Should we accept a symmetric accessibility relation for possible worlds semantics within metaphysics?

Possible worlds semantics is a common tool for doing metaphysics. An interesting question is the question of which properties should be ascribed to the accessibility relation between the possible worlds. Now, surely, we need reflexivity: each possible world can access itself. On standard Kripkean semantics, this guarantees that necessarily true propositions in each world are also true in that world, which is a principle without which possible worlds metaphysics becomes senseless. Moreover, we need the accessibility relation to be transitive. After all, if possible world w2 is possible from the perspective of possible world w1, that is, if w2 is accessible from w1, and if possible world w3 is possible from the perspective of possible world w2, that is, if w3 is accessible from w2, then, plausibly and reasonably, possible world w3 is also possible from the perspective of possible world w1, which means that w3 is also accessible from w1. Indeed, transitivity guarantees that necessity is stable or robust: if a proposition p is necessarily true in some possible world, then p is also necessarily necessarily true in that world. Now, what about symmetry? Should we also require that the accessibility relation be symmetric, meaning that if possible world w2 is accessible from w1, then w1 is also accessible from w2? This doesn't seem to be the case. Here is a proposed counterexample. Consider a possible world w1 and consider the conjunction p of natural laws in w1. Suppose that in w1 the natural laws are necessarily true, so that p is necessarily true in w1. Thus, p is true in each possible world accessible from w1, which includes w1 itself due to reflexivity. Now, consider a possible world w2 in which p is false and whose conjunction of natural laws, say q, thus differs from p, and in which the natural laws are merely contingently true. So, there is at least one possible world accessible from w2 in which q is false and whose conjunction of natural laws thus differs from q. In this example, since p is false in w2, w2 is, given Kripkean semantics, not accessible from w1. But plausibly, w1 is still accessible from w2. For while the natural laws in w2 are contingently true and have q as conjunction, it's from the perspective of w2 reasonably still metaphysically possible that they could have been necessarily true and have p as conjunction. That is, w1 is possible from the perspective of w2. In this specific plausible case symmetry thus does not hold. It follows that we should not accept that the accessibility relation is symmetric. Therefore, the most adequate modal logic for doing possible worlds semantics within metaphysics is S4 (reflexivity and transitivity) rather than the stronger S5 (reflexivity, transitivity and symmetry) or the weaker KT (only reflexivity).

The Extended Semantic Argument

This week I finalized a new paper titled A Semantic Argument Against the Existence of Universal Properties and Its Implications for the Likelihood of Theism. In this paper, I bring together the research of the last years regarding my semantic argument, and disclose numerous consequences of the argument's conclusion - including the implication that God likely exists. A preprint of the paper on PhilPapers is here available.

The Modal-Epistemic Argument: Wintein's Rebuttals Rebutted

Stefan Wintein has recently published rebuttals to my refutations of his objections to my modal-epistemic argument for God’s existence. In response to Wintein's rebuttals, I've written a new paper wherein I systematically address Wintein’s counterarguments and argue that they are all unsuccessful.

Strong and weak contingents

Let me split the class of contingents. A contingent thing is strongly contingent if it allows for near alternatives. A contingent thing is weakly contingent if it is not strongly contingent. So, an apple is strongly contingent since we can conceive of different apples that closely resemble it. If consciousness is contingent, then it is weakly contingent since nothing closely resembles it. Suppose that the origin of reality is contingent. Is it then strongly or weakly contingent?

Take again an apple. Now, there are near alternatives for this apple conceivable: a red instead of a green apple, a smaller or larger apple, an apple having a different shape. And so on. These variances are conceivable. It’s not even required to actually see other apples. That’s why an apple is strongly contingent. It’s not just contingent. It also allows for near alternatives. The origin of reality is indeed weakly contingent. Why? Well, it needs to be a regress of explanations ender. Therefore, it needs to be self-explanatory. But then it cannot have near alternatives. This gives us a good reason to consider consciousness as the origin of reality. For consciousness seems the only candidate for a causally efficacious entity that is weakly contingent.

Dual forms in metaphysics

Suppose that in true metaphysical propositions the concept of ‘object’ and ‘relation’ are convertible, so that for each true metaphysical proposition about objects and relations, there is a dual propositional form that holds as well. For example, the true dual form of “Objects participate into relations” would be “Relations bind to objects”. In that case the conclusion of my semantic argument (“No relation binds a quality to all objects”) would be the dual of Graham Harman’s core premise of his object-oriented ontology (“No object reveals its quality to all relations”).

Fábio Maia Bertato on my modal-epistemic argument

Fábio Maia Bertato has published a paper in Religious Studies on my modal-epistemic argument for the existence of God. In his paper he offers a thorough formalization of an observation that is closely related to an observation I made for example on p. 25 of my article A Modal-Epistemic Argument for the Existence of God and on twitter. Great to see the modal-epistemic argument further evolving. The first premise of my argument though does not say that "all possible truths can be known to human beings". As I explain in my original paper, the first premise says that all possible truths can be known to knowing beings, either human or non-human.

God’s necessary existence, the even weaker principle of sufficient reason and origin essentialism

Let God be defined as personal first cause. The even weaker principle of sufficient reason merely asserts that the existence of all actual beings is explainable. That is to say, for each being in the actual world there is a possible world in which that being exists and is explained. In what follows I show that (i) if God exists and (ii) if the even weaker principle of sufficient reason holds, God exists necessarily.

Suppose God exists. So God exists in the actual world. Consider a possible world in which the being referred to as 'God' in the actual world exists and is explained. Since it's God and thus uncaused in the actual world, it's also uncaused in the possible world in which its existence is explained. Therefore this explanation cannot be a causal explanation. But then the only explanatory option left is a sound ontological argument for its existence. This being thus exists necessarily. Hence God exists necessarily.

My argument actually needs origin essentialism as an additional premise. For the claim that the being referred to as ‘God’ in the actual world is also uncaused in the possible world in which it exists and is explained, follows if we accept the further premise that beings have their causes essentially. That is to say, if a being is caused in some possible world, then it has that cause in all possible worlds in which it exists.

Indeed, suppose for reductio ad absurdum that the being referred to as 'God' in the actual world is caused in the possible world in which it exists and is explainded. Since on origin essentialism this being has that cause essentially, it also has that cause in the actual world. But the being is uncaused in the actual world. After all, it's God in the actual world. We thus arrive at a contradiction. Hence the being is not caused in the possible world in which it is explained.

A class of causal principles

Take the principle that an infinite regress of causes is impossible. It's an instance of a class of principles Pt := 'An infinite regress of t-causes is impossible' where t stands for simpliciter, efficient, sustaining, composite and so on. For each t we can ask whether Pt holds. If Pt holds for some t, and if t is not equal to 'simpliciter', then all t-causal series are finite and have a first member. But this first member might not be uncaused. For it might have an s-cause where s is not equal to t.

The chair-cabinet

Consider a room that contains a chair, an empty table and an empty cabinet. Suppose that this list of things offers a complete description of what there is to be seen in the room. Imagine that you enter it. You look around and you see nothing more but those three things. Would it be possible for you to create a new material object in this room merely by looking around and using language? Such a verbal creatio ex nihilo obviously seems metaphysically impossible. Nevertheless, look intensively at the chair and the cabinet at the same time. Focus your attention on them as being together. While doing so introduce the word 'chair-cabinet' and fix it's reference by declaring that it refers to the chair and cabinet together. After this act of naming the context has changed. For it seems not unreasonable anymore to hold that the room now also contains a chair-cabinet that has the chair and the cabinet as its material parts. Here we have an example of an act that perhaps resembles to at least some extent an act of verbally creating something out of nothing.

It's metaphysically impossible to be alone (II)

In my previous post I explained that the conclusion of my semantic argument entails that it's metaphysically impossible to be the only object in the world. One might argue that this swift consequence is counterintuitive. Isn't it highly implausible that there are no possible worlds that contain only one object? No, it isn't. Take some possible world w. Since total nothingness is metaphysically impossible, there exists an object in w. This object is plausibly either a piece of matter, a mind or a piece of information. If it's material, it exists in space and time. Given that space or time or parts thereof have objecthood, w contains more than one object. If it's a mind, w also contains more than one object. For the mental contents of a mind count reasonably as mental objects. Finally, if the object is a piece of information, then it is discursive and therefore composed of multiple information elements. But then again w contains more than one object. So in all cases w contains more than one object. Since w was arbitrarily chosen, there are no possible worlds that contain only one object. Said consequence of my semantic argument thus isn't implausible.

An improved version of the semantic argument

The third premise of my semantic argument can be improved significantly. I realized this a few days ago while preparing my talk on the argument for the upcoming OZSW conference on 11 and 12 December at VU University in Amsterdam. Premise 3 of the argument can be stated as follows. If M1 and M2 are the meanings of two positive predicable generic expressions E1 and E2 such that RefSet(M1)=RefSet(M2), then M1=M2. In my paper I provide an elaborate defense of this premise. But in fact I do not need this premise. All I need for my argument to go through is the following weaker modified premise. If M1 and M2 are the meanings of two positive predicable generic expressions E1 and E2 such that RefSet(M1)=RefSet(M2), then 'x is E1 iff x is E2' is necessarily true. This modified premise is indeed weaker, since the original premise entails it but not vice versa. After all, E1 and E2 having the same meaning obviously entails that 'x is E1 iff x is E2' is necessarily true, while it is not the case that the necessary truth of 'x is E1 iff x is E2' suffices to conclude that E1 and E2 have the same meaning. Now, I suggest to replace the original premise 3 of my argument with the modified weaker premise. By doing so the conclusion of my argument, namely that there are no positive contingent universally held properties, still follows. Therefore, by doing so we have obtained a stronger version of the semantic argument, making its conclusion even more plausible.

A calculus for possible worlds

Many metaphysicians deploy the apparatus of possible worlds. Possible worlds are in principle complete descriptions of how the world is or could have been. So, there are many, perhaps infinitely many, possible worlds. One of them is the actual world. It reflects reality as it actually is. How do metaphysicians decide whether something may count as a possible world? In most, if not all cases, the only criterion used for deciding whether a proposed candidate may be considered a possible world is the criterion of conceivability. If we can conceive the sketched situation, then we may infer that it is indeed possible and therefore part of at least one possible world. Conceivability though is a rather vague notion. It seems to convey the idea that there must be a clear and comprehensible narrative that outlines how the situation in question could obtain. So what is needed is an reasonable recognizable account of how the situation could be actual. It is thus not enough to just stipulate a situation as being possible. Now, what I would like to propose is to develop a calculus for the generation of new possible worlds out of given ones. The conceivability criterion may become one of the rules of this calculus, but this is not necessary. Perhaps we are able to identify a set of rules for said calculus that together make the conceivability criterion superfluous. What sort of rules do I have in mind? Well, we need rules for the generation of new possible worlds out of existing ones. For example rules like the following one. If W1 is a possible world, and W2 can be construed by a finite, coherent and unproblematic pathway from W1 to W2, then W2 is a possible world as well. So, the actual world without the chair I'm sitting on is a possible world. And the same holds for the actual world without the planets Mars and Venus. Or a possible world in which there are twenty planets orbiting around the sun. And so on. We also need rules to ground an initial set of possible worlds. An example of such a grounding rule would be the rule that a world in which only God exists is possible. From these possible worlds we may construct many other possible worlds by using generation rules such as the aforementioned one. In this way we get for example a possible world in which God exists and brings a universe into being. Or a possible world in which God exists and creates a multiverse. And so on. It seems to me that we need a quite large number of generation and grounding rules in order to arrive at an adequate possible worlds calculus. It would be interesting to see how such a calculus looks like.

Metaphysics and language

Let us look at a general objection that is often leveled against the method of drawing metaphysical conclusions from claims about the structure of language. The objection is that ontological consequences simply cannot be deduced from claims about linguistics, however plausible. Any attempt to slide from the lingual to the ontological plane is by its very nature not permissible.

Is this objection convincing? Now, it seems reasonable to hold that the categorical structure of language reflects, at least to some extent, the categorical structure of the world. If this is so, linguistic categories provide us with insight into the world’s structure. That is to say, a conceptual analysis of the linguistic categories of language gives us clues on the type of entities that the world contains. For example, proper names and definite descriptions reflect the ontological category of objects, general terms reflect the category of properties and relational expressions that of relations.

Thus the fact that natural language contains proper names, definite descriptions, general terms, and relational expressions, is one of the reasons we have for believing that there are objects, properties, and relations in the world. That is to say, the linguistic structure of natural language reveals certain ontological categories.

Moreover, natural language seems not only to reflect reality’s ontological categories, it also seems to reflect a number of reality’s ontological patterns. Language for example points to the ontological pattern of objects having properties. After all, in language general terms are predicated of proper names and definite descriptions. Now, if it is reasonable to hold that language mirrors both reality’s categories and its patterns, then performing conceptual analysis of linguistic structures can provide us with (defeasible) insight into the ontological categories and patterns of the world. That is, from the structure of language ontological consequences can be derived.

Further, given that semantics is a part of the conceptual analysis of language as well, it follows that semantic theses can also have ontological consequences.[2] All in all, I conclude that the generic objection is unconvincing.

Two caveats: First, by maintaining that the structure of language reflect the structure of reality, one is not committed to the radical position that we must determine the structure of reality solely by analyzing the structure of language.[1] I do not claim that we know that there are objects (properties, relations) only because we know that there are proper names (general terms, relational expressions). I do not hold that metaphysical inquiry reduces to linguistic analysis.

Second, someone who holds that the structure of language reflect the structure of reality is also not committed to the even more radical position that there are objects (properties, relations) by virtue of there being proper names (general terms, relational expressions). I do not claim that the structure of reality is ontologically dependent on the structure of natural language.

[1] Miller (2002, pp. 67-68) accepts this radical position and holds that Frege accepted it as well. This view is shared by other contemporary metaphysicians, such as Hofweber (2009), who argues that metaphysical inquiry should consist of analyzing linguistic expressions, and Thomasson (2009), who holds that metaphysical questions are to be answered by analyzing the application conditions of the terms of our language.

[2] An example would be the famous argument for fatalism (the claim that whatever will happen in the future is already unavoidable) from the principle of bivalence – a fundamental principle of semantics according to which every proposition (including those about the future) is either true or false. I do not endorse this argument, but merely mention it as an example of how philosophers use semantic principles to argue for ontological claims.

- Miller, B. (2002). The Fullness of Being. Notre Dame: University of Notre Dame Press.
- Hofweber, T. (2009). Ambitious, Yet Modest, Metaphysics. In: Chalmers, D. J., Manley, D., & Wasserman, R. (eds.), Metametaphysics: New Essays on the Foundations of Ontology (pp. 260-289). Oxford: Oxford University Press.
- Thomasson, A. L. (2009). Answerable and Unanswerable Questions. In: Chalmers, D. J. et al., Metametaphysics: New Essays on the Foundations of Ontology (pp. 444-471). Oxford: Oxford University Press.

On fundamental physics

Fundamental physics is just experimental metaphysics. That is to say, up until now. For fundamental physics is slowly becoming non-empirical.

Is logic part of the world's ultimate structure?

"Everyone faces the question of what is 'real' and what is the mere projection of our conceptual apparatus, of which issues are substantive and which are 'mere bookkeeping'. [...] These are questions of structure: how much structure is there in the world? Unless one is prepared to take the verificationist's easy way out, and say that 'theories are the same when empirically equivalent', one must face difficult questions about where to draw the line between objective structure and conceptual projection. The ontological realist draws the line in a certain place: part of the world's distinguished structure is its [logical] quantificational structure. Those who regard ontological realism as 'overly metaphysical' should remember that they too must draw a line.

And in fact, the ontological realist can give a pretty convincing argument for his choice of where to draw the line. Quine's (1948) criterion for ontological commitment is good as far as it goes: believe in those entities that your best theory says exists. But in trying to decide how much structure there is in the world, I can think of no better strategy than this extension of Quine's criterion: believe in as much structure as your best theory of the world posits. The structure posited by a theory corresponds to its primitive notions - its 'ideology' in Quine's (1951) terminology - which includes its logical notions as well as its predicates.

[...] [N]otice this: every serious theory of the world that anyone has ever considered, employs a [logical] quantificational apparatus, from physics to mathematics to the social sciences to folk theories. Quantificationalism is as indispensable as it gets. This is defeasible reason to think that we're onto something, that quantificational structure is part of the objective structure of the world, just as the success of spacetime physics gives us reason to believe in objective spacetime structure.

[...] If you remain unconvinced and skeptical of ontology, what are your options? First, you could reject the notion of objective structure altogether. I regard that as unthinkable. Second, you could reject the idea of structure as applied to logic. I regard that as unmotivated.

[...] There are [...] alternatives to ontological realism. [...] [I]f you [are] tempted by one of the alternatives, think about one final thing. Is your rejection of ontological realism based on the desire to make unanswerable questions go away, to avoid questions that resist direct empirical methods but are nevertheless not answerable by conceptual analysis? If so, none of [...] [the alternatives] will give you what you desire. None of them lets you bypass debate over the ultimate structure of the world. Far from it: each is simply an alternative proposal about what that structure is like. Given each proposal there remain substantive metaphysical questions, namely those that can be raised in what the proposal grants to be fundamental terms. Furthermore, the very assertion that the proposed variety of structure, as opposed to the quantificational structure [...], is part of reality's objective structure seems itself to be incapable of being established by either straightforward empirical means or conceptual analysis. In fact, even a 'negative' thesis such as quantifier variance itself is a claim about the extent of the world's structure, and as such is as epistemologically problematic as any thesis in first-order metaphysics. Quantifier variance is 'just more metaphysics'.

[...] The point of metaphysics is to discern the fundamental structure of the world. That requires choosing fundamental notions with which to describe the world. No one can avoid this choice. Other things being equal, it's good to choose a set of fundamental notions that make previously intractable questions evaporate. [...] But no other than a positivist can make all the hard questions evaporate. If nothing else, the choice of what notions are fundamental remains. There's no detour around the entirety of fundamental metaphysics."

Ted Sider, “Ontological Realism,” in Metametaphysics: New Essays on the Foundations of Ontology, eds. David J. Chalmers, David Manley and Ryan Wasserman (Oxford University Press, 2009): pp. 416-420.

A metaphysical principle entailing theism? (II)

Recently I proposed a new argument for theism based upon a metaphysical principle connecting logic, knowledge and truth. Against this argument two specific objections can be proposed. In what follows I shall present and respond to both objections.

(A) It is also logically impossible to know that God exists (for someone could, even encountering God, believe that she is dreaming, or hallucinating, or being deceived). But then, by parallel reasoning, it also follows that, necessarily, God does not exist. And hence my argument fails. My response would be that it is not logically impossible to know that God exists. Take a possible world in which God exists. In this possible world there is a subject that knows that God exists, namely God. In that world God knows that God exists. So, it is not logically impossible to know that God exists.

(B) There might be some true mathematical Gödel sentence G that cannot be proven by any proper mathematical system. Hence, G is unknowable. But then not all truths are knowable, and therefore my principle (which entails that all truths are knowable) fails. My response would be that G is in fact knowable. For, there is a possible world in which G is known. Take again a possible world in which God exists. In that world God can be taken to know at least all mathematical truths by immediate intuition, and therefore God knows G as well.

A metaphysical principle entailing theism?

Take the following metaphysical principle, connecting logic, knowledge and truth: 'If it is logically impossible to know that p, then p is necessarily false'. This principle seems to be cogent. For, if a given proposition p could be true, then, plausibly, there is a possible world in which some subject knows that p is true. In other words, if in all possible worlds all subjects do not know that some proposition is true, then, plausibly, that is because that very proposition cannot in fact be true.

Well, on a cartesian view of knowledge, that is, to know p is to be certain that p is true, the above principle has an interesting consequence. For, take for p the proposition 'God does not exist'. It seems reasonable to hold that it is impossible to know that God does not exist. For, whatever the arguments against God, there will always be some (perhaps an extremely remote) possibility that God does exist after all, so that we can never truly say, on the cartesian view, that we know that God does not exist. But then it follows that it is necessarily false that God does not exist. Hence, it is necessarily true that God exists.

One might object that it is also impossible to know that God exists. And thus, by similar reasoning, it would follow as well that it is necessarily true that God does not exist. However, I would argue that there is a possible world in which some subject can truly say that he or she knows that God exists. Take a possible word in which God exists and in which there is an afterlife, such that all who enter the afterlife in that world will encounter the divine. In that case, those subjects who enter the afterlife will in fact know that God exists. So, it is not impossible to know that God exists.

Note that a similar move to reject the argument for theism is not open to the atheist. For, if God does not exist, then, plausibly, there is no afterlife. And besides, even if there would be an afterlife, then entering it would not bring a subject in the epistemic condition of knowing that God does not exist.

A renewed first cause argument (revised)

In the last few weeks I received helpful comments on my paper titled 'A Renewed Argument for the Existence of a First Cause'. This resulted in a revised version of my paper that is now again online available on my website.

In this paper I propose a renewed argument for the existence of a first cause, i.e. something uncaused from which all causal series eventually proceed. More specifically, I show that under some very general and sufficiently plausible conditions (regarding the nature of causation and mereological parthood) the existence of a first cause logically follows.

The paper starts with a discussion of the problems associated with two traditional paradigmatic first cause arguments, derived from Aquinas and Leibniz. It is shown that the proposed renewed argument does not face any of these problems.

The renewed argument does also not depend on the principle of sufficient reason, that is, it does not assume that there is an explanation of every truth. In fact, it does not depend on any of the weaker or restricted versions of this principle either, such as those proposed by Craig, Koons and Pruss.

Moreover, the argument does not rely on the much debated concepts of metaphysical or broadly logical possibility and necessity. It does not refer to these modal concepts at all.

A renewed first cause argument (preprint)

A preprint version of my academic paper titled 'A Renewed Argument for the Existence of a First Cause' is now online available on my website.

The article is intended for publication in a peer-reviewed scientific journal. Comments on the preprint are highly appreciated.

Concrete and abstract objects

In metaphysics a distinction is made between concrete objects and abstract objects. A paradigmatic example of a concrete object would be a table or a chair. Suppose that platonism with respect to mathematical objects is true. In that case numbers are the paradigmatic examples of abstract objects. Now, what makes a concrete object concrete? And what do we mean when we say that some object is an abstract object? I take it that, assuming God exists, any plausible definition of concrete and abstract objects must be compatible with these propositions:

- Concrete objects can be material (a table)
- Concrete objects can be immaterial (God or a human-mind)
- Concrete objects can be contingent (a table or a human-mind)
- Concrete objects can be necessary (God)
- Concrete objects can be located in time (a table)
- Concrete objects can be located at a single location (a table)
- Concrete objects can be located at multiple locations (tableware)
- Concrete objects can be outside time and space (God)
- Concrete objects can be caused (a table) or uncaused (God)
- Concrete objects can have causal powers (a rock, a mind)

- Abstract objects are always immaterial
- Abstract objects can be contingent (the property of being red)
- Abstract objects can be necessary (the number one, logical laws)
- Abstract objects can be located in time (the earth's equator)
- Abstract objects can be located at a single location (idem)
- Abstract objects can be located at multiple locations (red)
- Abstract objects can be outside time and space (the number two)
- Abstract objects can be caused (equator) or uncaused (two)
- Abstract objects can have causal powers (logical inference)

From this list it follows that we cannot simply define concrete objects as being objects that are located in time or space. The reason is that abstract objects can be located in time or space as well. Neither can we define concrete objects as objects having causal powers because some abstract objects also have causal powers. Further, we cannot define concrete objects as being material objects since there are also immaterial concrete objects. Because of this fact we cannot define abstract objects as being immaterial. It is also not possible to identify concrete objects with caused objects since abstract objects can be caused as well. Nor is it possible to identify concrete objects with contingent objects or abstract objects with necessary objects. After all, there are both necessary concrete and contingent abstract objects.

So, what features are left to define the notions of concrete and abstract objects? Should we say that concrete objects are accessible to empirical observation, whereas abstract objects are inaccessible to empirical observation? This is plainly false since a human-mind and God are not uncritically accessible to empirical observation. Similarly, the ultimate indivisible atomic building blocks of nature are arguably not empirically observable either. Moreover, the property of being red is in an important sense accessible to empirical observation. Should we hold that change is possible for concrete objects, whereas change is impossible for abstract objects? This is also obviously incorrect since for example the earth's equator can change.

It thus seems to me that there are in fact no ontological features left to provide an adequate definition of concrete and abstract objects. Therefore I propose to exclude the notions of concrete object and abstract object from metaphysics altogether.

The only genuine metaphysical distinction is the distinction between material objects and immaterial objects. If a certain type of platonism is true, we should divide the class of immaterial objects into the class of mental objects and the class of platonic objects. If platonism is false (which I believe is the case), all immaterial objects are mental objects. In that case objects such as the earth's equator, properties such as being red, logical laws or the number two exist only in our minds.