Logical positivism and predicate logic

Copyright 2016 Graham Berrisford. One of about 300 papers at http://avancier.website. Last updated 25/10/2018 18:04


Logical positivist philosophers focus on analysing the meaning of language used in philosophical propositions.

We use the words and grammar of a language to describe things.

The fluidity and imprecision of natural language enables human creativity and assists survival in a changing world.

But to specify a system in an unambiguous and testable way, an artificial language is needed.


Wittgenstein 1: his seven propositions. 1

Wittgenstein 2: Family resemblances and natural languages. 3

Ayer: Types and artificial domain-specific languages. 4

Carnap: The logical syntax of language. 5

Predicate logic. 6

Conclusions and remarks. 6



Wittgenstein 1: His seven propositions

Ludwig Wittgenstein (1889-1951) influenced the “Vienna circle” of logical empiricists (aka logical positivists).

In his main work, the Tractatus Logico-Philosophicus, set out seven propositions.

It is a tough read, but I gather the premises are as follows.

·         Philosophical propositions have logical structures.

·         Many propositions are poorly formulated; and futile debates arise from misunderstandings.

·         Philosophical disagreements and confusions can be resolved by analysing the use and abuse of language.


Proposition 1) The world is all that is the case

Wittgenstein’s first proposition includes these statements.

1 The world is all that is the case.

1.1 The world is the totality of facts, not of things.
1.2 The world divides into facts.


I understand this to say: the world is the totality of facts, and language is the totality of propositions.

The world and language are structured the same way.

Philosophy should confine itself to facts set out in well-structured propositions.


Here, we challenge the view that "facts" are the only reality.

The world is mysterious, unknowable as it is.

There are infinite ways to describe the world in terms of facts that are true enough.

Any description that helps us to predict or direct how the world moves forward must be a reasonable model of that world.

The better and more completely we can predict and direct reality, the closer our model to that reality.


Proposition 2) What is the case, the fact, is the existence of atomic facts

Wittgenstein’s second proposition includes a discussion of objects, form and substance.

2.02 Objects are simple.
2.021 Objects make up the substance of the world. That is why they cannot
be composite.


Here, we say our perceptions and descriptions of reality divide it into discrete chunks.

We can divide the universe in infinite ways - into stars, planets, the eyes in a peacock’s tail, the verses of a song.

With the exception of atomic physicists, people always perceive and describe composite objects and events.

What we regard as atomic differs according to the context.

What we regard as an atomic entity differs according to the context.

What we regard as an atomic event differs according to the context.


Wittgenstein goes on to say:

2.0271 Objects are what is unalterable and substantial; their configuration is what is changing and unstable.

Here, we say "facts" are atoms of description rather than atoms of reality.

All real objects are alterable, and they have a limited life time.

Every object is created, it may change during its existence, and it will be destroyed in the end.


Proposition 3) The logical picture of the facts is the thought

Does this confuse static facts with dynamic thinking?

Here, we say a thought is a process that involves creating and using logical models.

Thinking abstracts logical models from other logical models and from physical bio-electro-chemical models.


On meeting somebody we start thinking “What is that person’s name?”

That thought process explores our physical brain and might end in our expressing a logical “fact” -  a name, right or wrong.

Or it might continue unsuccessfully, eventually fizzling out when other thoughts occupy our thinking resources.


Proposition 4) The thought is the significant proposition

Proposition 4.003 says:

“Most of the propositions and questions of philosophers arise from our failure to understand the logic of our language.”

Later: In Philosophical Investigations Sect. 90 Wittgenstein said this.

“Our investigation is a grammatical one. Such an investigation sheds light on our problem by clearing misunderstandings away.

Misunderstandings concerning the use of words, caused, among other things, by certain analogies between the forms of expression in different regions of language.”


Here, we say our language is naturally messy and imprecise.

That almost all our thoughts and propositions are fuzzy rather than true or false.

Most if not all our propositions can be true or false only with reference to others within a limited descriptive ontology.


Proposition 5) Propositions are truth-functions of elementary propositions

Ditto. Here, we say our language is naturally messy and imprecise.


Proposition 6) The general form of a truth-function or proposition is…

Ditto. Here, we say our language is naturally messy and imprecise.


Proposition 7) Whereof one cannot speak, thereof one must be silent

Here, we say people are well advised to ignore this rule, because natural language is not reducible to true or false sentences.

So, we need feedback to refine our thoughts and our speech.

We say what is in our minds, partly to open our thoughts to inspection and correction - by ourselves and others who listen to us.

Wittgenstein 2: Family resemblances and natural languages

Eventually, Wittgenstein realised his “Tractatus” was self-contradictory, and developed an entirely different linguistics.

He turned his focus from the precision of language to the fluidity of language.

He dropped the metaphor of language “picturing” reality and replaced it with language as a tool.


In “Philosophical Investigations” (papers published posthumously) he articulated the concept of family resemblances.

He considered “games” as a set, which includes activities as varied as chess, archery and Super Mario.

He argued the set members have overlapping lists of features, but no single feature in common.


A biologist might propose every game is “an activity that serves as a direct or indirect rehearsal of skills useful to survival.”

But that does not matter here, since animals certainly do recognise things as resembling each other in a loose and informal way.

What does matter here is to understand the limitations of linguistics.

Natural language is loose; both words and grammar are very flexible.

No word, description or message has a universally-agreed meaning.

There is ambiguity and fuzziness in the meanings of words, and degrees of truth in how well a reality matches a description.

But for a system description to be holistic, unambiguous and testable, an artificial domain-specific language is needed.

Ayer: Types and artificial domain-specific languages

A. J. Ayer (1910 to 1989) was a philosopher who wrote about language, truth, logic and knowledge.

He rejected metaphysics and much philosophical discussion as meaningless, that is, not provable or disprovable by experience.

His “Language Truth and Logic” contains many examples in which he analysed the language used in propositions to expose flawed reasoning.


He pointed out that every coherent description of a thing or situation is a type definition.

“In describing a situation, one is not merely registering a [perception], one is classifying it in some way,

and this means going beyond what is immediately given.” Chapter 5 of “Language, truth and logic”


A type defines features (qualities, characteristics, attributes, properties) shared by things that instantiate or realise that type.

To describe something as the game is to imply it is the only one of that named type.

To describe something as a game is to imply it is one of a set containing many things of that named type.


As Wittgenstein indicated, the set of things people call a game is elastic, and it is difficult to agree a type that defines every set member.

However, words do give humans the ability to classify or typify things more formally and rigidly.

Within a bounded context, or domain of knowledge, we can determinedly fix the meanings of words.

A domain-specific language is an island of inter-related words with stable meanings, in the ever-unfolding evolution of natural language.

Words are treated as type names, and each type is defined by statements relating it to other types.



In the domain of mathematics, type names include: “number”, “division” and “remainder”.

Type definitions include: “An even number is a number that is divisible by two with no remainder.”


In the domain of physics, type names include “force,” “mass” and “acceleration”.

Type definitions include: “A force equals the mass of a body times its acceleration.”

(By contrast, in the language of management science, a force is a pressure acting on a business, such as competition or regulations.)


In any business domain, people define the rules of their specific business in terms of relations connecting types.

“An employee has a salary and may be assigned to a project.”

“An order is placed by a customer; a customer can place several orders.”


Again, for a system description to be holistic, unambiguous and testable, an artificial domain-specific language is needed.

Carnap: The logical syntax of language

Rudolf Carnap (1891 – 1970) was a member the Vienna circle who contributed to the philosophy of science and of language.

Carnap has been called a logical positivist, but he disagreed with Wittgenstein.

He considered that philosophy must be committed to the primacy of science and logic, rather than verbal language.


Carnap’s first major work, Logical Syntax of Language can be regarded as a response to Wittgenstein 's Tractatus.

He too rejected the sentences of metaphysics as pseudo-sentences, which prove to be either empty phrases or phrases which violate the rules of syntax.

the sentences of metaphysics are pseudo-sentences which on logical analysis are proved to be either empty phrases or phrases which violate the rules of syntax.

Of the so-called philosophical problems, the only questions which have any meaning are those of the logic of science.

To share this view is to substitute logical syntax for philosophy.”

— Carnap, Page 8, Logical Syntax of Language, quoted in Wikipedia.


He defined the purpose of logical syntax thus:

“The purpose of logical syntax is to provide a system of concepts, a language, by the help of which the results of logical analysis will be exactly formulable.”

Philosophy is to be replaced by the logic of science – that is to say, by the logical analysis of the concepts and sentences of the sciences,

for the logic of science is nothing other than the logical syntax of the language of science.

— Carnap, Foreword, Logical Syntax of Language, quoted in Wikipedia.


He defined the logical syntax of a language thus:

By the logical syntax of a language, we mean the formal theory of the linguistic forms of that language –

the systematic statement of the formal rules which govern it together with the development of the consequences which follow from these rules.

A theory, a rule, a definition, or the like is to be called formal when no reference is made in it either to the meaning of the symbols (for examples, the words) or to the sense of the expressions (e.g. the sentences), but simply and solely to the kinds and order of the symbols from which the expressions are constructed.

— Carnap, Page 1, Logical Syntax of Language, quoted in Wikipedia.


Carnap’s second major work, Pseudoproblems in Philosophy asserted that many metaphysical philosophical questions were meaningless.

His Principle of Tolerance says there is no such thing as a "true" or "correct" logic or language.

His concept of logical syntax is important in formalising the storage and communication of information/descriptions.

Computers require that logical data structures are defined using a formal grammar called a regular expression.

It is said that Carnap’s ideas helped the development of natural language processing and compiler design.


As I understand it, Carnap said:

A statement is only meaningful with respect to a given theory (a set of inter-related domain-specific predicate statements).

And only true to the extent it can be supported by experience or testing.

Predicate logic

The basic logic of statements is called propositional logic (or calculus).

A proposition is a statement that asserts a fact that may be true or false.

In natural (as opposed to mathematical) language it takes the form of a sentence: e.g. the sun is shining.

Simple propositions can be connected by connectives (and, or, not and if) into a compound proposition: e.g. pigeons fly and eat corn. 

Propositional logic is the foundation of first-order or predicate logic.

A predicate is a verbal phrase, with or without an object, which declares a feature of a subject.


Proposition (in the form of a predicate statement)



Verb phrase


A particular thing

or instance of a general type

A verb or verbal phrase

that either stands alone or

relates the subject to the object

A particular thing or a general type

related to the subject by the predicate.

The sun

is shining

A game of monopoly

results in

a winner with the largest credit amount

A game

is kind of


A game

is played by

one or more animate players

A game

results in

a measure of achievement

An order

is placed by

a customer

A customer

can place

several orders


Predicate statements can include variables.

And “a subject (or object)” can be read as “one instance in the set of subjects (or objects)”.

So, a type definition can be presented as a predicate statement.

E.g. “A game is a kind of activity and is played by one or more animate players and results in a measure of achievement.”

Conclusions and remarks

Here, we do not regard natural language as the basis for a philosophy of system theory.

Friedrich Nietzsche (1844 to 1900) was a philosopher whose metaphysical ideas influenced many Western intellectuals.

He took the view, called “perspectivism”, that our conceptualisations of the world are shaped by how we view it.


“Against positivism (the position ‘There are only facts’) I would say: no, there are precisely no facts, only interpretations.

To the extent that the word ‘knowledge’ has any sense, the world is knowable: but it is interpretable differently, it has… innumerable senses, ‘perspectivism.’

It is our needs that interpret the world: our drives and their to and fro.” Nietzsch


Some postmodernists read Nietzsch as saying there is no objective truth or accurate knowledge of the world.

Some interpret his assertion as meaning all descriptions of the world are equally valid.

Any appealing belief or poetic assertion carries the same weight as scientific evidence.


To some extent, different people do perceive the world differently from each other, and from birds, bats and bees.

But more importantly, their conceptualisations are shaped by testing them against reality.

All animal life depends on two facts: a) there is a real world, and b) only some descriptions of reality prove accurate enough when tested.


Here: natural language is a biological phenomenon, a product of evolution that gives us tools we need to communicate.

But it is imprecise, ambiguous and fluid.

The role of words in biology is to help us to remember and communicate descriptions that are true enough.

Words didn’t evolve to enable us to formulate perfectly true propositions (a concern of philosophers and mathematicians) they are not good for that.

Usually, any "state of matters" or "fact" that one animal recalls or communicates to another is a perception that has "degrees of truth".


To paraphrase von Foerster: “We live in the domain of types that we invented.”

The types idealise and symbolise the realities we observe and envisage.

These relations can be shown in the triangle you may now becoming familiar with.




<invent>                 <symbolise>

Human intelligences <observe & envisage> Realities


A type is a description; moreover, a description may be viewed as a type.

Every coherent description, even a very long and complex one, serves as a type definition.


In a domain-specific language, a word is a type name, and is defined by predicate statements relating it to other types. 7

Typically: “An instance of type A is related by this verb phrase to one or more instances of type B.”


But note the matching of a thing to a type can be incomplete.

A monothetic type, like “even number”, requires all instances of the type to have all its features.

A polythetic type, like “game” does not require that all instances of the type will have all its features.


And there can be degrees of truth in a predicate statement.

Newton’s laws describe the motion of things in the reality we normally experience.

The laws are true to the degree of accuracy we need, but only approximations, neither wholly true nor wholly false.

For more on fuzzy logic and fuzzy sets, try this link.





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