PROPOSITION
The word proposition can be a synonym for proposal, “the point to be discussed” or “demonstrated” (MW, art. Proposition). A proposition can be developed in a complex argumentative discourse, justifying the briefly expressed concrete proposition itself, see Argument – Conclusion.
In classical logic, a proposition is an independent 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 is defined by the specific contribution they make to the meaning of the terms or the propositions that they articulate
In grammar, a parallel distinction is made between the so-called full words, with 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 can be represented by its central element, the verb, accompanied by variables representing its complements. Variables are denoted ‘x’, ‘y’, or simply as empty places, “—”.
— Paul is sleeping: 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 “— gives — to —” or “x gives y to z”.
The same object can be attached to an infinite number of predicates, e.g. “— is a car”; “— is a means of transportation”; “— is an object that can be bought”; « — is a cause of pollution”… The discourse constantly creates new predicates, according to the interests of the speakers, such as “— was carried out on June 10, 2017”; “— is a car available for next Saturday’s trip”.
In the case of a predicate that admits several variables, one or more empty places can be filled by a constant. The predicate is then said to be partially saturated, which corresponds to a new predicate, for example, “Paul gives y (something) to z (someone)”, “x (someone) gives y (something) to John”, “Peter gives y (something) to John”.
In ordinary language, variables are preceded by quantifiers: any x, all y, some z, one w.
Constants are denoted by ‘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, see object of discourse
— Definite descriptions, or denoting phrases (the man with the green hat). The noun phrase can be complexified at will: the sitting man, the man pretending to look elsewhere.
This simple notation makes the skeleton of the sentence explicit, and is the basis for a more detailed semantic analysis of both its internal structure and its 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, that 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 signifying the true or the false, without any consideration of its meaning and conditions of use.
A proposition is unanalyzed if there is no information about its internal structure. Logical connectives and the laws of their combination are defined on the basis of such unanalyzed propositions.
A proposition is analyzed when its internal structure is taken into account. In classical logic, the analytic structure of a logical proposition is basically “Subject is Predicate”, “S is P”:
— The subject refers specifically (if it is a constant) or generally (if it is a variable) to the elements of a reference universe.
— The predicate says something about these elements.
— The proposition categorically unconditionally) affirms or denies that the predicate accepts the subject.
The 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, all beings or no 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« ; see syllogism:
The combination of quantity and quality produces four kinds of propositions:
A universal affirmative All S are P.
E universal negative No S are 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 the subject and the 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. Otherwise, the term is not distributed.
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, see paralogism.
3.4 The Presupposition of Existence
Some expressions such as “the unicorn”, “the present king of France”, “real dragons”, are misleading, in that they seem to be referential expressions although they do not refer to any existing being. This being the case, when such expressions are used as subjects of a proposition, that proposition cannot be said to be true or false, the present king of France is neither bald nor hairy. To avoid such puzzles, it is assumed that the reference universe of the subject term is not empty. S. Presupposition.
4. Immediate inference and logical square
4.1 Immediate inference
An immediate inference is a one-premise argument. It concludes 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. 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. In the second case, it is not.
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 connected by the following relations.
— Contrariety, between the affirmative universal A and the negative universal E. A and E are not simultaneously true, but can 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
— Contradiction, between:
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 is 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, the propositions E and I are convertible; cf. above, §3.2.
5. Immediate inference, quantifiers and terms
An 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 that characterize 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 that term. Argumentation by definition is therefore an immediate, substantial, semantic inference, about the meaning of the terms. Immediate inferences are formal; they are made not on the basis of full words, but on the basis of quantifiers.
Both types 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 deal with such these inferences often goes unnoticed. This does not mean, however, that the process is always error-free. Handling the such inferences correctly is part of the argumentative competence.