20.2. Proposition and Its Structure

In logic and philosophy, the term proposition (from the word "proposal") refers to both (a) the "content" or "meaning" of a meaningful declarative sentence or (b) the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence. The meaning of a proposition includes that it has the quality or property of being either true or false, and as such propositions are called truth bearers.

The existence of propositions in the abstract sense, as well as the existence of "meanings", is disputed by some philosophers. Where the concept of a "meaning" is admitted, its nature is controversial. In earlier texts writers have not always made it sufficiently clear whether they are using the term proposition in sense of the words or the "meaning" expressed by the words. To avoid the controversies and ontological implications, the term sentence is often now used instead of proposition to refer to just those strings of symbols that are truth bearers, being either true or false under an interpretation. Some scholars advocated the use of the term "statement", and this is the current usage in mathematical logic.

Aristotelian logic identifies a proposition as a sentence which affirms or denies a predicate of a subject. An Aristotelian proposition may take the form "All men are mortal" or "Socrates is a man." In the first example the subject is "men" and the predicate "are mortal". In the second example the subject is "Socrates" and the predicate is "is a man".

Often propositions are related to closed sentences to distinguish them from what is expressed by an open sentence. In this sense, propositions are "statements" that are truth bearers. This conception of a proposition was supported by the philosophical school of logical positivism.

Some philosophers argue that some (or all) kinds of speech or actions besides the declarative ones also have propositional content. For example, yes-no questions present propositions, being inquiries into the truth value of them. On the other hand, some signs can be declarative assertions of propositions without forming a sentence nor even being linguistic, e.g. traffic signs convey definite meaning which is either true or false.

Propositions are also spoken of as the content of beliefs and similar intentional attitudes such as desires, preferences, and hopes. For example, "I desire that I have a new car," or "I wonder whether it will snow" (or, whether it is the case that "it will snow"). Desire, belief, and so on, are thus called propositional attitudes when they take this sort of content.

The elementary structure of proposition

Proposition consists of subject and predicate. In traditional grammar, a predicate is one of the two main parts of a sentence (the other being the subject, which the predicate modifies). For the simple sentence "John is yellow" John acts as the subject, and is yellow acts as the predicate, a subsequent description of the subject headed with a verb.

In many current theories of linguistic semantics a predicate is an expression that can be true of something. Thus, the expressions "is yellow" or "is like broccoli" are true of those things that are yellow or like broccoli, respectively. This notion is closely related to the notion of a predicate in formal logic, which includes more expressions than the former one, such as nouns and some kinds of adjectives.

The most wide-spread propositions are categorical: S is P and S is not P

A categorical proposition contains two categorical terms, the subject and the predicate and affirms or denies the latter of the former. Categorical propositions occur in categorical syllogisms and both are discussed in Aristotle's Prior Analytics. Categorical propositions are part of deductive reasoning. For example: John serves on gasoline station (subject: John; predicate: serves on gasoline station). Some politicians are corrupt. (subject: politicians; predicate: corruptness). Nobody ever got fired for buying IBM. (subject: people; predicate: getting fired for buying IBM).

The subject and predicate are called the terms of the proposition. The subject is what the proposition is about. The predicate is what the proposition affirms or denies about the subject. A categorical proposition thus claims something about things or ways of being: it affirms or denies something about something else.

Categorical propositions are distinguished from hypothetical propositions (if-then statements that connect propositions rather than terms) and disjunctive propositions (either-or statements, claiming exclusivity between propositions).

There are only four classes (or categories) of categorical propositions: 1. Universal affirmative propositions; 2.Universal negative propositions; 3. Particular affirmative propositions; 4. Particular negative propositions. The systematic development of these four classes is credited to Aristotle.

Categorical propositions can be categorized on the basis of their quality, quantity, and distribution qualities. Quality refers to whether the proposition affirms or denies the inclusion of a subject to the class of the predicate. The two qualities are affirmative and negative. On the other hand, quantity refers to the amount of subjects in one class which are included in the other class. The first quantifier is the universal, "all". This means that every subject of one class has membership in the predicated class. The other quantifier is called a particular. It is an indefinite number, which could mean five, twenty or, perhaps, all, but always at least one. From quality and quantity four types of categorical propositions are designated alphabetically:

• A proposition is a universal affirmative: All S is P

• E proposition is a universal negative: No S is P

• I proposition is a particular affirmative: Some S is P

• O proposition is a particular negative: Some S is not P

Also there is special type of proposition that is called relative. The example: X R y (x has relation to y). X not R y (x does not have relation to y)

The logical operators

In logic, a logical connective (also called a logical operator) is a symbol or word used to connect two or more sentences (of either a formal or a natural language) in a grammatically valid way, so that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.

Each logical connective can be expressed as a function, called a truth function. For this reason, logical connectives are sometimes called truth-functional connectives. The most common logical connectives are binary connectives (also called dyadic connectives) that join two sentences whose truth values can be thought of as the function's operands. Also commonly, negation is considered to be a unary connective.

Logical connectives along with quantifiers are the two main types of logical constants used in formal systems such as propositional logic and predicate logic.

The commonly used logical connectives include:

Negation (not) (¬ or ~)

Conjunction (and) (&, or • )

Disjunction (or) ( ∨ )

Material implication (if...then) ( )

Biconditional (if and only if) ()

For example, the meaning of the statements it is raining and I am indoors is transformed when the two are combined with logical connectives:

It is raining and I am indoors (P & Q)

If it is raining, then I am indoors (P  Q)

It is raining if I am indoors (Q  P)

It is raining if and only if I am indoors (P  Q)

It is not raining (¬P)

For statement P = It is raining and Q = I am indoors.

In logic, quantification is the binding of a variable ranging over a domain of discourse. The variable thereby becomes bound by an operator called a quantifier. Academic discussion of quantification refers more often to this meaning of the term than the preceding one.

Quantification is used in both natural languages and formal languages. Examples of quantifiers in English are for all, for some, many, few, a lot, and no. In formal languages, quantification is a formula constructor that produces new formulas from old ones. The semantics of the language specifies how the constructor is interpreted as an extent of validity.

The two fundamental kinds of quantification in predicate logic are universal quantification and existential quantification. The traditional symbol for the universal quantifier "all" is "∀", an inverted letter "A", and for the existential quantifier "exists" is "∃", a rotated letter "E".

A truth table

А	В	А	В	АВ	АВ	АВ	АВ	АВ
T	F	F	T	F	T	T	F	F
F	T	T	F	F	T	T	T	F
F	F	T	T	F	F	F	T	T

Reasoning

Deductive reasoning is the reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises. A deductive argument is valid if the conclusion does follow necessarily from the premises, i.e., if the conclusion must be true provided that the premises are true. A deductive argument is sound if its premises are true. Deductive arguments are valid or invalid, sound or unsound, but are never true or false. An example of a deductive argument:

1. All men are mortal

2. Socrates is a man

3. Therefore, Socrates is mortal

The first premise states that all objects classified as 'men' have the attribute 'mortal'. The second premise states that 'Socrates' is classified as a man- a member of the set 'men'. The conclusion states that 'Socrates' must be mortal because he inherits this attribute from his classification as a man. Deductive reasoning is sometimes contrasted with inductive reasoning. Deductive arguments are generally evaluated in terms of their validity and soundness. An argument is valid if it is impossible both for its premises to be true and its conclusion to be false. An argument can be valid even though the premises are false. This is an example of a valid argument. The first premise is false, yet the conclusion is still valid.

1. Everyone who eats steak is a quarterback.

2. John eats steak.

3. Therefore, John is a quarterback.

This argument is valid but not sound. For a deductive argument to be considered sound the argument must not only be valid, but the premises must be true as well.

An argument is sound if and only if: 1. The argument is valid; 2. All of its premises are true. For instance:

All men are mortal.

Socrates is a man.

Therefore, Socrates is mortal.

The argument is valid (because the conclusion is true based on the premises, that is, that the conclusion follows the premises) and since the premises are in fact true, the argument is sound.

The following argument is valid but not sound:

All organisms with wings can fly.

Penguins have wings.

Therefore, penguins can fly.

Since the first premise is actually false, the argument, though valid, is not sound.