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Most people favour a correspondence theory of truth, according to which a proposition is true if it somehow corresponds to some aspect of reality. Quite what that relation of correspondence amounts to is hard to say.
It is common to refer to the aspects of reality to which a proposition might corresponds as ‘facts’.
That replaces one question by three
(1) What are facts?
(2) What is the relation of correspondence between a proposition and a fact?
(3) What is the relation between a fact and the world?
We might say that a fact is a true proposition about the world.
In his lectures on Logical Atomism Russell wrote:
"..when I speak of a fact – I do not propose to attempt an exact definition, but an explanation so that you will know what I am talking about – I mean the sort of thing that makes a proposition true or false"
Logic and Knowledge p. 182
That makes it very easy to answer question (2), but disqualifies correspondence as an explanation of truth.
It is also tricky to specify the relation between a fact and the state of affairs it refers to. A particular state of the world may support many facts.
Suppose the cat is sitting on the television set and looking around the room.
It would then be held to be a fact that the cat is sitting on the television. However that same arrangement of matter might equally be held to constitute a mammal sitting on a warm plastic box enclosing electronics. From the cat’s point of view the fact might be that the most important member of the household is surveying it’s territory from an elevated position while warming its bottom.
We think we can be fairly sure of ‘the facts’ because they can be checked by direct observation, so facts are often put forward as the basis of knowledge.
However there are many propositions which do not correspond to such readily accessible facts. For example ‘the nucleus of Helium4 contains two protons and two neutrons’ The theory on the basis of which we make that judgement was developed by observations made on other substances. Much of the work was done before helium was discovered.
Early in the twentieth century Russell and Wittgenstein developed a theory of Logical Atomism, according to which every fact is either an atomic fact, incapable of analysis into anything simpler, or it is somehow composed of atomic facts.
In the Tractatus Wittgenstein wrote:
"1 The world is everything that is the case.
1.1 The world is the totality of facts, not of things.
2 What is the case, the fact, is the existence of atomic facts"
The theory was never satisfactorily completed and Logical Atomism gave way to various theories about ‘sense data’
W.V.Quine and others suggested that what needs to be brought into agreement with observation is the whole body of our knowledge, not each individual belief separately. Many adjustments to our beliefs are based on a striving for consistency.
"Physical objects are known to us only as parts of a systematic conceptual structure which, taken as a whole, impinges at its edges upon experience."
"Our statements about external reality face the tribunal of sense experience not individually but as a corporate body"
From the introduction to Quine's Methods of Logic.
Quine argued that our body of knowledge is under-determined by experience so that there is more than one way of adjusting our beliefs to accommodate any surprising observation.
A contemporary example is the postulation of dark matter to account for gravitational anomalies. Cosmologists have proposed the existence of ‘dark matter’ to explain gravitational forces greater than can be explained by any observed matter. There is an alternative explanation, that current theories of gravitation are in error. Doubtless cosmologists have reasons for not pursuing that, but it is still a logical possibility.