Propositions

Grammar is the study of sentences. Logic is the study of arguments, which in turn are made up of sentences. Before you can start studying logic, you must first learn some of the basic principles of grammar.

What Is a Proposition?

Logic, as practiced by the students of Aristotle, deals with sentences of a particular kind: propositions. Aristotelian logic is sometimes called term logic or two-term theory because a proposition contains two terms: a subject and a predicate. Grammatically, the subject of a proposition consists of some noun or noun phrase. The predicate is a verb phrase that says something about that subject. The main verb in that verb phrase is in the indicative mood. In other words, the predicate of a proposition says something about the subject. In a proposition, the main verb in the predicate is in the indicative mood. Thus, the sentence is a statement of fact, which can be described as either true or false.

Notice that there are many sentences that are not statements of fact. Questions are not statements of fact. A question may prompt you to answer yes or no, but the question itself is neither true nor false. The verb in the predicate of a question is in the interrogative mood. Similarly, a command or request may be wise or foolish, and it may be obeyed or disobeyed. However, the command or request itself cannot be true or false. The verb in the predicate of a command or request is in the imperative mood.

In English, we sometimes use auxiliary verbs such as can, will, or should to express the mood of a verb. Modal auxiliary verbs can serve three basic kinds of functions. Epistemic modality deals with questions of how likely something is to be true or how certain we are of whether something is true. Deontic modality deals with what we think ought to be. We use deontic modality to express promises or threats, commands or requests, and wishes or desires. Dynamic modality deals with a subject’s ability or willingness to act. Sometimes, it is hard to tell what kind of modality is intended. For example, if a child says that he can go to the park, does he mean that he is physically capable of going to the park or that he has his parents’ permission to go to the park.

In general, statements that include modal auxiliary verbs are beyond the scope of an introductory course in logic. To deal with some kinds of modality, such as statements about what is possible, you need to use an advanced form of logic called modal logic. To deal with other kinds of modality, such as what you think ought to be, you enter the realm of rhetoric, which is the art of persuasion.

 Sentence Is it a proposition?
Poodles have curly hair Proposition. It has a subject (poodles) and a predicate (have curly hair) and the main verb in the predicate is in the indicative mood (statement of fact).
Poodles have wings Proposition. It has a subject (poodles) and a predicate (have wings) and the main verb in the predicate is in the indicative mood (statement of fact). Note that a proposition can be false.
Do poodles have wings? Not a proposition. It has a subject (poodles) and a predicate (do have wings), but the verb in the predicate is in the interrogative mood (question).
Clip that poodle’s wings! Not a proposition. It has an implied subject (you) and a predicate (clip that poodle’s wings), but the verb in the predicate is in the imperative mood (command or request).
Poodles can have wings Proposition, in modal logic. Modal logic deals with questions about what is possible or impossible and what is likely or unlikely.
Poodles should have wings Not a proposition. It has a subject (poodles) and a predicate (should have wings), but note the modal auxiliary verb (should) in the predicate. This sentence is an expression of an attitude about the idea of poodles having wings, not a statement about whether poodles do or do not have wings.

Truth Values

If a proposition’s predicate tells the truth about the proposition’s subject, then the proposition is true: its truth-value is true. If, on the other hand, the proposition’s predicate tells a lie about the proposition’s subject, then the proposition is false: its truth-value is false.

The Fourfold Scheme of Categorical Propositions

A proposition contains a subject (S) and a predicate ℗. Categorical propositions are propositions that say something about all or some members of a category. The four basic types of categorical propositions, labeled A, I, E, and O:

Code Quantifier Subject Copula Predicate Type Example
A All S are P universal affirmatives All poodles are reptiles
E No S are P universal negatives No poodles are reptiles
I Some S are P particular affirmatives Some poodles are reptiles
O Some S are not P particular negatives Some poodles are not reptiles

Note that if you know the truth value of one of these statements, you can use it to figure out the truth values of some of the other statements. The relationships between the truth-values of A, E, I, and O are shown in the Square of Opposition.

aristotlesquare

For example, If you know that the A statement is true, you know that the E statement is false. The relationships of the truth values of A, E, I and O are shown in the Square of Opposition. Note that the Square of Opposition is based on the assumption that the statements are about a category that contains at least one member.

Contrary statements—If a set of statements is contrary, they can all be false at the same time, but they cannot all be true at the same time. If you know that A is true, then you know that E cannot be true. Likewise, if you know that E is true, then you know that A is false. However, A and E can both be false at the same time.

Subcontrary statements—If a set of statements are subcontrary, they can all be true at the same time, but they cannot all be false at the same time. For example, I and O can both be true at the same time, but they cannot both be false at the same time.

Subaltern statements (implications)—If A is true, then I must be true (at least if the category that A refers to is not empty). If E is true, then O must be true (again, as long as the category that E refers to is not empty). However, the fact that I is true does not mean that A is true. Nor does the fact that O is true mean that E is true. If you make the argument if I then A, or if you make the argument if O then E, you make an error in reasoning called illicit subalternation. It is a form of overgeneralization (e.g., some horses are white, therefore all horses are white, or some horses are not white, therefore no horses are white). Illicit subalternation is an example of a logical fallacy, which is a bad argument.

Contradictory statements—Two statements that must have opposite truth values are called contradictory. If one of them is true, the other must be false, and vice versa. A is a contradiction of O, and E is a contradiction of I.