# Proposition

The word proposition may be a synonym of proposal, “the point to be discussed” or “demonstrated” (MW, art. Proposition). A proposition may be developed in a complex argumentative discourse, justifying the briefly expressed concrete proposition itself. S. Argument – Conclusion.

In classical logic, a proposition is an autonomous statement. Propositional logic considers concatenations of unanalyzed propositions P, Q, R…. Predicate logic considers a proposition analyzed in two terms, the subject and the predicate, “S is P”.

# 1. Term

In logic, a distinction is made between categorematic and syncategorematic terms. Categorematic terms function as subject names or concept names (predicates). Used without further clarification, the word term refers to a categorematic term.
Syncategorematic terms include negation, binary logical connectives (“&”, and, etc.) and quantifiers (“∀”, all, etc.). They cannot function as subject or concept names, they appear only in combination. They have no independent meaning; their meaning being defined by specific contribution they make to the meaning of the terms or proposition they combine with.

In grammar, a parallel distinction contrasts the so-called full words, having a full semantic content (verbs, substantives, adjectives, adverbs) and the so-called empty or grammatical words (such as linking words, discursive particles, auxiliaries…)

## 2. Predicate, variable, constant

A sentence may be represented by its pivotal element, the verb, accompanied by variables representing its complements. Variables are denoted ‘x’, ‘y’, or simply as empty places, “—”.

— Paul sleeps: To sleep is a one place predicate, written “ sleeps » or “x sleeps”. :
— Paul eats an apple: To eat is a two-place predicate, written “— eats —” or “x eats y”:
— Paul gave the apple to the lady in black: To give is a three-place predicate, written “— givesto —” or “x gives y to z”.

The same object can be attached to an infinite number of predicates, for example “ is a car”; “— is a means of transport”; “— is an object that can be bought”; « — is a cause of pollution”… Discourse constantly creates new predicates, according to the interests of the speakers, as “— was ​​carried out on 10 June 2017”; “— is a car available for next Saturday’s trip”.

In the case of a predicate admitting several variables, one or more empty places may be occupied by a constant. The predicate is then said to be partially saturated, which corresponds to a new predicate, for example, where “Paul gives y (something) to z (somebody)”, ​​“x (somebody) gives y (something) to John”, “Peter gives y (something) to John”.

In ordinary language, variables are expressed by indefinite phrases and pronouns: any, all, some, a (person)…”.

Constants are denoted ‘a’, ‘b’; in natural language, they are expressed by referring terms or phrases:

— Proper names (Peter), permanently attached to individuals.
— Pronouns (this the other, the next one). Their referential anchoring is based both on deictic maneuvers and on definite descriptions whose reference can be retrieved from the context. S. Object of discourse
— Definite descriptions, or denoting phrases (the man with the green hat). The noun phrase can be complexified at will: the seated man, the man who pretends to look elsewhere.

This simple notation renders explicit the skeleton of the sentence and is the basis of a more detailed semantic analysis of both its internal structure and external position in the broader discourse to which it belongs. Argument schemes are currently expressed in such a semi-symbolic notation.

# 3. Proposition

In classical logic, a proposition is a judgment, which can take only two values, true (T) or false (F); a proposition cannot be “more or less” true or false. A proposition is only a way of telling the truth or the false, without any consideration upon its meaning and conditions of use.

A proposition is unanalyzed if no information on its internal structure is available. Logical connectives and the laws of their combination are defined on the basis of such unanalyzed propositions.

A proposition is analyzed if its internal structure is taken into consideration. Classical logic considers that the analytic structure of a logical proposition is basically “Subject is Predicate”, “S is P”:

— The subject refers specifically (if a constant) or generally (if a variable) to the elements of a universe of reference.
— The predicate says something about these elements.
— The proposition categorically (without condition) affirms or denies that the predicate accepts the subject.

Capital letters A, B, C… P, Q, R… are used to denote both unanalyzed propositions and the subject and predicate in analyzed propositions.

## 3.1 Quality and quantity of a proposition

The quality of a proposition refers to its two possible dimensions, affirmative or negative.

The quantity of the proposition varies according to whether the subject refers to a being, certain beings or all beings of the universe of reference.

Quantifiers express the quantity. The quantifiers such as all (all N), or some (some N) express quantities. According to their quantity, propositions are:

Universals: all poets
Particular: a poet; some poets

Particular does not refer to a constant, a specific, known, individual. In its traditional form, logic does not deal with propositions predicating something from a determined individual, such as “Peter” or “this poet« ; S. Syllogism:

The combination of quantity and quality produces four kinds of propositions:

A     universal affirmative All S are P.
E     universal negative No S is P.
I      particular affirmative Some S are P.
O     particular negative Some S are not P

Traditionally, affirmatives are denoted by the letters A and I (two first vowels of the Latin verb AffIrmo “I affirm”) and the negatives by the letters E and O (nEgO, “I deny”).

## 3.2 Converse propositions

The converse proposition of a given proposition is obtained by swapping subject and predicate. The subject of the original proposition is the predicate of its converse proposition and the predicate of the original proposition is the subject of its converse proposition.

The quality (affirmative or negative) of the two propositions is the same.

The negative universal E and its converse are equivalent (they have the same truth conditions, cf. infra §4.2, Logical Square):

No P is Q ↔ no Q is P.

The positive universal E and its converse are not equivalent

all P are Q ≠  all Q are P.

## 3.3 Distribution of a term

A term is distributed if it says something of all the individuals belonging to the reference set. If not, the term is undistributed.
The terms preceded by the quantifier all are distributed. The terms quantified by some, many, almost all … are undistributed.
For example, in a universal affirmative proposition A, “All Athenians are poets”:

— The subject term, Athenians, is distributed.
— The predicate term, poet, is undistributed; the proposition only says that “some poets are Athenians”.

The notion of distribution is used by the rules of evaluation of the syllogism, S. Paralogism.

## 3.4 The presupposition of existence

Some expressions such as “the unicorn”, “the present king of France”, “real-life dragons”, are misleading, insofar as they appear to be referring expressions despite the fact they do not refer to any existing being. This being the case, when such phrases are used as subjects of a proposition, this proposition cannot be said to be true or false, the present King of France is neither bald nor hairy. To avoid such perplexities, it is assumed that the universe of reference of the subject term is assumed not to be empty. S. Presupposition.

# 4. Immediate inference and logical square

## 4.1 Immediate inference

An immediate inference is a one-premise argument, inferring from one proposition to another:

All the A are B, so some B are A

The two terms of this single premise are found in the conclusion, only the quantity of the proposition changes. While syllogistic inference requires a medium term (middle term), “im-mediate” inference does not need such a transition term. It is debatable whether immediate inference is a kind of reasoning.

Immediate inference is an inference, not a reformulation. The reformulation relation presupposes the identity of meaning between the two utterances it links:

Some A are B, so some B are A (conversion, see §3.2).
All the A are B, so some B are A (subalternation, see infra).

In the first case, the immediate inference corresponds to an equivalence. This is not true, however, of the second.

### 4.2 Logical square

The logical square expresses the set of immediate inferences between analyzed propositions of the subject-predicate form according to their quality, affirmative or negative, and the quantity of their subject (A, E, I, O, see above).

These four propositions are linked by the following relations.

Contrariety, between the affirmative universal A and the negative universal E. A and E are not simultaneously true, but may be simultaneously false. In terms of immediate inference, if one is true, then the other is false.

— Subcontrariety, between the particular affirmative I and the negative particular O. At least one of the two propositions I and O is true. They may be simultaneously true, but cannot be simultaneously false. In terms of immediate inference, if one is false, then the other is true

The universal negative E and the particular affirmative I.
The universal affirmative A and the particular negative O.

E and I cannot be simultaneously true or simultaneously false (only one of them is true). The same will be true for A and O. In terms of immediate inference, the truth of one immediately implies the falsity of the other, and vice versa.

­— Subalternation, between:

A and I, the universal affirmative and the particular affirmative.
E and O, the negative universal and the negative particular.

If the superaltern is true, its subaltern is true. Immediate inference:

Every S is P, so some S are P.

If the subaltern is false, its superaltern is false. Immediate inference:

It is false that some S are P, so it is false that every S is P.

The subaltern may be true and the superaltern false.

Moreover, propositions E and I are convertible; cf. supra, §3.2.

# 5. Immediate inference, quantifiers and terms

Immediate inference is an inference from a single premise. The two terms of the single premise are found in the conclusion (examples above). In the case of the syllogism@, the inference proceeds from two premises and three terms. The middle term functions as a “mediator”, an intermediary, between the major term and the minor term. In the case of immediate inference, the conclusion is “not-mediated” by a middle term.

From a cognitive point of view, argumentation by definition assigns to an individual the properties characterizing the class to which it belongs. From a linguistic point of view, argumentation by definition assigns to an individual designated by a name, all the elements of the linguistic definition of this term. Argumentation by definition is therefore an immediate, substantial, semantic inference, on the meaning of the terms. Immediate inferences are formal; they are not made on the basis of full words, but on the basis of quantifiers. Both kinds of inference function as semantic reflexes in ordinary discourse, linking natural statements, according to ordinary semantic intuition combined with contextual references based on the laws of discourse and the cooperative principle.

Because of their seeming obviousness, the way we handle such these inferences often goes unnoticed. This does not mean, however, that the process is always error free. Taking the correct approach to such inferences is part of the argumentative competence.