Abstract
This paper is an investigation in the use of truthmaker theory for exploring the relation of logic to world, and as a tool for metaphysics. A variant of truthmaker theory, which we call the simple theory, is defined and defended against objections. It is characterized formally, and its central features are derived. As part of this project, we give a formal metaphysics based on nondeterministic necessitation relations among possible entities. In what is called the fundamental theorem of truthmaking, it is shown that, as long as a logic is sound and complete, its inferential structure will be isomorphic to the necessitation structure of our metaphysics. We thus arrive at a purely structural logic–world relationship which can be used for metaphysical investigations. Other products of our investigation are a sound and complete semantics for first-order logic with identity and a solution to a result of Restall (Australian Journal of Philosophy, 74, 331–340, 1996) which has threatened to make truthmaker theory trivial.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The existence of a set of all possible entities does not by itself incur a contradiction as long as we keep object language and metalanguage firmly apart. Even if E were to contain all sets of ZFC, this does not mean that it has to contain all subsets of itself. E is a denizen of the metalanguage, while its elements belong to the object language. Cf. [27].
I hope that no confusion arises from the fact that ‘ ⇒’ is used both as a connective and as a metalinguistic predicate. ‘ ⇒’ has roughly the same meaning as Lewis’s fishhook symbol, but the latter is unfortunately, in the opinion of the current author, too ugly to use. For connections to derivability and logical consequence, see Section 5.
See, for example, [20].
Lewis actually goes even further, and assumes all sets of possible entities—even those that are in different worlds—to have sums. This leads to the conclusion that there are entities that are impossible, in the sense that they exist in no possible world. We will not have use for such impossible entities here.
Of course, the “usual” slingshot argument may be thought to be a problem for truthmaker theories as well. The most reasonable way to handle it is, to my mind, to deny that statements that differ by the replacement of coreferential singular terms have the same truthmakers. Cf. [21]. In doing this, it is important to keep the notions of referent and truthmaker apart: the referent of a singular term does not have to have any relationship to the truthmaker for statements including that term, and indeed it does not have to be an entity at all. For a semantics in which this is the case, see Section 6.
As far as I know, the first philosopher to point out the intimate connection between intuitionistic semantics and truthmaking was Michael Dummett, in The Logical Basis of Metaphysics [13].
Again, we should not be led to believe that the members of \(\mathfrak {P}\) are propositions. Just as we have represented possible worlds as sets of possible entities, we use sets of statements to represent propositions, whatever these might be.
The proof is exactly the same as for multiple-conclusion consequence relations, see Shoesmith & Smiley [28, p. 30].
See Shoesmith & Smiley [28, pp. 36–39] for a proof.
Note that we are talking about a sum here, rather than the sum, since it is generally not unique. In this it shares some properties with sums and products in category theory, rather than with those of universal algebra.
Most of the results appearing in this article are based on the author’s doctoral dissertation [1], for which he acknowledges the financial support of the Swedish science council. More detailed considerations about the role of truthmaker semantics in metaphysics and the philosophy of science can be found in that work.
References
Angere, S (2010). Theory and reality: metaphysics as second science. Ph.D. thesis, University of Lund. Available online from the author’s homepage at http://www.fil.lu.se/en/department/staff/StaffanAngere/.
Armstrong, D (1978). Universals and scientific realism. Cambridge University Press.
Armstrong, D (2004). Truth and truthmakers, Cambridge University Press.
Beall, J C, & Restall, G (2006). Logical pluralism. Oxford University Press.
Bigelow, J (1988). The reality of numbers: A physicalist’s philosophy of mathematics. Oxford University Press.
Boolos, G (1984). To be is to be the value of a variable (or some values of some variables). Journal of Philosophy, 81, 430–450.
Boolos, G (1985). Nominalist platonism. The Philosophical Review, 94, 327–344.
Carnap, R (1956). Meaning and necessity (second edition). University of Chigaco Press.
Daly, C. (2005). So Where’s the explanation? In Beebee, H, & Dodd, J (Eds.), Truthmakers: the contemporary debate. Oxford University Press.
Davey, B A, & Priestley, H A (2002). Introduction to lattices and order (second edition), Cambridge University Press.
Davidson, D (1969). True to the facts. Journal of Philosophy, 66, 748–764.
Dummett, M (1978). Truth and other enigmas. Harvard University Press.
Dummett, M (1991). The logical basis of metaphysics. Harvard University Press.
Dummett, M (1993). The seas of language. Oxford University Press.
Fox, J (1987). Truthmaker. Australasian Journal of Philosophy, 65, 188–207.
Horwich, P (1998). Truth (second edition). Oxford University Press.
Lewis, D (1986). On the plurality of worlds. Blackwell Publishers.
Lewis, D (1991). Parts of classes. Basil Blackwell.
Maurin, A -S, & Persson, J (2001). Realistic metaphysics: an interview with D. H. Mellor. Theoria, 67, 96–113.
Mulligan, K, Simons, P, Smith, B (1984). Truth-makers. Philosophy and Phenomenological Research, 44, 287–321.
Neale, S (2001). Facing Facts. Oxford University Press.
Read, S (2000). Truthmakers and the disjunction thesis. Mind, 109, 67–79.
Restall, G (1996). Truthmakers, entailment and necessity. Australian Journal of Philosophy, 74, 331– 340.
Rodriguez-Pereyra, G. (2005). ‘Why Truthmakers’ In Beebee, H, & Dodd, J (Eds.), Truthmakers: the contemporary debate. Oxford University Press.
Rodriguez-Pereyra, G (2006). Truthmaking, entailment, and the conjunction thesis. Mind, 115, 957–982.
Russell, B. (1956). The philosophy of logical atomism In Marsh, R C (Ed.), Logic and Knowledge: Allen & Unwin.
Scott, D (1971). On engendering an illusion of understanding. Journal of Philosophy, 68, 787–807.
Shoesmith, D J, & Smiley, T (1978). Multiple conclusion logic. Cambridge University Press.
Wittgenstein, L (1922). Tractatus logico-philosophicus. Routledge & Kegan Paul Ltd.
Wójcicki, R (1988). Theory of logical calculi. Kluwer Academic Publishers.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Angere, S. The Logical Structure of Truthmaking. J Philos Logic 44, 351–374 (2015). https://doi.org/10.1007/s10992-014-9323-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-014-9323-9