On “Is”

<<τὸ ὂν λέγεται πολλαχῶς…>> (Bill Clinton and Aristotle)

“Is” (to be) is a tricky word, and I think the ambiguous nature of this word has led to some misunderstandings of some of the arguments I present, which are typically written in Free Logic. “Is” has multiple meanings, and some of the meanings are more “ontologically committing” or “existentially loaded” than others. Some common logical notation that gets translated as “is” in ordinary language include: 1) “(∃x)”, 2) “=”, 3) “Px”, and 4) “≝”, and I would like to emphasize that they are not syntactically equivalent, and do not function in logical arguments in the same way.

1) The “is” of existential quantification: There is an x, e.g. there is something green, or (∃x)Gx,. This can be interpreted as a “particular quantifier” indicating that there is at least one individual x. Depending on the domain of discourse, the existential quantifier can be more or less ontological committing. One could say, there is a fictional detective that Arthur Conan Doyle wrote about, and use the existential quantifer, and one would not be committed to the reality of fictional beings, i.e. (∃x)(Fx & Wax) [read: there is an x such that x is fictional and Arthur Conan Doyle wrote about x], s = x satisfies the formula in this case, where “s” means “Sherlock Holmes”.

2) The “is” of identity: (x = y), e.g. Tully is Cicero, or (t = c). Sometimes the “is” of identity is combined with the existential quantifier to make strong existential claims, e.g. there is a planet named Venus: (∃x) [Px & (x = v)]. There are rules around identity that are, themselves, metaphysically complicated, and it is controversial how those rules should apply to logic. For instance, it is sometimes granted that (∀x)(x = x) can be introduced at any stage of an argument simply because everything is self identical. Also, if a = b, then b can be substituted for a in an argument in some, but not all, contexts. The contexts were such substitutions cannot occur are called “referentially opaque contexts”. For example, Clark = Superman. Lois believes Superman = Superman. But it doesn’t follow that Lois believes Superman = Clark.

3) The “is” of predication:  x is purple, or simply Px. This “is” is not very existentially committing, but merely ascribes properties to individuals, on could say Sherlock, where “s” is Sherlock, and “B” is the predicate “Brave”: Bs. In “Free Logic” to make strong “existentially loaded” or “existentially committing” claims, you might specify “Real Existence” as a kind of predicate said of an individual. This might run contrary to “Kant’s Dictum” that existence is not a real predicate, but alternative ways of forming existential claims about what exists in the world are problematic for other reasons. When I construct ontological arguments, I tend to use Free Logic. This is because free logic allows you to quantify over things that may or may not exist in reality, which is needed, if one is not to beg the question in ontological arguments.

4) The “is” of definition: for example, the name “God” is “the x such that x is perfect”, or g ≝ (ɿx)Px.  I might stipulate such a definition in an argument by writing “D1: God is perfect.” This is not an existentially committing sentence, but a stipulation of the meaning of a term. Definitions are not really propositions in the fullest sense, as they are not true or false, but merely what one means when one uses a term in a proposition. As such, a definition is usually assessed in terms of clarity and coherence rather then whether it is true.  The scholastics would make this point by saying that definitions pertain to the first act of the mind, not the second.  Explicitly adding predicates into a definition in order to prove that the thing defined has those predicates can be question-begging, this would include adding “real existence” as a predicate in the definition, e.g. A shmunicorn is a unicorn that exists, therefore shmunicorns exist would constitute a question-begging proof. Adding “existence” directly into a definition also entails that the thing defined would exist necessarily, since one can add necessity to any conclusion derived from zero premises. It would be unclear and possibly incoherent to say that shmunicorns exist of necessity, so such a proof should not command assent. My ontological arguments for God are never zero-premise, and always require one or more premises to reach the conclusion.

So this can help us to disambiguate.  Consider the following sentence: “There is an individual who is the author of this blog and who is Daniel, who is the only son of James and Kathy Vecchio, and who is.”

Axy ≝ x is the author of y
Sxyz ≝ x is the only son of y and z
d ≝ (ɿx)Sxjk
j ≝ James Vecchio
k ≝ Kathy Vecchio
b ≝ Vexing Questions blog

(∃x){[Axb ∧ (x = d)] ∧ E!x}

There are a lot of “ises” in that expression, but we can now see how each has its own function.

Posted on July 30, 2019, in Logic and tagged , , . Bookmark the permalink. Leave a comment.

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