that nothing other than a human plays guitar. 'Only Ss are Ps' means 'non-Ss that are Ps don't exist'.
Existential statements are the exact opposites of universal statements. More specifically:
'Some Ss are Ps' and 'No Ss are Ps' are exact opposites (contradictories);
'Not all Ss are Ps' and 'All Ss are Ps' are exact opposites;
'Not only Ss are Ps' and 'Only Ss are Ps' are exact opposites.
The exact opposite of the claim that things of a certain kind do
not exist is the claim that things of that same kind do exist. Thus the
interpretations for all six main types of categorical statements are as
follows:
| All Fs are Gs | = | Fs that are not Gs don't exist |
| Only Fs are Gs | = | Non-Fs that are Gs don't exist |
| No Fs are Gs | = | Fs that are Gs don't exist |
| Not all Fs are Gs | = | Fs that are not Gs do exist |
| Not only Fs are Gs | = | Non-Fs that are Gs do exist |
| Some Fs are Gs | = | Fs that are Gs do exist |
Universal Statements: Existential and Hypothetical interpretations
The interpretations Aristotle gave for universal statements differ from the modern interpretations summarized above. According to Aristotle, universal statements imply existence. For example, the statement that all politicians are liars would be interpreted by Aristotle as meaning not only that truthful politicians don't exist, but also that politicians (hence lying politicians) do exist. Likewise, the statement that no Martian invader has been taken captive would be interpreted by Aristotle as meaning not only that captive Martian invaders don't exist, but also that Martian invaders do exist.
The modern interpretation is usually called the "hypothetical" interpretation because it is equivalent to treating universal statements as universalized conditional (:hypothetical") statements:
all Fs are Gs = if any thing is an F, it is a G = Fs that are not Gs don't exist;
only Fs are Gs = if any thing is a G, it is an F = non-Fs that are Gs don't exist;
no Fs are Gs = if any thing is an F, it is not a G = Fs that are Gs don't exist.
Proponents of the Aristotelian interpretation argue that people often reason from universal premises to existential conclusions, indicating that the universal statements have existential implications. Advocates of the modern interpretation handle such cases by saying that the reasoners are making use of unstated premises, something we do quite frequently anyway. A person who argues that some animals have fleas because all dogs are animals and all dogs have fleas is making use of the unstated premise that dogs exist, which need not be stated since everyone already knows that it is true. Modernists also point to the many cases in which we can apparently known that all Fs are Gs, or that no Fs are Gs, without knowing whether Fs even exist. For example, a professor can apparently know that all students who make As on every assignment will make an A in the course without knowing whether there are any such students, and a gardener can apparently know that no unicorn has eaten his radishes without knowing that unicorns exist.
I will proceed on the assumption that the hypothetical interpretation is the correct one for universal statements.
The Recipe For Interpretations
For categorical statements in any of the six forms discussed above, interpretation can easily be produced by the following procedure.
First, select the terms to be used in the interpretation according to the quantifier used. If the quantifier is 'all' or 'not all', use the subject as is, but negate the predicate. If the quantifier is 'only or 'not only', use the predicate as is, but negate the subject. If the quantifier is 'no' or 'some' use both the subject and predicate without negating either. Use these terms in the first, "_____s that are _____s", part of the interpretation. It doesn't matter which term goes in which position because, for example, Fs that are Gs are necessarily the same things as Gs that are Fs.
For the remainder of the interpretation, use "don't exist" if the quantifier is universal ('all', 'only' or 'no'), "do exist" if it is existential ('not all', 'not only', or 'some').
This recipe is summed up in the following table:
| All Ss are Ps | Not all Ss are Ps | Ss that are not Ps... |
| Only Ss are Ps | Not only Ss are Ps | Non-Ss that are Ps... |
| No Ss are Ps | Some Ss are Ps | Ss that are Ps... |
| ...don't exist | ...do exist |
The connection between interpretations of categorical statements
and the Venn diagrams for the statements is quite simple.
The first, "_____s that are _____s", part of the interpretation
indicates which part(s) of the diagram to mark.
The second, "don't exist/do exist", part of the interpretation indicates
which type of mark to use.