i would need some help in understanding Yablo's theory of subject matter as expressed in his book "Aboutness". Yablo has uploaded an appendix on his academia.edu profile that serves as a sort of summary of the main formal issues of the book and he applies some formal concepts on propositional calculus.
The source is: https://www.academia.edu/2604116/Aboutness_Theory
Specifically, i do not understand the following passage:
9.1 Truthmakers in PC
A partial valuation ν of a given propositional language is an assignment of truth-values to some but not necessarily all of the language’s atoms. A partial model of A is a partial valuation each of whose classical extensions makes X true. A minimal model χ of X is a partial model of X none of whose proper submodels are partial models of X. (Notation: α is a minimal model of A, β is a minimal model of B, and so on.) A truthmaker for sentence X is a minimal model of X. A falsemaker for X is a minimal model of ¬X, aka a minimal counter model of X. A truthmaker for X in ν is a minimal model χ of X such that χ⊆ν. A falsemaker for X in ν is a minimal countermodel χ of X such that χ⊆ν."
a) What are the classical extensions of a partial valuation on a sentence A in a given propositional language? I think they should be considered as all the valuations that assign to a given atomic formula p the value"true", but correct me if it is wrong.
Shouldn't a truthmaker be a set of valuations in which the sentence X is true, rather then a single valutation that gives X the value "true"?
b) What does it mean that a minimal model for X is a partial model that does not have any proper sub-models that are partial model of X? What is a proper sub model of a partial valutation?
In this case, I thought that a minimal model should be a set of valuations that assign X the value "true" and it is the smallest such set, not having any set with valuations that is included in that minimal set. But I think I still do not get the point, so I ask you about this.
10.1 Aboutness in PC
X ’s subject matter is the m whose cells are made up, for each minimal model χ of X, of the classical models “above” (including) χ. X ’s subject anti-matter is the m whose cells are the classical models above X ’s minimal.
c) What does it mean that m has as cells the classical models "above" each minimal model of X? Inuitively I thought that a subject matter could be considered the set that has as elements all the sets that contain valutations that makes X "true", but I do not get the sense of introducing models "above" a certain minimal model for X
Thanks in advance for all the help.
See 1.1.: $v$ is a "classical valuation" if it satisfies the rule for negation: $v(¬A)=\text T \text { iff } v(A)=\text F$, that means that: "A proposition $A$ is true in world w iff $w \in A$, otherwise false".
A partial model of $A$ is a partial valuation that can be extended (using classical truth tables) to be a model of $A$, i.e. to satisfy A.
A truthmaker is a model that is minimal.
See Aboutness, page 61:
This correspond to the valuation $v(p)=\text T$ and $v(q)=\text F$ and is a model for the formula $(p \land \lnot q)$.
It is minimal because if we remove one of the two assignment, what is left is no more a model of formula $(p \land \lnot q)$, because we have not enough information to compute the truth value of the formula.
Now, using this simple example, we have that:
It is not a partial model of $(p \land \lnot q)$ because the extension $z$ of $v$ such that $z(p)=v(p)$ and $z(q)=\text T$ does not make $A$ true.
Every extension $w$ of $v$ that is a model must contain $w(p)=\text T$ and $w(q)=\text F$.
Thus, $v_1$ such that $v_1(p)=\text T, v_1(q)=\text F, v_1(r)=\text T$ is a partial model of formula $(p \land \lnot q)$.
The minimal one is $v'(p)=\text T, v'(q)=\text F$.