What are the laws of logic, and are they universal? Are the laws of logic something that exists “out there” and our symbolic and syntactical conventions merely a way of describing it? Or do our logical propositions and assertions dictate the truth? This may seem like an easy question to answer, but not everyone agrees.
Willard van Orman Quine famously rejected the cherished notion of analyticity. He pointed out that analyticity is defined by reference to synonymy, which is then defined by reference to analyticity, thus making the intensionality of analyticity circular. However, Quine was also skeptical of the idea of intensionality, and this, he said, was because it lacked analyticity. And so, his own argument seems to result in circularity. Ludwig Wittgenstein, on the other hand, contends that analytic statements do not say anything about our world, but about our language. For instance, the proposition “all bachelors are unmarried men” is not a necessary truth, but a way of defining the language we’re using. It’s through a proposition like “all bachelors are unmarried men” that we define the rules of our language, not determine any semantic truths about the world.
This is where Wittgenstein made his distinction between ‘criterial’ and ‘symptomatic’ statements – or, more familiarly, definitional and empirical (analytic and synthetic). For instance, when we say “this ruler is 1 meter long” we are making a criterial statement: we define what the length of a meter is by how long the ruler is – a meter is defined as “the length of this ruler.” But, to say “this table is 1 meter long” is symptomatic: it is just the case that, because we found that this table is as long as our ruler, that we say the table is a meter long.
In this way, it is possible for things to be redefined. If what we now call 1.25 meters was redefined to be a meter (for whatever reason – perhaps it’s found to be more practical), it could, conceivably, be the case that 1 meter is now redefined to this new length. However, this would not change the empirical fact about the speed of light: just because we say it is a different number, due to the rescaling of units, that does not make the speed of light slower – the number by which we measure the speed of light of symptomatic in that it is a consequence of how fast the speed of light just is in reality, sort of like a contravariant change of basis in a vector transformation.
But what about the rules of logic: are these criterial or symptomatic? For instance, we might say that “all bachelors are unmarried men” says something about the linguistic criteria our language – namely the definition of “bachelor” – but we might also say that it also corresponds to something ‘out there’ in reality. If we accept the definitional criteria about the semantic value of all the words in the proposition “all bachelors are unmarried men” then, by going out into the world, if we actually met every person in the world, we would reliably find that all of the bachelors have the property of being unmarried men in common.
How is it we would come to semantic value in the first place? Semantic value is produced by virtue of a definition that requires only conceptual understanding and the laws of logic alone. The concept of the unmarried man is named by the words “unmarried man” and “bachelor” but the concept itself is instantiated in the real world. Knowing the concept, regardless of what words are used to name it, is the semantic value of the proposition. A semantic understanding of this concept would allow a person to predicate it of subjects in a truth-functional way: if I know that “bachelor” is the word that names the concept unmarried man, and I know that person x is an extension of the concept unmarried man, then I know, by the laws of logic alone, that I can predicate “bachelor” of x – “x is a bachelor” is a true proposition, not because “bachelor” is defined in our language as the words “unmarried man” but because, in a metalinguistic sense, “bachelor” names the concept unmarried man.
Semantic value, then, is not gained from acquaintance with the referent of a concept. One does not have to know even a single unmarried man to know what the referent of the concept is. Knowing the members of the concept’s extension requires synthetic a posteori work: we might say that “x is a bachelor” and believe it to be true, but further justification is required for this belief to be knowledge. If we substitute a random name for x such as “Ralph is a bachelor” then the proposition has no semantic value and therefore no truth value. Something more needs to be known of Ralph as a person, because the name Ralph itself (the actual word) is not a bachelor – “the word Ralph is a bachelor” is not true. The concept that the word “bachelor” names must be predicated of an object for the proposition to be meaningful and therefore to have truth value.
What this means is that, analytic propositions do not just define language, since the language itself must conform to the real world. We define “bachelor” to name the concept of unmarried man, we do not define it to name the words “unmarried man.” Saying “all bachelors are unmarried men” is not the same as “all bachelors are bachelors” because in the first the word “bachelor” is a name which, in isolation, has as its referent the word “bachelor,” which has only the word “bachelor” in all its forms (spoken, written, thought, etc.) as its extension, whereas “unmarried man” is a name whose referent is the concept unmarried men, which has extensionality ranging over all men who are unmarried. Thus, while the actual words uttered (or written, or thought, etc.) are criterial, they must conform to some symptom of reality: if I ask how many wudjadoos are in this article, there is no way to know what I mean unless the criterial definition of “wudjadoos” is applied to some concept which exists in the world.
Perhaps another way to think of analytic propositions is as a form of Tarskian T sentences:
“The word [bachelor] refers to the concept of unmarried men” is true if and only if the word [bachelor] refers to the concept of unmarried men.
Where, in this instance, the object language in quotes assigns the name “bachelor” to the concept of unmarried men, and this is said to be true in the metalanguage just in case that the word “bachelor” has the concept of unmarried men as its semantic value. If the T sentence is evaluated as true, then the proposition “all bachelors are unmarried men” is semantically analytic, even if it is not necessary that the word “bachelor” must name the concept of unmarried men. And so, proposition “all bachelors are unmarried men” is not a tautology: it is significant in that it evaluates the semantic value of the word “bachelor.” The copula are (or is in the case of singular) assigns a name to the concept of unmarried men, not to the words “unmarried men.” And so
∀x(Bx→Ux)
where B = bachelor and U = unmarried man, means that the word “bachelor,” which is defined to name the concept of unmarried men, can thereby be used to predicate subjects of the concept which it names.
Criterial reassignment can be the consequence of falsification, resulting in a paradigm shift. This can be seen in the law of gravity, where it was once defined by Newton’s equation:
F = (GMm)/r2
It used to be that to say what gravity is, one would say that gravity is the force that obeys the above equation. This was found to be inadequate, but gravity was not redefined immediately, since Newton’s equations still produced accurate predictions in most cases – it reduced uncertainty, thereby giving information. But, the falsification of Newton’s law of gravity was what made it possible for Einstein to redefine gravity: both paradigms conformed to reality in some way (they give varying amounts of information about reality), but ultimately they are a norm of representation – a way for us to conceptualize, model, and think about reality. But, in order for these representations to be informative, they must reduce uncertainty: a new theory of gravity that gave even worse predictions than Newton’s theory would not be adopted, because the criterial ruler, in this case, is reality itself. We judge the information a theory gives by how close it is to the information inherent in reality, the concept that exists ‘out there’ so to speak.
One may protest to this picture in a Quinean fashion, in saying that this creates a “myth of the museum” for the concepts being named: there is (at the very least the possibility of) indeterminacy in the semantic value of names: the concept of unmarried men may have slightly different meanings to different people. There is not some Platonic ideal of the concept unmarried man out there in the world that acts as a reference for the words “unmarried man” or “bachelor” or any synonym or translation of these words. But this concept does exist as information: the ontological “substance” of the concept is its ability to determine certain behaviors and contexts that can, in principle, be predicted. For instance, knowing that a person x is a bachelor would allow, in principle, for the prediction that there is nobody who calls x their husband; there will be no authentic government or ecclesiastical records of a marriage pertaining x that can be found; if x asserts any propositions about being married or about their spouse, those propositions (at the time they are asserted) will reliably be found to be false. Thus, the concept of unmarried man that is out there in the world is the information inherent in the semantic value of the concept.
An objection to this information view may be that a word can name different concepts. For instance, the word “love” can name the concepts of platonic love, familial love, or romantic love. And so, when someone says “A loves B” there is, in principle, different predictions that could be made about how A and B interact. However, this is not a problem of the information ontology, but an indication that the word “love” is a high-entropy word: to say “A loves B” does not reduce the uncertainty about the relationship between A and B enough to make accurate predictions, it only reduces the uncertainty enough to make somewhat more general predictions about the relationship between A and B. Knowing only that “A loves B” might allow, in principle, for predictions about whether A will make efforts to spend time with B, even if it is not in principle reliably predictable in what activities A and B will partake.