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Sunday, October 13, 2013

Truth-making and Reference-making: Part 2

Part 2: A Quasi-Formal Account


I will now define truth-makers for truth-functional compounds and quantified sentences. To define truth-makers for them, we  will also require the notion of a false-maker: We say that x false-maker for a subject-predicate sentence p iff x is a reference-maker for p's subject term but is not a reference-maker for p's predicate term.  For relational sentences, an ordered n-tuple o is a false-maker for an n-ary relational sentence p iff the objects ordered in o are each reference-makers for one of p's subject terms but o is not a reference maker for p's predicate term. 

We then define:

1a) An object x is a truth-maker for ~p iff x is a false-maker for p. A set s is a truth-maker for ~p iff some member of s is a false-maker for p. 

1b) An object x is a false-maker for ~p iff x is a truth-maker for p. A set s is a false-maker for ~p iff some member of s is a truth-maker for p. 


2a) An object x is a truth-maker for p & q iff x is a truth-maker for p and for q. A set s is a truth-maker for p & q iff some member of s is a truth-maker for p and some member of s is a truth-maker for q. 

2b) An object x is a false-maker for p & q iff x is a false-maker for p or for q. A set s is a false-maker for p & q iff some member of s is a false-maker for p or if some member of s is a false-maker for q. 


3a) An object x is a truth-maker for p v q iff x is a truth-maker for p or for q. A set s is a truth-maker for p v q iff some member of s is a truth-maker for p or if some member of s is a truth-maker for q. 

3b) An object x is a false-maker for p v q iff x is a false-maker for p and for q. A set s is a false-maker for p v q iff some member of s is a false-maker for p and some member of s is a false-maker for q. 


4a) An object x is a truth-maker for p --> q iff x is a false-maker for p or a truth-maker for q. A set s is a truth-maker for p --> q iff some member of s is a false-maker for p or if some member of s is a truth-maker for q. 

4b) An object x is a false-maker for p --> q iff x is a truth-maker for p and a false-maker for q. A set s is a false-maker for p --> q iff some member of s is a truth-maker for p and some member of s is a false-maker for q. 


5a) An object x is a truth-maker for p <--> q iff x is a truth-maker for p and for q, or if x is a false-maker for p and a false-maker for q. A set s is a truth-maker for p <--> q iff some member of s is a truth-maker for p and some member of s is a truth-maker for q, or if some member of s is a false-maker for p and some member of s is a false-maker for q. 

5b) An object x is a false-maker for p <--> q iff x is a truth-maker for p and a false-maker for q, or if x is a false-maker for p and a truth-maker for q. A set s is a false-maker for p <--> q iff some member of s is a truth-maker for p and some member of s is a false-maker for q, or if some member of s is a false-maker for p and some member of s is a truth-maker for q. 


6a) For a given, restricted domain of discourse: An object x is a truth-maker for a universally quantified sentence iff x is the only object in the domain and x is a reference-maker for the open sentence bound by the quantifier (I take open sentences to be predicates, and thus covered by what was said above). A set s is a truth-maker for a universally quantified sentence iff every member of s is a reference-maker for the open sentence bound by the quantifier.

6b) For a given, restricted domain of discourse: An object x is a false-maker for a universally quantified sentence iff x is not a reference-maker for the open sentence bound by the quantifier. A set s is a false-maker for a universally quantified sentence iff some member of s is not a reference-maker for the open sentence bound by the quantifier.



7a) For a given, restricted domain of discourse: An object x is a truth-maker for an existentially quantified sentence iff x is a reference-maker for the open sentence bound by the quantifier. A set s is a truth-maker for an existentially quantified sentence iff some member of s is a reference-maker for the open sentence bound by the quantifier.

7b) For a given, restricted domain of discourse: An object x is a false-maker for an existentially quantified sentence iff x the only object in the domain and x is not a reference-maker for the open sentence bound by the quantifier. A set s is a false-maker for an existentially quantified sentence iff no member of s is a reference-maker for the open sentence bound by the quantifier.


There may be some additional complications concerning open sentences which are truth-functionally complex and/or which involve nested quantifiers, but I take it that they can be accounted for in much the same way as above. (And remember, I only said that this was a quasi-formal account.) In any case, this much should suffice for the purposes of Part 3

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