One common fallacy is the fallacy of composition, where one argues from the fact that each part of a thing has a certain feature to the conclusion that the whole thing has that feature. For instance, one could argue that every brick of the house is cube-shaped, therefore the house is cube-shaped. Or one could argue that each part of one's brain is unconscious, therefore the whole brain is unconscious. These inferences are fallacious.
However, I think it is worth noting that not all inferences from properties of parts to properties of the whole are invalid. If each part of a wall is made entirely of stone, then the whole wall is made entirely of stone. Similarly, if each part of the ball is entirely red, then the whole ball is entirely red. And so on.
Contingency seems to be like this, at least in this case. So here's an argument that the universe must be contingent:
Sunday, August 31, 2014
Friday, August 29, 2014
Quantifier Variance and the Semantics of Quantifiers
In my previous post I explained the basic idea behind quantifier variance. Now I want to criticize it. In particular, I said I want to point out some problems with the quantifier variantist's simultaneously affirming the following two statements:
(i) the different quantifiers behave the same logically; and
(ii) the different quantifiers have different meanings.
Let's do a little basic semantics. Let's define the truth function τ[ψ]U,g relative to models U and g for the cases of quantified formulas ψ as follows. The following definitions are true for all models M, all variable assignments s, all variables x, and all formulas φ. If a formula is not assigned to T it is assigned to F:
τ : {<ψ,U,g>|ψ is a formula, U a model, g a var. assign.} → {T,F}
(i) the different quantifiers behave the same logically; and
(ii) the different quantifiers have different meanings.
Let's do a little basic semantics. Let's define the truth function τ[ψ]U,g relative to models U and g for the cases of quantified formulas ψ as follows. The following definitions are true for all models M, all variable assignments s, all variables x, and all formulas φ. If a formula is not assigned to T it is assigned to F:
τ : {<ψ,U,g>|ψ is a formula, U a model, g a var. assign.} → {T,F}
- (τ-∀): τ[∀xφ]M,s = T ⇔ for all variable assignments s′, if for all variables v, s(v) ≠ s′(v) ⇒ v = x, then τ[φ]M,s′ = T
- (τ-∃): τ[∃xφ]M,s = T ⇔ for some variable assignment s′, for all variables v, s(v) ≠ s′(v) ⇒ v = x, and τ[φ]M,s′ = T
Monday, August 25, 2014
Basics of Quantifier Variance
When I say that there are tables is it unambiguous what I'm saying? Quantifier variantists say no. Or at least they would say that in certain contexts it is not. In particular, the sentence is ambiguous when we are engaging in metaphysical debate about the existence of the table, as in the following case.
Consider the debate between what I will call compositionalism and anti-compositionalism. Compositionalism is the thesis that there are composite material objects, while anti-compositionalism is the thesis that there are not. Take the case of a world with just a table and its parts, and suppose we are considering a form of compositionalism which says there are tables. Assume further that there are exactly n atoms which, according to this form of compositionalism, are proper parts of the table. Note that we are using a philosophical definition of 'atom', according to which an atom is a material object which has no proper parts. Anti-compositionalism says there is no table; there are just the n atoms.
In essence, compositionalism says (A) there are n+1 distinct things (viz. the n atoms, plus the table), while anti-compositionalism says (B) there are n things and there are no more than n things. Note that (A) and (B) can be adequately translated into a quantified language which only contains variables, quantifiers, sentential connectives, and the identity sign with the usual interpretation. For example, (A) would be translated as follows:
∃x1∃x2...∃xn((x1≠x2 ∧ ... ∧ x1≠xn+1) ∧ (x2≠x3 ∧ ... ∧ x2≠xn+1) ∧ ... ∧ (xn≠xn+1))
Consider the debate between what I will call compositionalism and anti-compositionalism. Compositionalism is the thesis that there are composite material objects, while anti-compositionalism is the thesis that there are not. Take the case of a world with just a table and its parts, and suppose we are considering a form of compositionalism which says there are tables. Assume further that there are exactly n atoms which, according to this form of compositionalism, are proper parts of the table. Note that we are using a philosophical definition of 'atom', according to which an atom is a material object which has no proper parts. Anti-compositionalism says there is no table; there are just the n atoms.
In essence, compositionalism says (A) there are n+1 distinct things (viz. the n atoms, plus the table), while anti-compositionalism says (B) there are n things and there are no more than n things. Note that (A) and (B) can be adequately translated into a quantified language which only contains variables, quantifiers, sentential connectives, and the identity sign with the usual interpretation. For example, (A) would be translated as follows:
∃x1∃x2...∃xn((x1≠x2 ∧ ... ∧ x1≠xn+1) ∧ (x2≠x3 ∧ ... ∧ x2≠xn+1) ∧ ... ∧ (xn≠xn+1))
Thursday, August 21, 2014
Pure Actuality
Many scholastic theologians, most notably Aquinas, make the claim that God is "pure actuality." This is supposed to do a lot of philosophical and theological "work"; it is by showing that there exists a being which is pure actuality that Aquinas is able to deduce many of the divine attributes. However, it is not immediately clear what this even means if one is not familiar with the metaphysical context of medieval philosophy.
A charitable interpreter who has read some medieval philosophy may be able to see how scholastics use this claim and identify certain inferences from this claim as being valid and others not. But it'd be nice if we had a more precise characterization of what it means to say God is 'pure actuality', so that we can see if all that Aquinas says follows actually does follow from this claim. Moreover, once we have a precise characterization of what Aquinas is even asserting, we can begin to more clearly assess the plausibility of the claim itself and whether Aquinas has established it. I propose the following definition:
For completeness and wider scope of application, I also propose the following definitions of a thing's being 'composed of' or 'having' actuality and potentiality:
A charitable interpreter who has read some medieval philosophy may be able to see how scholastics use this claim and identify certain inferences from this claim as being valid and others not. But it'd be nice if we had a more precise characterization of what it means to say God is 'pure actuality', so that we can see if all that Aquinas says follows actually does follow from this claim. Moreover, once we have a precise characterization of what Aquinas is even asserting, we can begin to more clearly assess the plausibility of the claim itself and whether Aquinas has established it. I propose the following definition:
- x is pure actuality if and only if for all (intrinsic) P, if x is P then x is actually P.
For completeness and wider scope of application, I also propose the following definitions of a thing's being 'composed of' or 'having' actuality and potentiality:
- x is composed of potentiality if and only if for some (intrinsic) P, x is P and x is potentially P
- x is composed of actuality if and only if for some (intrinsic) P, x is P and x is actually P.
Sunday, August 25, 2013
Thoughts on the Grounding Objection to Molinism
So, I want to get a bit more clear on what the grounding objection to Molinism is saying. As far as I can tell at this moment, the grounding objection seems to go something like this.
The anti-Molinist says that some general statement about the relation between grounding and truth such as the following holds:
(A) If some proposition is true then there is an entity which grounds its truth.
It seems in this context 'grounds the truth of p' just means 'is the truthmaker of p'. The objector to Molinism then proposes:
(B) There could be no entity to ground the truth of CCF's.
Of course from A and B it follows that all CCF's, if they are meaningful, are necessarily false. Hopefully this is all a correct representation of the objection.
The anti-Molinist says that some general statement about the relation between grounding and truth such as the following holds:
(A) If some proposition is true then there is an entity which grounds its truth.
It seems in this context 'grounds the truth of p' just means 'is the truthmaker of p'. The objector to Molinism then proposes:
(B) There could be no entity to ground the truth of CCF's.
Of course from A and B it follows that all CCF's, if they are meaningful, are necessarily false. Hopefully this is all a correct representation of the objection.
Wednesday, July 3, 2013
Essence and Ontological Dependence
This is my term paper from my independent study last quarter on ontological dependence. I will say beforehand that I did not have enough time to make it great, and there is a lot more I could have said. However, I believe it contains a relatively good summary of Kit Fine's position, and I think the stuff toward the end about causation is somewhat original (albeit sketchy). So hopefully someone will find it interesting and useful.
I. Introduction – Examples and What We Want
In many areas of philosophy, as well as common discourse, it is normal to say that one thing depends on another. Moreoever, one of these uses of the word 'depends' is a distinctly ontological sense, as opposed to, say, a notion of epistemological dependence or logical dependence. I will use the term 'dependence' throughout this essay to stand for this particularly ontological notion, unless otherwise stated. So for instance, we might say that a composite depends on its constituents. Or we might say a smile depends on the mouth of which it is a smile. Or that a hole depends on the thing which it is a hole in. This is a philosophical datum and the only reason one would deny it or feign incomprehension seems to be hard-headedness.
I. Introduction – Examples and What We Want
In many areas of philosophy, as well as common discourse, it is normal to say that one thing depends on another. Moreoever, one of these uses of the word 'depends' is a distinctly ontological sense, as opposed to, say, a notion of epistemological dependence or logical dependence. I will use the term 'dependence' throughout this essay to stand for this particularly ontological notion, unless otherwise stated. So for instance, we might say that a composite depends on its constituents. Or we might say a smile depends on the mouth of which it is a smile. Or that a hole depends on the thing which it is a hole in. This is a philosophical datum and the only reason one would deny it or feign incomprehension seems to be hard-headedness.
Wednesday, June 19, 2013
Two Inadequate Arguments for a Finite Past
In this post I will consider two arguments which have at times been brought up in connection with the Kalam Cosmological Argument (KCA), which I will call the "subtraction argument" and the "argument from traversing an infinite," the former of which I have heard from Dr. William Lane Craig. The KCA goes as follows:
1. Whatever begins to exist has a cause.
2. The universe began to exist.
3. So the universe has a cause.
The arguments in question are designed to defend the second premise, which is presumably implied by the past's being finite. I should note that I think the second premise is true and there is strong evidence in favor of its truth. Alexander Pruss has given an excellent argument here, to which I have heard no compelling reply. I also think there is very strong scientific indication of the premise's truth, which Craig has adequately demonstrated. I just don't think these two arguments demonstrate its truth.
The "argument from traversing an infinite" goes something like this:
1*. If the past were infinite, one would have to cross an infinite temporal distance to get to the present moment.
2*. If one had to cross an infinite temporal distance to get to the present moment, then one could not get to the present moment.
3*. So if the past were infinite, then one could not get to the present moment.
4*. But we are at the present moment.
5*. So the past is not infinite.
The argument requires some unpacking. First of all, to say the past is finite is to say there was a beginning of time, and to say the past is infinite is to say there was no beginning of time. Second, 'temporal distance' means the length of time between one moment and another. There is a perfectly good way to define finite temporal distance. If we take our measure of time as a second, we can assign the current time the number 0, the time one second ago -1, the time two seconds ago -2, and so forth. To find the temporal distance from one time t1 to another t2, we take the number assigned to t1 and the number assigned to t2, and take the absolute value of the difference between the two. For instance, take the time 1000 seconds ago. To find the temporal distance from that time to the present time you take the absolute value of -1000 minus 0, which is of course 1000 seconds. Pretty simple.
However, problems begin to arise when we start to talk about an "infinite temporal distance." This phrase is ambiguous, and depending on which interpretation of this phrase we take it will either cause problems for premise 1* or for premise 2*. First, the phrase could mean something analogous to the way finite temporal distance has been defined above. However, infinity is not a real number, so you simply cannot define an infinite temporal distance the same way as above. There is no number "-infinity" from which you can subtract, say, -5. So if this is what is meant, then premise 1* appears to be false, since no real sense can be given to an infinite distance in this way.
On the other hand, crossing an infinite temporal distance could just mean that the set of all the numbers assigned to the seconds is infinitely large. This makes perfectly good sense of the phrase, but then in that case it is not clear why premise 2* is true. As Thomas Aquinas points out, there being an "infinite temporal distance" in this sense is perfectly consistent with all the temporal distances from the past to the present being finite, where "temporal distance" is defined as it was earlier:
"Passage is always understood as being from term to term. Whatever bygone day we choose, from it to the present day there is a finite number of days which can be passed through. The objection, however, is founded on the idea that, given two extremes, there is an infinite number of mean terms." [ST Ia q.46 a.2]
So for instance, the distance from the present to one second ago is 1 second, the distance from the present to two seconds ago is 2 seconds, etc. and so on forever and ever back into time. Hence, no matter how far you go back in time, the distance in the way I've defined above from any given past moment to the present will be finite, and thus you will only have to cross a finite number of seconds to get to the present moment. But of course any finite number of seconds can at least in principle be crossed; hence, premise 2* is false.
So much for the "traversing an infinite" argument then. The "subtraction argument" goes something like this:
1'. If the past were infinite, then an actual infinity would be possible.
2'. If an actual infinity were possible, then one could perform subtraction on infinities.
3'. But if one can perform subtraction on infinities, then one will get contradictory results.
4'. So if the past were infinite, then one would get contradictory results.
5'. So the past is not infinite.
This seems to be one of the arguments William Lane Craig gave in his debate against Peter Millican. Let me first point out an ambiguity in the phrase "actual infinity," after which I'll assess the argument under each interpretation. Here are two possible meanings of the phrase "actual infinity":
(ACT1) An actual infinity exists just in case for some time, at that time there exist distinct concrete objects such that the size of the set containing all and only them is infinite.
(ACT2) An actual infinity exists just in case there is some set containing only distinct concrete objects whose size is infinite.
Some elaboration is in order. First off, both definitions presume when speaking of actual infinities that we are dealing with concrete objects. While my understanding is that Craig does not believe there are infinitely many numbers (he is a nominalist), presumably his argument doesn't presuppose this view; Craig only wants to rule out the possibility of infinitely many concrete objects. Now as for the definitions themselves, the difference between (ACT1) and (ACT2) is that in (ACT1) you only have an actual infinity when all the concrete objects exist at the same time. In (ACT2) you could have finitely many objects at t2, finitely many at t1, and so forth, yet if you take a set containing concrete objects from different times, and the times go back to infinity, you will still have an actual infinity. So both of these definitions make fine sense. However, the assessment of the argument will depend on which interpretation we take.
Let's deal with the first definition, (ACT1). Given our definition of actual infinity in (ACT1), premise 1' does not appear to be true, or at least not obviously true. It is consistent with holding that the past is infinite that at each time there are only finitely many concrete objects. And if you believe only objects in the present moment exist, then 1' is definitely not true. The fact that there were objects at each time in the eternal past in no way implies an infinite collection of simultaneously-existing objects.
My main concern is with 2' and 3' though. Take 2', since it is also ambiguous to a certain extent. The problem is it is not immediately clear what is meant by "perform subtraction on infinities." Craig acknowledges that the ordinary operation of subtraction is not defined for "infinity". If he did mean this, i.e. the ordinary operation of subtraction, then clearly 3' would be true but 2' would have no support at all. So it cannot mean the ordinary operation of subtraction.
Now, Craig gives us an example to support 2' and 3', and this may help us understand what he means. Suppose we have an infinite number of coins. Then we can take away all the coins except three of them. And in this sense we can be said to perform subtraction on infinity, i.e. taking away some number of things from an infinitely large collection. This definition of performing subtraction on infinities makes sense. But then why is 3' true? Craig says, considering our infinite number of coins, that you can take away infinitely many coins and be left with 3 coins, and thus infinity minus infinity = 3; but you can also take away infinitely many coins and be left with 2 coins, and thus infinity minus infinity = 2; hence, 2 = 3, which is our contradiction.
The problem with this argument is that it runs on an equivocation: We agreed that we are not using "performing subtraction" or "minus" in the normal sense of the arithmetical operation, since this just makes no sense. So "infinity minus infinity = 3" must simply be shorthand for saying "taking away infinitely many objects from an infinite collection leaves us with 3", and similarly with "infinity minus infinity = 2." But then if "2 = 3" means that 2 is identical to 3, then it certainly does not follow that 2 = 3; all that follows is that you can take away an infinite number of things and be left with 3, and also take away an infinite number of things and be left with 2, and this is certainly not a contradiction! It only looks like a contradiction when we are illicitly inferring "2 = 3", as if the phrase "infinity minus infinity = 2" were using "minus" and "=" in the same way as "5 minus 3 = 2." It would be like if I had infinitely many pennies and dimes, and I said, "infinity minus infinity = a penny, infinity minus infinity = a dime, so a penny = a dime." Clearly I am making an illicit inference here, and for the same reason Craig's argument makes an illicit inference as well.
Now, this whole time I have been working under the assumption that throughout the argument "actual infinity" is meant in the sense of (ACT1). But under interpretation (ACT2) the situation is even worse, since it is not clear 2' is true. It seems that in order to "subtract" infinitely many coins in the sense defined above, all of them must exist at the same time. But if "actual infinity" is taken in the sense of (ACT2), then it is not required that all of the infinite number of coins exist at the same time, and thus 2' has no support. And of course, with the exception of premise 1', all the same criticisms I have just given apply equally well under (ACT2). So, interpreted charitably, the argument seems to be a failure, with the primary problem being in premise 3'.
I should note one more time that, in spite of all my criticisms of these two arguments, I think there are good reasons for thinking the KCA is sound. I just don't think these are among them.
1. Whatever begins to exist has a cause.
2. The universe began to exist.
3. So the universe has a cause.
The arguments in question are designed to defend the second premise, which is presumably implied by the past's being finite. I should note that I think the second premise is true and there is strong evidence in favor of its truth. Alexander Pruss has given an excellent argument here, to which I have heard no compelling reply. I also think there is very strong scientific indication of the premise's truth, which Craig has adequately demonstrated. I just don't think these two arguments demonstrate its truth.
The "argument from traversing an infinite" goes something like this:
1*. If the past were infinite, one would have to cross an infinite temporal distance to get to the present moment.
2*. If one had to cross an infinite temporal distance to get to the present moment, then one could not get to the present moment.
3*. So if the past were infinite, then one could not get to the present moment.
4*. But we are at the present moment.
5*. So the past is not infinite.
The argument requires some unpacking. First of all, to say the past is finite is to say there was a beginning of time, and to say the past is infinite is to say there was no beginning of time. Second, 'temporal distance' means the length of time between one moment and another. There is a perfectly good way to define finite temporal distance. If we take our measure of time as a second, we can assign the current time the number 0, the time one second ago -1, the time two seconds ago -2, and so forth. To find the temporal distance from one time t1 to another t2, we take the number assigned to t1 and the number assigned to t2, and take the absolute value of the difference between the two. For instance, take the time 1000 seconds ago. To find the temporal distance from that time to the present time you take the absolute value of -1000 minus 0, which is of course 1000 seconds. Pretty simple.
However, problems begin to arise when we start to talk about an "infinite temporal distance." This phrase is ambiguous, and depending on which interpretation of this phrase we take it will either cause problems for premise 1* or for premise 2*. First, the phrase could mean something analogous to the way finite temporal distance has been defined above. However, infinity is not a real number, so you simply cannot define an infinite temporal distance the same way as above. There is no number "-infinity" from which you can subtract, say, -5. So if this is what is meant, then premise 1* appears to be false, since no real sense can be given to an infinite distance in this way.
On the other hand, crossing an infinite temporal distance could just mean that the set of all the numbers assigned to the seconds is infinitely large. This makes perfectly good sense of the phrase, but then in that case it is not clear why premise 2* is true. As Thomas Aquinas points out, there being an "infinite temporal distance" in this sense is perfectly consistent with all the temporal distances from the past to the present being finite, where "temporal distance" is defined as it was earlier:
"Passage is always understood as being from term to term. Whatever bygone day we choose, from it to the present day there is a finite number of days which can be passed through. The objection, however, is founded on the idea that, given two extremes, there is an infinite number of mean terms." [ST Ia q.46 a.2]
So for instance, the distance from the present to one second ago is 1 second, the distance from the present to two seconds ago is 2 seconds, etc. and so on forever and ever back into time. Hence, no matter how far you go back in time, the distance in the way I've defined above from any given past moment to the present will be finite, and thus you will only have to cross a finite number of seconds to get to the present moment. But of course any finite number of seconds can at least in principle be crossed; hence, premise 2* is false.
So much for the "traversing an infinite" argument then. The "subtraction argument" goes something like this:
1'. If the past were infinite, then an actual infinity would be possible.
2'. If an actual infinity were possible, then one could perform subtraction on infinities.
3'. But if one can perform subtraction on infinities, then one will get contradictory results.
4'. So if the past were infinite, then one would get contradictory results.
5'. So the past is not infinite.
This seems to be one of the arguments William Lane Craig gave in his debate against Peter Millican. Let me first point out an ambiguity in the phrase "actual infinity," after which I'll assess the argument under each interpretation. Here are two possible meanings of the phrase "actual infinity":
(ACT1) An actual infinity exists just in case for some time, at that time there exist distinct concrete objects such that the size of the set containing all and only them is infinite.
(ACT2) An actual infinity exists just in case there is some set containing only distinct concrete objects whose size is infinite.
Some elaboration is in order. First off, both definitions presume when speaking of actual infinities that we are dealing with concrete objects. While my understanding is that Craig does not believe there are infinitely many numbers (he is a nominalist), presumably his argument doesn't presuppose this view; Craig only wants to rule out the possibility of infinitely many concrete objects. Now as for the definitions themselves, the difference between (ACT1) and (ACT2) is that in (ACT1) you only have an actual infinity when all the concrete objects exist at the same time. In (ACT2) you could have finitely many objects at t2, finitely many at t1, and so forth, yet if you take a set containing concrete objects from different times, and the times go back to infinity, you will still have an actual infinity. So both of these definitions make fine sense. However, the assessment of the argument will depend on which interpretation we take.
Let's deal with the first definition, (ACT1). Given our definition of actual infinity in (ACT1), premise 1' does not appear to be true, or at least not obviously true. It is consistent with holding that the past is infinite that at each time there are only finitely many concrete objects. And if you believe only objects in the present moment exist, then 1' is definitely not true. The fact that there were objects at each time in the eternal past in no way implies an infinite collection of simultaneously-existing objects.
My main concern is with 2' and 3' though. Take 2', since it is also ambiguous to a certain extent. The problem is it is not immediately clear what is meant by "perform subtraction on infinities." Craig acknowledges that the ordinary operation of subtraction is not defined for "infinity". If he did mean this, i.e. the ordinary operation of subtraction, then clearly 3' would be true but 2' would have no support at all. So it cannot mean the ordinary operation of subtraction.
Now, Craig gives us an example to support 2' and 3', and this may help us understand what he means. Suppose we have an infinite number of coins. Then we can take away all the coins except three of them. And in this sense we can be said to perform subtraction on infinity, i.e. taking away some number of things from an infinitely large collection. This definition of performing subtraction on infinities makes sense. But then why is 3' true? Craig says, considering our infinite number of coins, that you can take away infinitely many coins and be left with 3 coins, and thus infinity minus infinity = 3; but you can also take away infinitely many coins and be left with 2 coins, and thus infinity minus infinity = 2; hence, 2 = 3, which is our contradiction.
The problem with this argument is that it runs on an equivocation: We agreed that we are not using "performing subtraction" or "minus" in the normal sense of the arithmetical operation, since this just makes no sense. So "infinity minus infinity = 3" must simply be shorthand for saying "taking away infinitely many objects from an infinite collection leaves us with 3", and similarly with "infinity minus infinity = 2." But then if "2 = 3" means that 2 is identical to 3, then it certainly does not follow that 2 = 3; all that follows is that you can take away an infinite number of things and be left with 3, and also take away an infinite number of things and be left with 2, and this is certainly not a contradiction! It only looks like a contradiction when we are illicitly inferring "2 = 3", as if the phrase "infinity minus infinity = 2" were using "minus" and "=" in the same way as "5 minus 3 = 2." It would be like if I had infinitely many pennies and dimes, and I said, "infinity minus infinity = a penny, infinity minus infinity = a dime, so a penny = a dime." Clearly I am making an illicit inference here, and for the same reason Craig's argument makes an illicit inference as well.
Now, this whole time I have been working under the assumption that throughout the argument "actual infinity" is meant in the sense of (ACT1). But under interpretation (ACT2) the situation is even worse, since it is not clear 2' is true. It seems that in order to "subtract" infinitely many coins in the sense defined above, all of them must exist at the same time. But if "actual infinity" is taken in the sense of (ACT2), then it is not required that all of the infinite number of coins exist at the same time, and thus 2' has no support. And of course, with the exception of premise 1', all the same criticisms I have just given apply equally well under (ACT2). So, interpreted charitably, the argument seems to be a failure, with the primary problem being in premise 3'.
I should note one more time that, in spite of all my criticisms of these two arguments, I think there are good reasons for thinking the KCA is sound. I just don't think these are among them.
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