(1) “Whatever is green is coloured.” (2) “Whatever is square is rectangular.” (3) “The sum of three and two is five.” (4) “Necessarily, if ‘p or q’ is true and q is false, then p is true.”
Intuitively, the statements directly above do seem to be known to be true without evidence, observation, experience, data, etc. That is, they seem to be a priori in nature.
However, readers will now need to know what the word “intuitively” (in this context at least) means and whether intuition is reliable at all.
More technically and relevantly, the very idea of a priori knowledge will need to be tackled.
The first thing to say about statements (1) to (4) is that even if they can be known to be true a priori, then they’re still very different from each other.
For a start, (1) and (2) seem to be conceptual statements — even if (2) is geometrical and (1) isn’t. (4) Is a logical statement in which the symbolic letters don’t count (i.e., their semantic content doesn’t need to be known). And (3) is an arithmetical (possibly metamathematical — see later section) statement.
What’s more, because we’ve brought on board conceptual, arithmetical, geometric and logical truths, can it still be argued that all these statements can be known to be true in exactly the same way?
(1) “Whatever is green is coloured.”
Two immediate things can be asked about this statement:
(1) Is it about the world? Or:
(2) Is it about the concepts [green] and [coloured]?
Two further statements (i.e., rather than questions) can now be made:
(1) Whatever is green (i.e., any green thing in the world), must also be coloured.
(2) Contained in the concept [green] is the concept [coloured].
At first sight, the statement “Whatever is green is coloured” appears to be a purely conceptual truth. That is, if something is green, then it must be coloured. That’s simply because green is a colour.
If we taken on board (2) above, then, it can be said that if the concept [coloured] is contained in the concept [green], then we don’t need to (as it were) check the world to see if this statement is true. Thus, surely it’s an a priori truth.
That said, questions have been asked about the a priori status of this precise statement (i.e., “Whatever is green is coloured”), as well as about others like it (such as “Whatever is square is rectangular”, which will be discussed in a minute).
So what is it for one concept to be contained in another concept?
Surely the word “contained” is purely metaphorical. (Can it be anything else?)
The philosopher W.V.O. Quine famously denied that the statement “Whatever is green is coloured” (as well as others like it) is what is called analytic. Or, more correctly, Quine argued that the word “analytic” (or “analyticity”) was never satisfactorily explained (or defined) in ways which weren’t circular. (See Quine’s ‘Two Dogmas of Empiricism’.)
So do we require the notion of analyticity in order to claim that the statement “Whatever is green is coloured” is an a priori truth?
Well, now it must be explained what an analytic statement is.
It was the logical positivists who popularised the term analytic to explain a priori knowledge. So as the oft-used phrase has it:
Analytic statements are deemed to be (as Quine himself put it) “true by virtue of meanings and independently of fact”.
This position on analytic statements was also supposed to have the important consequence that even though an analytic statement is known to be true a priori, a special faculty of intuition isn’t actually required in order to know that. Instead, it is known to be true (at least partly) because the meanings of the words within the analytic statement are already understood.
Thus, there doesn’t seem to be anything intuitive about understanding the terms “green” and “coloured”. Consequently, there’s nothing intuitive about understanding the whole statement (i.e., “Whatever is green is coloured”) either . Such understanding is simply a consequence of understanding the meanings of the words and the nature of the grammatical (or syntactical) construction of the whole sentence.
Still, in our case, the statement “Whatever is green is coloured” does indeed seem to be true by virtue of meanings and independently of fact.
As already hinted at, it is evident that the person who knows that the statement “Whatever is green is coloured” is true must have already learned the English language and must therefore understand what the statement means. So he must also know the meanings of the words “green” and “coloured”, as well as the words “whatever” and “is”.
Of course, all apriorists happily accept all this.
Apriorists believe that a priori knowledge is about what occurs only after human beings have acquired a language, and then also understood the meanings of individual words and whole sentences.
This was (at least partly) summed up by Immanuel Kant (1724–1804). He wrote:
“Although all our knowledge begins with experience, it does not follow that it arises from experience.”
So, in simple terms, a priori knowledge is independent of current experience or any requirement for new (or further) knowledge. Yet a priori knowledge still isn’t independent of all experience.
So, having acquired the English language, etc., such a person doesn’t need to (as it were) go elsewhere to know that the statement “Whatever is green is coloured” is true.
On this account, then, acts of deduction, the recognition of tautologies, the truths of basic mathematical statements, etc. don’t require new knowledge or new experiences either.
More relevantly, all this also applies to statements (1) to (4) above.
(2) “Whatever is square is rectangular.”
This is similar to (1) above.
Just as it’s the case that if something is green, then it must also be coloured. Similarly, if something is square, then it must also be rectangular.
[The fact that all squares are rectangles, but not all rectangles are squares, doesn’t impact on this statement’s a priori status.]
It is impossible for x to be square and not be a rectangle.
So is this a truth about the world?
Yet, even if it is a truth about the world, then human beings have still created contingent natural-language terms to describe these aspects of the world. Humans have also made it a geometrical (even grammatical) rule that all squares must be classed as rectangular.
What’s more, and in a Platonist sense, there are no perfect squares in the world (or in nature). Thus, does that mean that such terms and rules are (as it were) abstractions and/or simplifications? And is it these factors which determine the a priori nature of the statement “Whatever is square is rectangular”?
These are complex metaphysical questions.
However, these questions don’t seem to impact on the a priori status of the statement, “Whatever is square is rectangular”. That seems to be because whatever the correct metaphysics of squares is, this statement can still be known to be true without consulting the world. That is, no reader of this statement (if he or she understands it) need actually check any squares (or what are taken to be squares) out there in the world. Thus, in that sense, the reader must assume (as it were) Platonic squares, as well as Platonic rectangles.
(3) “The sum of three and two is five.”
Interestingly, what’s being discussed here isn’t this example of arithmetic:
3 + 2 = 5
It’s this statement:
“The sum of three and two is five.”
This means that it’s not only the arithmetical equation embedded with the sentence (i.e., “three and two is five”) that’s known to be true a priori, but the entire sentence (i.e., “The sum of three and two is five”).
In any case, there are many philosophical takes on mathematical truth, and whichever one is adopted, it doesn’t seem to have any impact on the a priori status of this statement.
For example, say that one believes in the mathematical constructionist (i.e., rather than the narrower and more precise mathematical constructivist) position which has it that the statement “The sum of three and two is five” (or even the statement “3 + 2 = 5”) is taken to be true purely because of the meanings we give the words “two”, “three” “sum” and “equals” (or the meanings we give the symbols “2”, “3”, “+” and “=”), and the rules we’ve established (via convention) which make it the case the 5 is the sum of 3 and 2. Thus, this statement is still known to be true a priori regardless of one’s metaphysics. So even within a constructionist philosophy, there may still be a place for a priori knowledge — or even a priori truth.
Thus, whenever anyone who knows a little arithmetic reads this statement, he will know that it’s true. It doesn’t seem to matter that he may also believe that it’s true-by-convention (or due to “historical accident”): he still doesn’t need to consult the world to establish its truth (or, perhaps, only its correctness).
(4) “Necessarily, if ‘p or q’ is true and q is false, then p is true.”
This can be seen to be true even without knowing what the letters p and q stand for (or what their semantic content is). In fact, it doesn’t even matter what they p and q stand for, as long as they stand for propositions (or statements) which are taken to be either true or false.
Thus, the a priori reasoning here is purely logical.
That is, the statement “p or q” means that either p or q is true. That is, p and q aren’t both true. Similarly, it isn’t the case that both p and q are false.
Thus, since we’re then told that q is false, and either p or q must be true, then p must be true.
So what about the opening word “necessarily”?
The word “necessarily” is modal in nature. It tells us that the statement which follows it must be true. That is, there’s no possibility of it being false. It is, therefore, a tautology or a logical truth. In other words, the statement is true simply because of its logical form or grammar.
In terms of a priori knowledge, then, this statement can be known to be true simply by reading and understanding it. That is, the semantic content of the symbols p and q needn’t be known. More mundanely, one needn’t check the world, do research, find evidence, observe anything, read a history book, etc. in order to establish the truth (or falsehood) of this statement. However, (as argued earlier) one must have previously learned a little logic, the English language, etc. in order to understand the sentence which expresses the statement under consideration.
Yet, in a sense, all this also applies to the statement “Snow is white”.
After all, once one learns that snow is [always?] white, then when one comes across the statement “Snow is white”, we can say it is “true” without further research, finding evidence of new snowfalls, “checking the facts”, etc.
In that case, then, how does the statement “Snow is white” differ from the statement “Necessarily, if ‘p or q’ is true and q is false, then p is true”?
Well, the statement “Snow is white” isn’t logical. It’s empirical. It’s also contingently true, rather than necessarily true. In other words, it could have been the case that snow isn’t white. (Or could it?)
On the other hand, there is no situation in which the statement “Necessarily, if ‘p or q’ is true and q is false, then p is true” could be false. This statement, then, is necessarily true in that its negation would be self-contradictory. Or, in a different jargon, the logical statement is true at every possible world.
So although the statement “Snow is white” can be assented to without further evidence, it still required evidence to be established in the first place. Yet that doesn’t apply to the statement with p’s and q’s in it. That statement isn’t empirical in nature — even if the semantic content of both p and q can be taken to have empirical content.
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