# "df." as an abbreviation

I'm reading a book and I stumbled upon the following paragraph, in context of doing a conceptual analysis, to quote:

Ml: Motherhood is the conjunction of femaleness and parent-hood.
M2: To be a mother is to be a female parent.
M3: Being a mother is being a female parent.
M4: 'Mother' means the same as 'female parent'.
M5: x is a mother at t = df. x is a female parent at t.

For the life of me I cannot figure out what that "df." in the last sentence supposed to mean.

The Book: Confrontations with the Reaper

• The symbol seems to be = with df or perhaps df. as a suhscript, and many situations can't handle the subscript. It is so unusual that it must be considered non-standard. Oct 14 '20 at 18:23

I disagree with Greybeard. It is true that this notation is from mathematics (or logic), but here

= df.

means: "is equal to by definition".

It is no longer usual to see this in mathematics, but other areas have retained it.
A web search found LINK1 and LINK2. At LINK3 it is shown as = with subscript "Def", and attributed to Cesare Burali-Forti, 1894.

Parse

x is a mother at t = df. x is a female parent at t.

as

[x is a mother at t] =df. [x is a female parent at t].

That is,

[x is a mother at t] is by definition the same as [x is a female parent at t]

We haven't seen the original (we haven't even been told what book it is), but I would guess it is printed as an equal sign "=" with "df." as a subscript.

• Can you add a reference? This answer makes a lot of sense bur I’ve never seen this notation, so it would be good to have an authoritative citation. Oct 14 '20 at 12:10
• How do you explain "t = df."? Oct 14 '20 at 13:40
• In support of this, I appear to have found the book (added link), and it prefaces "M5" with the comment "[...] I happen to prefer to formulate analyses as definitions [...]" Oct 14 '20 at 17:54
• The correct way of rendering this symbol has def or df. above the equals sign, like this: ≝ . The only possible excuse for its appearing with df. next to the equals sign in the text that the OP has seen is that the typesetter was not able to reproduce the sign properly, and resorted to this as a close-enough approximation. Oct 14 '20 at 21:27
• I didn't add the book because well it's very pessimistic for the times. Thanks a lot for your research and answer! Oct 15 '20 at 8:10

The symbol that the author actually intended to appear there is:

# ≝

It means that what is of one side of it is defined as what is on the other side. Its ‘home’ is in formal logic, where the expressions on both sides of it would be formulated by using logical symbols, as in, for example:

M(x,y)≝P(x,y)&F(x)

Once such a definition is introduced, it can be invoked in any proofs that follow, as the ground for treating the expressions on the two sides of the symbol as interchangeable.

Professional philosophers, when writing for philosophically trained audiences, which can be presumed to be familiar with the symbol, sometimes use it informally to connect ordinary English expressions, and that was done in the example that the OP encountered.

The symbol looks like an equals sign with def or df. above the top line. It, however, occasionally happened in the past that the typesetters were not equipped to render this symbol properly; they then resorted to placing df. next to the equals sign, which is what the OP has seen. While that may have been the best available substitute under the circumstances, it is confusing, because it may leave an impression that df. is a part of the expression on the right side of the symbol, rather than a part of the symbol. The problem should not appear in anything that is published today, as the present-day software makes it easy to render the symbol correctly.

df originates in the mathematical operation of differentiation.

df = delta 'f' where 'f' represents a particular function (condition or state), and delta f = one point in time on the line of f with respect to 't' another variable - in this case, time, which is symbolised as 't'.

From the context "f" seems to "being female" or "the female".

M5: x is a mother at t = df. x is a female parent at t.

Here both sentences are the same, the second merely substitutes "female parent" for "mother".

This gives us

M5: x (the woman/subject) is a mother at a specific time where that time, t. is a specific point in her life as a woman.

x (the woman/subject) is a female parent at t, i.e. a specific point in her life as a woman.