, the
is called an
apparent variable14 Mar 2025 -
$\require{amsmath}$
$\require{amssymb}$
$\require{amsfonts}$
Last updated 3/7/25
We are acquanted with “…sensible qualities, relations of space and time, similarity, and certain abstract logical universals”
In a proposition of either the two forms , the
is called an
apparent variable
A class is all the objects satisfying some propositional function.
Classes are logical fictions, and a statement which appears to be about a class will only be significant if it is capable of translation into a form in which no mention is made of the class.
We cannot treat “class” as a primitive idea.
Constructions are not all philosophical analyses, ontological reductions, proposals for regimentation of ordinary notions for specific purposes, though elements of these all appear in some constructions. If anything, a construction should primarily be seen as a replacement for an ordinary notion. That replacement does provide an explicit notion where the original may have been imprecise, but it can have very different logical and epistemological properties from the original. In particular, a construction has precise, provable features where the original is vague or leads to skepticism.
…constructions are not simply restricted to entities that might be seen as classes of other entities.
…we should say that it is talk about certain entities that is constructed rather than the entities, or even names for the entities.
A description is a phrase of the form “the term which etc.”
A description may be of two sorts, definite and indefinite (or ambiguous). An indefinite description is a phrase of the form “a so-and-so,” and a definite description is a phrase of the form “a so-and-so” (in the singular).
The term having the relation R to x…where R is any one-many relation.
Propositions which contain no apparent variables we call elementary propositions
a function, all whose values are elementary proposition, is called an alementary function.
Russell speaks of entities as metaphysical, psychological, constructed, structured.
Facts are the sort of things that are asserted or denied by propositions, and are not properly entities at all in the same sense in which their constituents are. This is shown by the fact that you cannot name them.
The notion of function need not be confined to numbers, or to the uses to which mathematicians have accustomed us ; it can be extended to all cases of one-many relations, and ” the father of x ” is just as legitimately a function of which x is the argument as is ” the logarithm of x.” Functions in this sense are descriptive functions. As we shall see later, there are functions of a still more general and more fundamental sort, namely, propositional functions ; but for the present we shall confine our attention to descriptive functions, i.e. ” the term having the relation R to x” or, for short, ” the R of x,” where R is any one-many relation.
We will give the name of generalization to the process which turns
into
or
All such as contain apparent variables.
Definitions…in which an object is defined with reference to sets whose existence in turn depend on the object ot be defined, were called impredicative by Poincare
does not have meaning by itself, but requires some supplementation in order to acquire a complete meaning.
An expression not analyzed into names in the (pictoral) logical form of a sentence
We say that x is an “individual” if x is neither a proposition nor a function
the terms of elementary propositions other than functions
We may explain an individual as something which exists on its own account; it is then obviousl ynot a proposition, since propositions…are incomplete symbols, having no meaning except in use.
Only individuals can, properly speaking, be named.
When Russell says that an entity is “linguistic”, he means that it is the subject matter of logic, and not an ordinary concrete particular. It does not “exist” in the same sense as ordinary individuals.
Russell seems to imply that a “logical fiction” and “incomplete symbol” are equivalent
A function that contains no “apparent” (bound) variables. This also means that a matrix contains no quantifiers.
First form - rejection of universals. Second form - rejection of abstract objects.
a non-predicative function of the nth order is obtained from a predicative function of the nth order by turning all the arguments of the n-1th order into apparent variables.
non-predicative functions always result from such as are predicative by means of generalization.
An assymetric, transitive, connected relation.
[Predicative functions of individuals]…together with those derived from them by generalization, will be called first-order functions.
We will give the name of first-order propositions to such as contain one or more apparent variables whose possible values are individuals, but contain no other apparent variables. First order propositions are not all of the same type, since…two propositions which do not contain the same number of apparent variables cannot be of the same type.
We will give the name first-order matrices to such as have only individuals for their arguments.
A second-order matrix is one which has at least one first-order matrix among its arguments, but has no arguments other than first-order matrices and individuals
A second-order function is one which either is a second-order matrix or results from one by applying generalization to some (not all) of the arguments to a second order matrix.
A second-order proposition is one which results from a second-order matrix by applying generalization to all its arguments.
- A function of the first order is one which involves no variables except individuals, whether as apparent variables or as arguments.
- A function of the (n+1)th order is one which has at least one argument or apparent variable of order n, and contains no argument or apparent variable which is not either an individual or a first-order function or a second-order function or … or a function of order n.
The order of a type simble is given by the following recursive definition: (i) The type symbol “o” has order 0. (ii) A type symbol
has order n+1 if the highest order of the type symbols
is n.
By the order of a typed relation in a type hierarchy we mean the order of the type symbol which is its type.
The order of a term [(variable or a constant)] is the order of the type symbol that is its superscript. For an abstract t, let n be the highest order of all its variables (bound or free) and its constants. If there is a bound variable y of order n in t, then t is of order n+1. Otherwise, the order of t is n (in this case t mus thave som efree variable or constant of order t).
…a predicative function which is true of the object
A predicative function of x is one whose values are propositions of the type next above that of x, if x is an individual or a proposition, or that of values of x if x is a function. It may be described as one in which the apparent variables, if any, are all of the same type as x or of lower type; and a variable is of lower type than x if it can significantly occur as argument to x, or as argument to an argument to x, etc.
A function is said to be predicative when it is a matrix. It will be observed that, in a hierarchy in which all the variables are individuals or matrices, a matrix is the same thing as an elementary function. ‘Matrix’ or ‘predicative function’ is a primitive idea. The fact that a function is predicative is indicated, as above, by a note of exclamation after the function letter.
We will define a function of one variable as predicative when it is of the next order above that of its argument, i.e., of the lowest order compatible with its having that argument. If a function has several arguments, and the highest order of function occurring among the arguments is the nth, we call the function predicative if it is of the n+1th order, i.e., again if it is of the lowest order compatible with its having the arguments it has.
A function is said to be predicative when it is a matrix
“Matrix” or “predicative function” is a primitive idea
A ‘property of x’ may be defined as a propositional function satisifed by x
A proposition may be defined as: What we believe when we believe truly or falsely.
We mean by a “proposition” primarily a form of words which expresses what is either true or false.
I think the word “proposition” should be limited to what may, in som sense, be called “symbols,” and further to such symbols as give expression to truth and falsehood.
Owing to the plurality of objects in a single judgement, it follows that what we call a ‘proposition’ (in the sense in which this is distinguished form the phrase expressing it) is not a single entity at all. That is to say, the phrase which expresses a proposition is what we call an ‘incomplete symbol’; it does not have meaning by itself, but requires some supplemention in order to acquire a complete meaning.
Propositions can only be asserted or judged (not named)
Propositions are not properly entities, in virtue of belonging to a different logical type than the real, or basic individuals in the world.
The constituents of propositions are all real objects, not linguistic symbols…
By a ‘propositional function’ we mean something which contains a variable x, and expresses a proposition as soon as a value is assigned to x. That is to say, it differs from a proposition solely by the fact that is ambiguous: it contains a variable of which the value is unassigned.
A propositional function is simply an expression containing an undetermined constituent, or several undetermined constituents, and becoming a proposition as soon as the undetermined constituents are determined.
A “Propositional function,” in fact, is an expression containing one or more undetermined constituents, such that, when values are assigned to these constituents, the expression becomes a proposition. In other words, it is a function whose values are proposition.
propositional functions do presuppose a totality of propositions, which in turn presuppose universals and particulars.
Introduces orders into type theory
The viewpoint which accords to things which are known or perceived an existence or nature which is independent of whether anyone is thinking about or perceiving them.
We may regard a relation, in the sense in which it is required for our purposes, as a class of couples
A singular or Russellian proposition is modeled as an n-tuple consisting of a relation (propositional function) and n-1 individuals.
The doctrine that general terms signify many different things, rather than one single thing like a universal or a class
Semantical realism claims that truth is a semantical relation between language and reality.
Not gramatticaly ill-formed
One class is said to be “similar” to another when there is a one-one relation of which the one class is the domain, while the other is the converse domain.
…propositional and functional variables have intensional values but the values of the individual variables remain extensional
There is a type i, the type of individuals. If
are types, there is a type
.
The Substitutional Theory was a part of Russell’s efforts to avoid type theory. It failed because of cantor diagonal paradox.
Whatever may be an object of thought, or may occur in any true proposition, or can be counted as one, I call a term.
A man, a moment, a number, a class, a relation, a chimaera, or anything else that can be mentioned, is sure to be a term…
We shall give the name of a truth-function to a function
whose argument is a proposition, an dwhose truth-value depends only upon the truth-value of its argument.
…[A]s a rough-and-ready indication of what we mean by a “type,” we may say that individuals, classes of individuals, relations between individuals, relations between classes, relations of classes to individuals, relations between classes, relations of classes to individuals, and so on, are different types.
In general, a universal can be characterized as some one thing that certain similar, but numerically distinct, objects have in common. The “having” here is analyzed in different ways.
…universals are what are predicated “of” objects in propositions.
Particulars and universals differ in…that each particular has a unique spatio-temporal location, while universals can be located in each of many spatially and temporally separated objects.
The Universe consists of objects having various qualities and standing in various relations.
Every proposition containing all asserts that some propositional function [open sentence] is always true; and this means that all values of the said function are true, not that the function is true for all arguments, since there are arguments for which any givne function is meaningless, i.e., has no value. Hence we can speak of all of a collection when and only when the ocllection forms part or the whole of the range of significance of some propositional function, the range of significance being defined as the collection of those arguments for which the function in question is significant, that is, has a value.
The principle of vicious circle says that the values which are meaningless for a given open sentence must be those involving the type of supposedly illegitimate self-reference previously discussed.
