The Successive Addition Argument (SAA) is one of the key arguments espoused by William Lane Craig for the thesis that the universe began to exist. Recently, Alex Malpass (2021) has developed a challenge to the SAA by way of constructing a counterexample that originates in the work of Fred Dretske. In this paper, I show that the Malpass-Dretske counterexample is in fact no counterexample to the argument. Utilizing a distinction between properties of members and properties of collections, I argue that Malpass’ counterexample has no bearing on the soundness of the SAA. I also develop a novel parity argument against Malpass’ argument that I demonstrate can only be resolved by way of the aforementioned analysis

Dagher, Ibrahim

The Successive Addition Argument (SAA) is one of the key arguments espoused by William Lane Craig for the thesis that the universe began to exist. Recently, Malpass, Mind, 131(523), 786–804 (2021) has developed a challenge to the SAA by way of constructing a counterexample that originates in the work of Fred Dretske. In this paper, I show that the Malpass-Dretske counterexample is in fact no counterexample to the argument. Utilizing a distinction between properties of members and properties of collections, I argue that Malpass’ counterexample has no bearing on the soundness of the SAA. I also develop a novel parity argument against Malpass’ argument that I demonstrate can only be resolved by way of the aforementioned analysis

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 1 Properties, Collections, and the Successive Addition 1 Argument: A Reply to Malpass  2  3 Abstract 4 The Successive Addition Argument (SAA) is one of the key arguments espoused by William 5 Lane Craig for the thesis that the universe began to exist. Recently, Alex Malpass (2021) 6 has developed a challenge to the SAA by way of constructing a counterexample that 7 originates in the work of Fred Dretske. In this paper, I show that the Malpass-Dretske 8 counterexample is in fact no counterexample to the argument. Utilizing a distinction 9 between properties of members and properties of collections, I argue that Malpass’ 10 counterexample has no bearing on the soundness of the SAA. I also develop a novel parity 11 argument against Malpass’ argument that I demonstrate can only be resolved by way of the 12 aforementioned analysis. 13  14 Keywords: Kalām; Successive Addition Argument; Infinity; Philosophy of Time; Cosmological Arguments for 15 Theism 16  17 1. Introduction  18  19 The Kalām cosmological argument, as championed by William Lane Craig 20 (1979), is an argument largely concerned with demonstrating that the temporal 21 series of events cannot regress infinitely. One argument Craig offers in support of 22 this thesis, as titled by Alex Malpass (2021), is the ‘Successive Addition Argument’, 23 or the SAA. The SAA is stated by Craig (1979, p. 103) as follows:  24  25 (1) The temporal series of events is a collection formed by successive addition;  26 (2) A collection formed by successive addition cannot be an actual infinite; 27 (3) Therefore, the temporal series of events cannot be an actual infinite.  28  29 Malpass (2021) has recently produced a novel challenge to the SAA, based on the 30 work of Dretske (1965). Malpass argues that there exists a clear counterexample to 31 (2), and thus the argument is not sound. After investigating six potential objections 32 Craig might offer, and finding them all wanting, Malpass concludes that his 33 argument constitutes a genuine counterexample to (2), and thus that the SAA is 34 unsound. 35 In this paper, I respond to Malpass on behalf of the SAA, arguing his 36 counterexample does not render the argument unsound. My contention is that 37 Malpass’ argument is either invalid in virtue of a shift in scope, or else irrelevant to 38 the truth of (2) in virtue of making a claim about the properties of members 39 successively added, rather than the properties of collections formed by successive 40 addition.     41 I will proceed as follows. In §2, I explicate Malpass’ argument from his 42 counterexample. In §3, I argue that Malpass’ counterexample is either of an invalid 43 form or else irrelevant to the truth of (2). In §4, I further this claim by developing a 44 parity argument that can only be satisfactorily resolved by accepting the disjunction 45 in §3. I conclude in §5.  46  2  47 2. Malpass’ argument 48  49 Malpass begins building his counterexample to (2) by asking us to imagine a 50 man named George who starts to count numbers at some time t.1 George counts at a 51 constant rate of one number per second, and never stops. Thus, each finite number 52 n is such that George will count n. So, the cardinality of the numbers that George 53 will count is just the cardinality of the natural numbers, ℵ0. From this setup, 54 Malpass (2021, p. 3) constructs the following argument: 55   56 (4) It is possible that George starts counting now and will never stop;  57 (5) If George starts counting now and will never stop, then for each natural 58 number n, George will count n;  59 (6) If George will count each natural number, then George will count ℵ0-many 60 numbers; 61 (7) Therefore, it is possible that George will count ℵ0-many numbers.  62  63 However, (7) appears to be a clear counterexample to (2). Prima facie, if it is 64 possible that I will count x-many numbers, then it is also possible that I will form a 65 collection by successive addition with x-many members. Thus, the fact that it is 66 possible that George will count ℵ0-many numbers means that George can form an 67 actually infinite collection by successive addition. Yet, (2) is the claim that 68 collections formed by successive addition cannot be actually infinite. So, Malpass’ 69 argument seems to undermine (2).  70  71 3. The scope shift fallacy  72  73 Malpass’ argument has two critical premises. These are that: 74   75 (5) If George starts counting now and will never stop, then for each natural 76 number n, George will count n;  77 (6) If George will count each natural number, then George will count ℵ0-many 78 numbers.  79  80 Both of these will be important in our discussion, and thus it is important to be 81 clear on what they mean: especially the consequents. The consequent of (5) is fairly 82 simple to formalize. Let G(n) abbreviate ‘n is counted by George’. Then what 83 Malpass has in mind in the consequent of (5) is that each n is such that there is a 84 future time t at which n is counted by George. In terms of the tense operator ‘F’ for 85 ‘it will at some future time t be the case that’, this can be stated as:  86  87 (8) (∀n) F(G(n)) 88  89  1 The setup for this counterexample originates in Dretske (1965).  3 The antecedent of (6) is the same. But what about the consequent of (6)? Here 90 things are not so clear. It’s not immediately obvious what Malpass means by the 91 phrase ‘George will count ℵ0-many numbers’. On the one hand, if we take ‘it will be 92 that 𝜙’ to mean that ‘there is a future time t such that 𝜙’, the consequent of (6) can 93 be read as the claim that ‘there is a future time t at which George counts ℵ0-many 94 numbers’. More formally,  95  96 (9) F (∀n)(G(n)) 97  98 But this cannot be the correct analysis. For Malpass explicitly notes that there is 99 no time at which George counts ℵ0-many numbers, as his counting literally ‘takes up 100 all the time in the world’ (2021, p. 15). Furthermore, moving from (8) to (9) blatantly 101 commits a scope-shift fallacy, and Malpass notes that ‘if the Dretske argument 102 involved a shift like this it would be bad news’ (Malpass 2021, p. 7). Thus, (9) cannot 103 be what Malpass has in mind in the consequent of (6). 104 What, then, does (6) amount to? Another analysis of (6) shows itself if we 105 examine how the premise is defended. Malpass notes that (6) is an application of 106 the following broader rule of inference:  107  108 (R) If each element in a set S has property P, and the cardinality of the 109 elements of S is X, then the cardinality of the elements that are P is also [at 110 least] X. (2021, p. 8) 111  112 (R), in the context of (6), is applied as such: if each natural number in the set of 113 natural numbers instantiates the property ‘will be counted by George’ and the 114 cardinality of the set of natural numbers is ℵ0, then the cardinality of the natural 115 numbers that instantiate the property ‘will be counted by George’ is also at least ℵ0. 116 As Malpass says: ‘if each natural number will be counted, and there are ℵ0-many 117 natural numbers, then ℵ0-many natural numbers will be counted’ (2021, p. 15). 118 So, it appears that when Malpass affirms that the cardinality of the natural 119 numbers that will be counted by George is also ℵ0, Malpass is not falsely claiming 120 that there is some future time where George counts ℵ0-many natural numbers. 121 Rather, what is being said is that there exist ℵ0-many natural numbers each of 122 which instantiates the property ‘will be counted by George’. Otherwise, both (6) and 123 the broader rule (R) are clearly faulty. 124 Thus, let us suppose the proper reading of (6) is just that there are ℵ0-many 125 natural numbers instantiating this property. If this is what is meant by Malpass, 126 then his argument faces a serious problem. Namely, (6) is not telling us anything 127 importantly over and above what is said by (5), the premise that each number is 128 such that George will count it. Indeed, it seems all (6) tells us that is not already 129 contained in (5) is that there exist ℵ0-many such numbers.  130 But if all (6) does in the context of Malpass’ argument is alter the relevant 131 quantification––such that we are entitled to the further claim that there are ℵ0-132 many numbers instantiating the property ‘will be counted by George’––then we are 133 left wondering how the argument can be considered in any sense a counterexample 134  4 to (2). Recall that (2), the initial premise Malpass is attempting to challenge, is the 135 claim that no collection formed by successive addition can be actually infinite. What 136 (2) amounts to is not the claim that there cannot be an actually infinite number of 137 x’s such that each x instantiates the property ‘is successively added’. This claim is 138 demonstrably false. Any actually infinite number of elements is such that each 139 element instantiates the property ‘will be successively added’ (provided the 140 elements occur at future times). Instead, (2) must be understood as the stronger 141 claim that there cannot be a collection of an actually infinite number of x’s that 142 instantiates the property ‘is formed by successive addition’.  143 This should not come as a surprise––after all, Craig is talking about collections 144 and what properties we may ascribe to them throughout the SAA. (1) is the claim 145 that the temporal series of events is a collection that is formed by successive 146 addition. (2) is the claim that such collections cannot be actually infinite. His 147 conclusion is that the collection of the temporal series of events is not actually 148 infinite.  149 Thus, a counterexample to (2) cannot merely be one where ℵ0-many x’s each 150 instantiate the property ‘is successively added’––rather, it must be one where a 151 collection of ℵ0-many x’s is formed by successive addition. One might rightfully ask 152 at this point what it means to say of a collection that it is formed by successive 153 addition. To my mind, it is just our earlier (9): a collection of x’s is formed by 154 successive addition just in case there is a future time t at which, for all x’s in the 155 collection, x has been successively added. Craig is claiming that a certain collection, 156 namely, the temporal series of events, cannot have a certain property, namely, being 157 formed by successive addition, whilst having a cardinality of ℵ0. What Malpass’ 158 argument shows (if it is not interpreted to be invalid) is that it is indeed possible for 159 ℵ0-many x’s to instantiate this property. That, though, is perfectly compatible with 160 the SAA.  161 To defeat the claim that a collection that is formed by successive addition cannot 162 be actually infinite it does not suffice to give an example of ℵ0-many x’s that each 163 are successively added. One must give an example of a collection of ℵ0-many 164 members which is formed by successive addition. That George will count ℵ0-many 165 numbers, in the sense that ℵ0-many numbers instantiate the property ‘will be 166 counted by George’, entails that it is possible that there are ℵ0-many elements that 167 instantiate the property ‘will be successively added’. But that is all it entails. It 168 certainly does not entail that a collection with ℵ0-many members instantiates the 169 property ‘will be formed by successive addition’. Put simply, if Craig is arguing that 170 collections cannot exemplify certain properties, a proper counterexample must be 171 one where a collection exemplifies those properties. No premise in (4)–(7) entails 172 that any collection has any such property.  173 Thus, the problem for Malpass’ argument is disjunctive: either it is invalid in 174 virtue of moving from a claim about the properties of members to a claim about the 175 properties of collections, or else falls short of the mark of a proper counterexample 176 to (2) because it is concerned only with a claim about the properties of members.  177  178  5 4. A parity 179  180 Malpass’ argument relies crucially on the claim that (7) is a counterexample to 181 the premise that a collection formed by successive addition cannot be an actual 182 infinite. To put more forcefully the idea that Malpass’ counterexample is not 183 relevant to the truth of (2), consider how one would argue against the more 184 evidently false claim that it is possible that the numbers George will have counted 185 are actually infinite. Indeed, Malpass agrees that ‘it is false that George will have 186 counted infinitely many numbers’ and makes it explicit that ‘George will not have 187 counted every number’ (2021, pp. 12, 14). So, the following is a claim Malpass must 188 accept: 189   190 (C) The numbers that George will have counted cannot be an actual infinite. 191  192 But now imagine that someone attempts to challenge (C) by way of the following 193 familiar line of argumentation:  194  195 (4*) It is possible that George starts counting now and will never stop; 196 (5*) If George starts counting now and will never stop, then for each natural  197   number n, George will have counted n; 198 (6*) If George will have counted each natural number, then George will have  199   counted ℵ0-many numbers; 200 (7*) Therefore, it is possible that George will have counted ℵ0-many numbers. 201  202 (4*)–(7*), despite paralleling (4)–(7), form an unpalatable argument for Malpass, 203 as (7*) is agreed to be blatantly false. And it also seems as though the guilty 204 premise in this argument is (6*)––it does not follow from the mere fact that each 205 number will have been counted by George that George will have counted ℵ0-many 206 numbers. But if this is correct, then (6*) and (6) employ the same faulty form of 207 inference, and so Malpass’ own argument is unsound.   208 However, like Malpass, the defender of (6*) can claim that the phrase ‘George 209 will have counted ℵ0-many numbers’ simply means that ℵ0-many numbers 210 instantiate the property of ‘will have been counted by George’.  211 The obvious response here is that if (6*) is to be understood as such, (7*) will not 212 constitute a relevant counterexample to (C): that each natural number instantiates 213 the property ‘will have been counted by George’, such that there are ℵ0-many 214 numbers with this property, does not undermine the fact that the numbers that 215 George will have counted cannot be actually infinite. Again, this is because what is 216 meant by ‘the numbers that George will have counted cannot be actually infinite’ is 217 that there is no time where the collection of numbers George has formed by his 218 counting is actually infinite. Pointing out that each natural number is such that it 219 will have been counted is irrelevant. 220 It might be objected here that I have treated the sentences ‘George will count ℵ0-221 many numbers’ and ‘George will have counted ℵ0-many numbers’ as broadly 222  6 symmetrical: under either interpretation they will both be false or both be true. 223 However, perhaps this is problematic, since Malpass understands these phrases to 224 be asymmetric:  225  226 The reason George will count every number, but George will not have counted 227 every number, isn’t just that there are infinitely many numbers; it is because 228 there is no point in time which follows his counting. (Malpass 2021, pp. 14-15, 229 emphasis in original) 230  231 Dretske concurs:  232  233 It is true that at any stage of his task George will not yet have counted some 234 numbers. But, clearly this fact is not relevant to whether he will count to 235 infinity; it only shows that he never will have counted to infinity. (Dretske 236 1965, p. 100, emphasis in original) 237  238 But in the context of (4)–(7) and (4*)–(7*), whence the difference between the 239 fact that George will count every number and the fact that George will have counted 240 every number? If Malpass thinks that the latter is false because there is no time 241 following George’s counting, it can equally be said that the former is false because 242 there is no time terminating George’s counting, and so it is not the case that George 243 will count every number. Unless, of course, what Malpass has in mind is that there 244 need be no time terminating George’s counting: it suffices that there are infinitely 245 many numbers, each of which instantiates the property ‘will be counted by George’. 246 Yet, again, by the same token it can be said that there need be no time following 247 George’s counting: it suffices that there are infinitely many numbers, each of which 248 instantiates the property ‘will have been counted by George’. The point here is that 249 Malpass is faced with two possible options with respect to (6) and (6*): either they 250 make the fallaciously-motivated claim that there is a time at which George counts 251 (or will have counted) ℵ0-many numbers, or they are simply claiming that there are 252 ℵ0-many numbers that each instantiate the property ‘will be counted by George’ or 253 ‘will have been counted by George’.  254 The salient point here is that Malpass cannot consistently hold that it is not 255 possible that George will have counted ℵ0-many numbers and that (4)–(7) 256 successfully defeat the SAA. I have proposed a disjunctive remedy: either both (6*) 257 and (6) are false, or both (7*) and (7) are not relevant counterexamples.  258  259 5. Conclusion   260  261 In total, then, the SAA is not defeated by Malpass’ argument. As  262 outlined herein, the crucial third premise of the argument––that if George will 263 count every number then he will count ℵ0-many numbers––can either be read such 264 that it is a fallacious rule of inference, or otherwise facilitates a conclusion that is 265 no way a counterexample to (2). By my lights, this provides ample reason for 266 defenders of the SAA to reject Malpass’ counterexample to (2).  267  7 References 268 Craig, William Lane 1979, The Kalām Cosmological Argument (London:  269 Macmillian). 270 Dretske, Fred 1965, ‘Counting to Infinity’, Analysis 25:99. 271 Malpass, Alex 2021, ‘All the time in the world’, Mind; fzaa086,  272 https://doi.org/10.1093/mind/fzaa086  273 

Properties, Collections, and the Successive Addition Argument: A Reply to Malpass

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Properties, Collections, and the Successive Addition Argument

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