One analogue of rigidity for general terms: how about we think of it as a rigid connection between properties (or alternatively, an additional connection between the "rigid" term and a further property).
So for example, 'water' in the first instance picks out the property of being water, which is tied rigidly to the property of being (mainly composed of) H20.
Or 'cat' is tied in the first instance to the property of being a cat, and that is tied rigidly to the property of being an animal.
But now there is another kind of thing which seems different, and comes up with sentences like 'John has the property we talked about yesterday'.
Suppose the property we talked about yesterday was the property of having the property that is discussed in Book A. And suppose Book A contains a discussion of the property of redness. Now 'has the property we talked about yesterday' rigidly designates the property of being a property we talked about yesterday, but it also non-rigidly designates the property of having the property that is discussed in Book A, as well as the property of redness.
This motivates the picture of, behind a predicate, a stack of properties, where the top one is designated rigidly, and the ones below not.
Problem: we might want to say that 'cat' rigidly designates a certain kind of animal. And I may then want to rephrase that as: 'cat' rigidly ascribes animality.
But doesn't the 'rigid' bit here fall away? Take the phenomenological, underlying-nature-neutral counterpart of 'cat' - 'catty thing'. Now even if in our world all the catty things are cats, it doesn't sound right to say that 'catty thing' ascribes the property of being a cat at all - it's not that it ascribes that property, only non-rigidly.
So now it is beginning to look like the distinction we are after here is between a term merely covering things with property P, vs. ascribing to them property P.
But then that seems wrong when we go back to 'John has the property we discussed yesterday', since if what we talked about yesterday was the property of redness, there is a sense in which that sentence ascribes redness to John.
This is hell!
But this whole problem, occupying the last few paragraphs, perhaps only arises from mixing together two different analogues of rigidity that we get when we look at predicates.
It may be protested that 'John has the property ...' is not a property ascription syntactically at all, but rather a 2-place relational statement with a non rigid second term.
Be that as it may, we can still classify 'has the ...' as a predicate and can still talk about a rigid/non-rigid distinction. And so I think we need to recognise that there are at least two quite different things going on here - two different things which are a bit similar to rigidity/non-rigidity as applied to names.
One is the difference between 'has the property discussed earlier' and 'is red'. Another is the difference between 'is water' and 'is watery' (one brings being composed of H20 along with it in counterfactual scenario descriptions, and the other doesn't), or 'is a cat' and 'is a catty thing'.
One reason the second analogue may be counterintuitive if presented as a kind of rigidity is that in the case of singular terms, rigidity is associated with simplicity (both syntactic and semantic), but in the case of predicates (let's look at 'is gold', 'is water', 'is a cat' and put aside 'has the property...') it's the opposite. The "non-rigid" predicates just don't take any further property along with them, but the rigid ones do. I.e. 'is a catty thing' or 'is catlike' just picks out one property, but 'is a cat' is tied to the further property of being an animal.
Actually, the 'any' in 'any further property' is probably wrong! Maybe all predicates rigidly take some further properties along with them. So this second sort of "rigidity" we can talk about in connection with general terms should be thought of as relative to whatever further property is in question. (For instance, 'is a pencil' is arguably counterfactually locked to 'is a physical object'. So 'is a pencil' rigidly picks out physical objects - we might want to say something like that.)
One thing that is emerging here is that the 'has the property discussed earlier' vs. 'is red' thing is one distinction which pattern-matches with Kripke's discussion of rigidity as applied to names, but there is also another thing going on - predicates dragging further properties along with them in counterfactual scenario descriptions - which actually corresponds better with Kripke's informal applications of the notion of rigidity to general terms.
Now it looks like the general-term-"rigidity" considerations in Naming and Necessity are actually closer to the "necessity of constitution" and similar considerations than they are to the "necessity of identity" considerations.
Background Reading:
- Section 4.2. of the SEP entry for Rigidity: https://plato.stanford.edu/entries/rigid-designators/#ObjAppRigTerForKinPro
Also:
- Kripke, Saul (1980). Naming and Necessity. Harvard University Press.
- Soames, Scott (2002). Beyond Rigidity: The Unfinished Semantic Agenda of Naming and Necessity. Oxford University Press
- Salmon, Nathan (2004). Are general terms rigid? Linguistics and Philosophy 28 (1):117 - 134.
- Linsky, Bernard (2006). General Terms as Rigid Designators. Philosophical Studies 128 (3):655-667.
- Martí, Genoveva & Martínez-Fernández, José (2011). General terms, rigidity and the trivialization problem. Synthese 181 (2):277 - 293.
- Schwartz, Stephen P. (2002). Kinds, general terms, and rigidity: A reply to LaPorte. Philosophical Studies 109 (3):265 - 277.
- de Sa, Dan López (2007). Rigidity, General Terms, and Trivialization. Proceedings of the Aristotelian Society 107 (1pt1):117 - 123.
- Marti, Genoveva (2004). Rigidity and General Terms. Proceedings of the Aristotelian Society 104:131-148.
- Zouhar, Marián (2009). On the Notion of Rigidity for General Terms. Grazer Philosophische Studien 78 (1):207-229.
- Orlando, Eleonora (2014). General terms and rigidity: another solution to the trivialization problem. Manuscrito 37 (1):49-80.
- Gómez-Torrente, Mario (2004). Beyond Rigidity? Essentialist Predication and the Rigidity of General Terms. Critica 36 (108):37-54.
- Kosterec, Miloš (2018). Criteria for Nontrivial General Term Rigidity. Acta Analytica 33 (2):255-270.
- Soames, Scott (2002). Beyond Rigidity: The Unfinished Semantic Agenda of Naming and Necessity. Oxford University Press
- Salmon, Nathan (2004). Are general terms rigid? Linguistics and Philosophy 28 (1):117 - 134.
- Linsky, Bernard (2006). General Terms as Rigid Designators. Philosophical Studies 128 (3):655-667.
- Martí, Genoveva & Martínez-Fernández, José (2011). General terms, rigidity and the trivialization problem. Synthese 181 (2):277 - 293.
- Schwartz, Stephen P. (2002). Kinds, general terms, and rigidity: A reply to LaPorte. Philosophical Studies 109 (3):265 - 277.
- de Sa, Dan López (2007). Rigidity, General Terms, and Trivialization. Proceedings of the Aristotelian Society 107 (1pt1):117 - 123.
- Marti, Genoveva (2004). Rigidity and General Terms. Proceedings of the Aristotelian Society 104:131-148.
- Zouhar, Marián (2009). On the Notion of Rigidity for General Terms. Grazer Philosophische Studien 78 (1):207-229.
- Orlando, Eleonora (2014). General terms and rigidity: another solution to the trivialization problem. Manuscrito 37 (1):49-80.
- Gómez-Torrente, Mario (2004). Beyond Rigidity? Essentialist Predication and the Rigidity of General Terms. Critica 36 (108):37-54.
- Kosterec, Miloš (2018). Criteria for Nontrivial General Term Rigidity. Acta Analytica 33 (2):255-270.
