Symposium: Speech of Phaedrus

Phaedrus sets the tone for the Symposium, complaining no poet praises Love. From Phaedrus, we learn Love is ancient – not the oldest – and one of the earliest gods to exist. Love has no parentage, though Love is said to have started to exist at some time. Phaedrus rests on authority in this origin myth, adding both humanity and the gods stand in awe and praise of Love.

We also learn Love is the greatest good for humans. Here, it seems Phaedrus is providing something of an argument, though as you’re no doubt aware he’s sees no reason to provide support for his claims, or consider potential counterexamples or difficulties. Rather, Phaedrus claims Love is the greatest guidance or motivator for humans, because whether lover or beloved, being shamed in a lover or beloved’s eyes is something we all seek to avoid and being admired is something we all seek. In fact, Phaedrus claims avoiding shame and seeking admiration effected through the lover-beloved pairing is a much better guidance or motivator than anything deriving from kinship, wealth, or even honor. Putting this point another way, if we are to seek to achieve great things, we require great guidance and motivation, and Love provides the best source of such guidance and motivation.

I pause here to point out the intuitive plausibility of this claim. Most of us no doubt can empathize with the sting of hearing those words from a lover “I’m disappointed in you.” Feeling that you’ve disappointed one you love – even if that love falls short of the sort of lover-beloved relationship Phaedrus has in mind here – is not enjoyable, and one feeling the sting is likely spurred to ensure they are not stung similarly in the future. In other words, the recipient will likely change their behavior to avoid disappointing their love. On the other hand, most of us likely know how good it feels to be admired by a lover, and to admire. Admiration by a lover spurs one to seek out further admiration, by achieving great things perhaps. We may even do quite drastic, perhaps unhappy, things for to acquire admiration and avoid shame. But I’m getting ahead of myself; let’s return to Phaedrus.

Phaedrus illustrates his understanding of Love as a great motivator with the example of the army of lovers. He claims, hyperbolically, that an army of lovers would be invincible, perhaps capable of taking over the world. I can’t help but think of Thebes’ Sacred Band, elite troops who loved, fought, and often died together, who respected one another as lovers might. They posed a considerable threat on the battlefield, from what I understand. I’m not sure which came first, the Symposium or the band. It’s not that important though. What is important is that this thought experiment seems well-motivated.

Phaedrus goes on to claim Love is the reason we are willing to make great sacrifices, with the greater sacrifice in the right context leading to the greater blessings from the gods. He provides three examples to illustrate. The first is of a lover Alcestis who sacrifices herself to save the life of her beloved – her husband. Alcestis is returned to life by the gods, a blessing provided for her great sacrifice. Note too in this example, the callback to how poorly kinship pales as a motivator for great sacrifice, as the husband’s parents are not even willing to sacrifice their lives for their son.

Contrast this with Orpheus, a lover who only caught a glimpse of his beloved, since he wasn’t willing to sacrifice himself. The gods did not praise Orpheus, but punished him with a mere image of his beloved. This is because Orpheus was unwilling to do what a lover should: sacrifice.

Where both Alcestis and Orpheus are examples of a lover sacrificing or not, Phaedrus’ third example is that of Achilles who he understands as the beloved of Patroclus. Even so, Achilles sacrificed himself by avenging the death of Patroclus by killing Hector, and consequently the gods gave him one of the highest prizes – the Isle of the Blessed. This is so even though Achilles was – as Phaedrus claims – the beloved and not the lover.

In fact the gods, Phaedrus claims, delight more with a beloved cherishes their lover, than when the lover cherishes the beloved. I suspect the point here trades on loving not being a symmetric relation. That is, just because x loves y it doesn’t follow that y loves x. Anyone can love, and one who loves may be motivated to do rather unacceptable things if that love is unrequited. This should be expected, as love is – again – a great motivator. But it seems paradigmatic cases Phaedrus has in mind of loving are those where love is symmetric, i.e. where the lover is loved in return. This is perhaps why the gods delight more with a beloved who cherishes their lover, than with a lover who cherishes the beloved. The latter may be had too easily, while the former secures a good.

Before closing his speech, Phaedrus says something rather puzzling: the lover is more like a god than the beloved. This is so because the lover is inspired by the gods. I can think of two ways to understand this passage.

  1. The implication here is that the beloved is not inspired by the gods. I think this is a problematic reading of the passage. If the lover is inspired by the gods, while the beloved is not, but the gods delight and bestow more honors on the beloved, then it seems the gods praise something more than what they inspire. More concretely, the gods praise Achilles the beloved for his sacrifice more than they praise Alcestis the lover for hers, though they make similar sacrifices. But Alcestis was inspired by the gods in her sacrifice, since she was a lover. There seems tension here, since this seems to imply that the gods praise something as greater than themselves. I take this consequence to speak against reading the passage as having the implication that the beloved is not inspired by the gods.
  2. But we can mitigate by claiming the beloved is not directly inspired by the gods, though the beloved is indirectly inspired. Achilles – after the death of Patroclus – acts as a lover would act, and so acts as if he’s inspired by a god. Because Achilles sacrifices himself, the way a lover would despite the fact that he is not a lover, he is more praiseworthy than Alcestis. Note: on this reading it is important only to claim Achilles acts as a lover. We can’t, for instance, go so far as to say Achilles – in acting as he does – becomes a lover. This is because if Achilles becomes a lover through his action, then Alcestis – who was already a lover – should receive just as much praise. Since she doesn’t, according to Phaedrus, we seem limited to saying Achilles acts as a lover would, but is not himself a lover. Ultimately then, the implication from the passage should be that the beloved is not inspired directly by the gods, but is inspired indirectly.

Summary aside, there are patent worries one should have about Phaedrus’ characterization of Love. Most clearly, Phaedrus simply assumes that Love guides lovers towards things that are good. This is not obviously true. We can illustrate the point in several ways.

  1. Consider first a lover who is not cherished by his beloved. It is easy to imagine a lover doing all sorts of terrible things for the sake of the beloved, because they aren’t cherished or perhaps because the beloved cherishes someone else.
  2. Consider second Phaedrus’ army of lovers not directed at admirable ends, but instead, say, genocide in the name of racial purity. Put another way, were Nazi’s lovers, I’d hope Phaedrus is incorrect about whether the resulting force was “invincible.” An army of lovers may achieve great things, but pure motivation need not be directed at a good end. 
  3. Consider third Achilles and Patroclus. Achilles seemed motivated by wrath and revenge rather than love. Indeed, it seems his love was an instrument for his wrath rather than the other way around, i.e. love was the justification but vengeance was the end. I take Achilles’ desecration of Hector’s body after killing him – parading him in view of his family and Troy – illustrates this point. Perhaps more telling is the fact that the gods had to intercede to force Achilles to stop, i.e. give Hector’s body to his father Priam for proper burial. Surely then the gods did not find this action praiseworthy. This again illustrates that Love understood by Phaedrus has no valence, it’s directed – but not much more.

Objections aside, I think Phaedrus’ speech is valuable for three reasons.

  1. First, Phaedrus provides Love a motivational character, which is taken up by subsequent speakers in the Symposium.
  2. Second, Phaedrus seems to play the role of a foil for later speakers. This is particularly apparent with the subsequent speech of Pausanias who begins his speech by making a philosophical distinction, something Phaedrus noticeably does not do throughout his speech. Phaedrus instead prefers to make claims, rely on myth, and basically play the role of a rhetorician. Pausanias doesn’t merely show Phaedrus as being a mere rhetorician by making philosophical distinctions where Phaedrus didn’t, but also tells by distinguishing between two sorts of love: one better than the other. I take this to be Pausanias picking up on the lack of direction towards the good that Phaedrus’ account of Love employs. In that, I think Pausanias is correct to make this distinction, as Love perhaps should be more than pure motivation; it should be directed towards something good.
  3. Third, Phaedrus’s speech isn’t merely a foil, but is a natural starting point for discussion to follow. Phaedrus’ account is wrong, but it’s by virtue of realizing his mistakes that we make progress towards the truth. Isn’t it plausible masses of people can be moved by mere rhetoric of the sort exhibited by Phaedrus – without reflecting much on its content? Of course. This is common enough in our lives today. It is common enough now, and likely was common enough then, to be worthy of being addressed directly. Phaedrus provides a case to dispute, but in doing so provides our base camp from where we begin our ascent towards understanding the nature of Love. It’s a starting point for dissent, which is a starting point for ascent.

Virtue in Rags: Virtue Requires Friendship

In Chapter 9 of the Nichomachean Ethics Aristotle defends - with a rather tortured argument - the claim that a virtuous friend is naturally desirable for a virtuous individual. I’ve attempted to extract his argument here (Let “John” and “Sally” designate distinct virtuous individuals):

(1) John exists
(2) If x exists then x perceives/thinks
(3) If x perceives/thinks then x perceives that x perceives/thinks
(4) If x perceives that x perceives/thinks then x perceives that x exists
(5) Hence, if x exists then x perceives that x exists                                                                                                     (from 2-4)
(6) John perceives that John exists                                                                                                                                     (from 1,4)
(7) If x exists then x's existence is intrinsically good/pleasant for x
(8) John's existence is intrinsically good/pleasant for John                                                                                    (from 1,6,7)
(9) If z is intrinsically good/pleasant for x & x perceives z, z seems good/pleasant for x
(10) Hence, John's existence seems good/pleasant for John                                                                                    (from 6,8,9)
(11) If y is x's friend, then whatever is intrinsically good/pleasant for x is intrinsically good/pleasant for y
(12) Sally is John's friend
(13) Hence, whatever is intrinsically good/pleasant for John is intrinsically good/pleasant for Sally   (from 11,12)
(14) Hence, John's existence is intrinsically good/pleasant for Sally                                                                   (from 8-13)
(15) If y is x's friend, then whatever seems intrinsically good/pleasant for x seems intrinsically good/pleasant for y
(16) Hence, John's existence seems intrinsically good/pleasant for Sally                                                          (from 12,15)
(17) If y is x's friend, then x is y's friend
(18) Hence, John  is Sally's friend                                                                                                                                         (from 12,17)
(19) Hence, Sally's existence is intrinsically good/pleasant for Sally                                                                   (repeat 1-8 replacing "John" with "Sally")
(20) Hence, Sally's existence seems intrinsically good/pleasant for Sally                                                         (repeat 1-10 replacing "John" with "Sally")
(21) Hence, whatever is intrinsically good/pleasant for Sally is intrinsically good/pleasant for John   (from 11,18)
(22) Hence, Sally's existence is intrinsically good/pleasant for John                                                                   (from 19-21)
(23) Hence, Sally's existence seems intrinsically good/pleasant for John                                                          (from 15,18)
(24) If z is intrinsically good/pleasant for y & z seems intrinsically good/pleasant for y, then z is desirable to y
(25) Hence, Sally's existence is desirable to John                                                                                                          (from 22,23,24)
(26) Hence, John's existence is desirable to Sally                                                                                                          (from 14,16,24)

Virtue in Rags: Vicious Friends

Several arguments in Aristotle's Nicomachean Ethics – in particular Chapter 9 – suggest Aristotle thinks virtuous individuals need friends because, as a second self, friends provide a route to self-knowledge. That sounds correct, but seems too limited. Consider the following: During a long-term relationship years ago, I tried in vain to be friendly with my lover's best friend, call her E. E and I did not get along, and were around each other often. Reflection on why suggested to me that E and I shared many qualities, but those qualities we shared were qualities I didn't like about myself, e.g. pride, aggressiveness, etc. Meeting with E reminded me of what I disliked about myself. In fact, I learned quite a bit about myself that I didn’t like from E, who I wouldn’t consider a friend.

Now, neither of us was virtuous, and it doesn’t seem Aristotle would count this relationship as a friendship, e.g. neither of us felt goodwill towards the other, etc. Nevertheless, E provided an effective route to self-knowledge because she was, in a sense, a second self – reflecting vicious qualities of mine at the time. This leads me to think where Aristotle thinks virtuous individuals need friends because their second-self provides self-knowledge, less-than-virtuous individuals might need less-than-virtuous relationships – not necessarily friendships - because these second-selves also provide self-knowledge.    

Moderate Modal Skepticism

Philosophers frequently motivate claims as possible based on conceivability.[1] It is then natural to wonder whether conceiving is a reliable method for generating justified beliefs in possibility claims. Yablo argued[2] conceiving proposition p as possible provides evidence that p is, in fact, possible.[3] Less optimistic philosophers, such as van Inwagen,[4] claim conceiving proposition p as possible provides no evidence that p is possible. If van Inwagen is correct, philosophical arguments relying on possibility claims motivated solely by conceivability are suspect. Given how widespread such philosophical arguments seem to be, van Inwagen’s claims are worth examining in detail. 

            In Section 1 of this paper, we extract and evaluate van Inwagen’s argument for Moderate Modal Skepticism, the view that while agents have justified beliefs in some – rather commonsensical – modal claims, agents do not have justified beliefs in many other – rather philosophical – modal claims. Having outlined van Inwagen’s position, in Section 2 we note a long-standing objection to van Inwagen’s argument – that it can be generalized to undermine justified beliefs in commonsensical modal claims as well as philosophical modal claims – fails, but a related worry – that van Inwagen’s argument relies on an under-motivated distinction between basic and non-basic modal claims – does not. Two responses are offered on behalf of van Inwagen, though neither are entirely satisfying. Additionally, we note van Inwagen’s argument implausibly requires agents justified in believing a given modal claim is true in every case know the modal claim is true. Having observed costs of van Inwagen’s characterization of the relationship between conceivability and possibility, in Section 3 we examine Yablo’s well-known alternative, which does not rely on an obscure distinction between basic and non-basic modal claims, and which allows justified belief and knowledge concerning modal propositions to come apart. Since there are independent reasons to prefer Yablo’s proposal to van Inwagen’s as an appropriate analysis of the link between conceivability and possibility, and since – pace van Inwagen – Yablo’s proposal does not entail Moderate Modal Skepticism, we conclude the various philosophical arguments targeted by van Inwagen are not threatened by his skeptical thesis.

[1](Putnam, 1980)’s super-Spartans; (Putnam, 1975)’s twin-earth; (Lewis, 1980)’s pained Martians; (Jackson, 1986)’s Mary; (Chalmers, 1996)’s zombies, etc.
[2]Cp. (Chalmers, 2002); others tie modal knowledge to counterfactual reasoning (Williamson, 2007), (Kroedel, 2017).
[3](Yablo, 1993).
[4](Van Inwagen, 1999).

2018 North American Summer School in Logic, Language, and Information

NASSLLI was a blast! I got to catch up with old friends (Anastasia!), make a few new ones (Anatha! Seth!) and - among other things - enjoy Rineke Verbrugge masterfully dissect shifts in rational knowledge attributions in the Friends episode The One Where Everybody Finds Out, using Kripke models. Other highlights include debating whether logic is the right tool to characterize counterfactual attitudes over lunch with Valentin Goranko, being mesmerized by Patrick Blackburn's entertaining and persuasive case for Hybrid Logic, and enjoying Paolo Santorio and Justin Khoo explore triviality results concerning conditionals and probabilities.

Also, the Carnegie Mellon campus is beautiful.  

Aristotle's Tense Friendships

An interpretive puzzle arises from a straightforward reading of Aristotle’s analysis of friendship. Shortly after claiming friendship involves mutually reciprocated goodwill for another’s sake, Aristotle claims those involved in friendships based on use or pleasure do not bear goodwill to their friends for their own sake, but instead only for the sake of what is – respectively - advantageous or pleasant.[1] Many proposals have been offered to ease the interpretive tension.[2] In this paper, I arbitrate between two and propose a third. The Standard Reading treats goodwill for the other’s sake as a defining feature of friendship based on virtue, with use and pleasure friendships resembling this form in other ways, but involving goodwill only for the sake of what is advantageous or pleasurable. On this reading, Aristotle either misspoke in his initial presentation of what varieties of friendship require, or – perhaps more charitably – dropped the requirement that all forms of friendship involve goodwill towards another for their own sake as he refined his characterizations of the lesser forms.[3] In contrast, the Goodwill Reading[4] treats goodwill for the other’s sake as a feature of all forms of friendship discussed by Aristotle, though they are nevertheless differentiated based on their respective objects. On this reading, Aristotle’s later remarks concerning the lesser forms of friendship are perhaps meant to merely emphasize the crucial role use and pleasure play in the corresponding forms of friendship, but were not meant to undermine each form of friendship involving goodwill towards others for their own sake. Arbitrating between these two readings stands to clarify Aristotle’s intended analysis of varieties of friendship while simultaneously providing a foundation on which alternative interpretive proposals may be evaluated. 

In Section 1, we examine Aristotle’s discussion of varieties of friendship further, extracting salient details. Here too we outline and motivate the Standard Reading of Aristotle’s discussion, and note the Standard Reading appears to treat most friendships as based entirely on egoistic motivation. These observations inspire seeking an alternative. In Section 2, we contrast the Standard Reading with the Goodwill Reading, which we also outline and motivate. We then pose several objections to the latter reading. In particular, we undermine the Goodwill Reading insofar as it relies on Aristotle’s definition of friendship from the Rhetoric, and observe this reading entails various relationships Aristotle explicitly counts as friendship fail to count as friendships. Having posed trouble for the Goodwill Reading, rather than retreat to the Standard Reading, we extract lessons from the preceding discussion and gesture at a prima facie promising synthesis of these distinct readings that provides a more nuanced solution to the interpretive puzzle than its predecessors.

[1](1156a11-3; 1164b10; 1167a14).
[2]See (Nehamas, 2010) for an overview of interpretive options. See (Whiting, 1991) among others, for more detail.
[3](Pakaluk, 2005, pg. 270-1); (Nehamas, 2010, pg. 220). The characterization of the Standard Reading here is from (Cooper, 1999), and shares much in common with the first option (Nehamas, 2010, pg. 220-1) considers as a solution to the interpretive puzzle. [4](Cooper, 1999, pgs. 312-35).

Binary Relations in OWL: Generic and Specific

A General Challenge

OWL is a semantic web ontology language based on a guarded binary fragment of first-order logic (FOL). Restricting FOL to this guarded fragment provides computationally valuable properties, such as decidability. As a consequence of this restriction, however, OWL is less expressive than FOL. In particular, FOL sentences involving ternary relations cannot be straightforwardly expressed in OWL. Ternary relations, however, are a natural way to capture certain relationships among entities of a given domain during and over time. To illustrate, consider three examples from the biological domain. First, any brain is always part of the same host. That is, specific brains are permanently related to their host. Call this type of relatedness Permanent Specific Relatedness. Second, organic tissue is always composed of cells, but not necessarily the same collection of cells. That is, generic cells are permanently related to a given tissue. Call this Permanent Generic Relatedness. Lastly, certain organisms have wings at some time, but not necessarily at all times, of their development. That is, wings are temporarily related to a given organism. Call this Temporary Relatedness. In FOL, these varieties of relatedness are easily distinguishable with ternary relations (letting ‘R’ stand for a relevant ternary relation):

(1)   ∀(x)∃(y)∀(t) [R(x,y,t)]           Permanent Specific Relatedness
(2)  ∀(x)∀(t)∃(y) [R(x,y,t)]           Permanent Generic Relatedness
(3)  ∀(x)∃(t)∃(y) [R(x,y,t)]            Temporary Relatedness

Not so in OWL. A well-known general challenge for any ontology represented in OWL modeling a domain in which these varieties of relatedness are considered distinct, is to differentiate (1), (2) and (3) with OWL’s limited resources.
Attempts to address the general challenge range from reification to extending the expressiveness of the logic underwriting OWL. We will not rehearse the advantages and disadvantages of these much-discussed proposals. Rather, we examine a recent attempt to address the general challenge in the context of the widely-used Basic Formal Ontology (BFO).

A Particular Challenge

BFO is a top-level, domain-neutral ontology used by biologists, among others, to provide a common starting point for the creation of domain ontologies in various areas of science. Given its wide purview, BFO is motivated to distinguish the preceding varieties of relatedness. BFO currently has two distinct formal language implementations, one employing FOL, BFO-FOL, and the other OWL, BFO-OWL. Where needed, BFO-FOL distinguishes among the varieties of relatedness with ternary relations along the lines of (1), (2), and (3). Clearly BFO-OWL must adopt a different strategy, making the general challenge salient for this implementation. The general challenge is made more difficult in this context, however, since developers adopt as a design principle that characterizations of BFO concepts in BFO-OWL be translatable into BFO-FOL. Hence, the general challenge in this context is to distinguish motivated varieties of relatedness represented in BFO-FOL with the limited resources of BFO-OWL, while ensuring there is a translation from the latter to the former. It is this particular version of the general challenge we address in what follows.
Only certain relationships among concepts in BFO are in purview of our particular challenge. Our motivating examples each plausibly concern parthood, and BFO adopts distinct primitive mereological relations among entities, one between continuant entities, and the other between occurrent entities. Roughly, continuant entities do not have temporal parts, and are disjoint from occurrent entities which do. Given the intended reading of the occurrent mereology in BFO, e.g. occurrent entities never gain or lose parts, distinguishing the preceding varieties of relatedness for the relation is unmotivated. This is reflected in BFO-FOL, where the occurrent parthood relation, named occurrentPartOf and restricted to entities of the class Occurrent, is binary and so has a straightforward translation from BFO-FOL into BFO-OWL. On the other hand, given the intended reading of the continuant mereology in BFO and observing our motivating examples are plausibly characterized as parthood among continuant entities at times, distinguishing among varieties of relatedness for this parthood relationship is desirable. In BFO-FOL, the continuant parthood relation, named continuantPartOfAt, restricts the first two entities to instances of the class Continuant and the third to instances of the class Temporal Region, a subclass of Occurrent. Since ternary, continuantPartOfAt does not have a straightforward translation from BFO-FOL to BFO-OWL. Hence, providing a translation of BFO-FOL’s characterizations of (1), (2), and (3) in BFO-OWL is within our particular challenge.

Temporally Qualified Continuant Strategy

There have been attempts to address the particular challenge. The “Graz Release” of BFO, for example, proposed an underdeveloped first pass solution - replacing ternary relations in BFO-FOL with tensed binary relations in BFO-OWL - which has since become a starting point for addressing the particular challenge. A recent proposal found here builds on the Graz recommendations introduces what advocates call temporally qualified continuants to BFO-OWL, an ontologically neutral class of computational artifacts. In more detail, advocates make the following recommendations for BFO-OWL:

(i) Universally tensed binary relation corresponding to the ternary continuantPartOfAt relation of BFO-FOL
(ii) Temporally qualified continuant individuals time-stamped with the temporal regions over which a corresponding continuant exists
(iii) Classes and Relations linking temporally qualified continuants to other BFO-OWL entities, such as temporal regions

Call this the Temporally Qualified Continuant proposal, or TQC. Each commitment deserves discussion.
Commitment to (i) stems directly from the Graz recommendation. However, where the Graz proposal advocated introducing two tensed binary relations to BFO-OWL for each ternary relation in BFO-FOL, reflecting universal and existential quantification over temporal indices, TQC adopts only the universally quantified versions. Proponents of TQC claim only universally tensed relations are needed to distinguish among varieties of relatedness in the presence of (ii) and (iii). To illustrate (i), assume for some pair of continuants, John and John’s Hand, it is the case that John’s Hand is continuant part of John for as long as John’s Hand exists. According to (i), this fact may be characterized in BFO-OWL with a universally tensed binary relation named continuantPartOf∀, satisfied (roughly) when the first and second entities stand in the continuant parthood relation at any temporal region at which the first entity exists.
Commitment (ii) is best introduced by example. Assume continuant John lives over 85 years. Then according to (ii), John has a corresponding temporally qualified continuant, John2000-2085, with accompanying 2000-2085 time-stamp. Additionally, John2000-2085 may have temporally qualified sub-continuants, corresponding to smaller time-stamps over which John exists. For example, John2000-2085 may have a temporally qualified continuant corresponding to JohnTuesdayFebruary22-2017, with a time-stamp smaller than and intuitively contained within John2000-2085.  Prima facie, commitment to (ii) appears to blur the BFO distinction between continuant and occurrent entities, but advocates of (ii) deny any ontological commitment. Rather, temporally qualified continuants are mere ad hoc entities designed to solve the particular challenge.
Concerning (iii), observe that formally, introducing temporally qualified continuant individuals hides complexity in a mere name, e.g. “John2000-2085” is indistinguishable from “John2000-2285”. Recovering complexity requires appealing to formal machinery, such as predicates and relations. Proponents of TQC reuse native BFO concepts to that end where possible. As above, we may introduce without much loss BFO concepts as they are characterized in BFO-FOL. For example, TQC reflects the BFO-FOL class Material Entity, restricted to certain continuant entities which have matter as parts, such as John. Instances of Material Entity bear the useful BFO-FOL hasHistory binary relation to unique instances of the BFO-FOL class History, occurrent sums of processes transpiring in the spatiotemporal region a Material Entity occupies. TQC extends BFO-FOL, however, by adopting the temporally qualified continuant class Tqc, instances of which bear the binary tqcOf relation to a corresponding instance of Material Entity. Each Tqc may be divided into instances of the novel class Phase which bear the novel phaseOf relation to the Tqc (an inverse hasPhase is defined as one would expect). Relevant histories of material entities are then bridged to phases through the binary hasOccurrentPart of BFO-FOL. Diagrammatically (borrowed from the cited paper):

                                         Figure 1…

                                         Figure 1. Overview of TQC

Altogether, TQC represents material entity John who lives between 2000 and 2085 as having a unique history, the sum of all processes occurring in the spatiotemporal region John occupies, which, we may say without loss of generality, has phase occurrent parts, such as John2000, John2001, John2002, etc., that are the phases of a Tqc John2000-2085, itself the tqcOf John. 

TQC Strategy Applied

Adding these commitments to BFO-OWL permits the following characterizations of varieties of relatedness involving continuant parthood (we include characterizations in BFO-FOL for comparison):

(FOL-1)     ∀(x)∃(y)∀(t)(continuantPartOfAt(x,y,t))           
(TQC-1)  ∀(x)∃(y)(continuantPartOf∀(x,y))    
(FOL-2)    ∀(x)∀(t)∃(y)(hasContinuantPartAt(x,y,t))          
(TQC-2)  ∀(x)∃(y)∃(z)∃(w)∃(u)(hasHistory(x,y) & hasOccurrentPart(y,z) & phaseOf(z,w) & hasContinuantPart∀(w,u))
(FOL-3)    ∀(x)∃(t)∃(y)(hasContinuantPartAt(x,y,t))           
(TQC-3)  ∀(x)∃(y)∃(z)∃(w)∃(u)(hasPhase(x,y) & hasOccurrentPart(y,z) & phaseOf(z,w) & continuantPartOf∀(w,u))

Recall, our motivating example for Permanent Specific Relatedness, reflected in BFO-FOL as (FOL-1), is every brain is always continuant part of the same host. Observe, (TQC-1) captures this relationship by appealing only to the binary tensed universal continuant parthood relation. Turning to Permanent Generic Relatedness, reflected in BFO-FOL with the inverse of continuantPartOfAt, named hasContinuantPartOfAt, as (FOL-2), our motivating example was every tissue has some cell as part at all times. Observe, with (TQC-2) this becomes every tissue has a history which has a phase as occurrent part, which is the phase of some temporally qualified continuant that has some continuant cell part at all times. Turning finally to Temporary Relatedness, reflected in BFO-FOL as (FOL-3), our motivating example was organisms having wing parts at some, but not all, portions of their development. Observe with (TQC-3) this becomes every organism has a history with a phase as occurrent part that is the phase of some temporally qualified continuant which is itself continuant part of some continuant wing.
In short, proponents of TQC claim the binary universal tensed relation is adequate for (1), and distinguish (2) and (3) in terms of whether a given temporally qualified continuant has a continuant part or is a continuant part of some relevant continuant. The TQC strategy thus appears to address one aspect of our particular challenge.

Relationship to BFO-FOL

However, our particular challenge requires an adequate characterization of varieties of relatedness in BFO-OWL be translatable into BFO-FOL. With respect to BFO-FOL, the path to (ii) can be understood as a series of relation parametrizations resulting in additions to the domain.  Roughly, to parametrize a relation is to define a lower arity relation with satisfaction conditions dependent on the higher arity relation. To illustrate, consider the sentence “John’s hand is part of John on Tuesday” characterized with the BFO-FOL continuantPartOfAt relation:

(I) continuantPartOfAt(John’s Hand, John, Tuesday)

Which, assuming standard first-order semantics, is satisfied (roughly) iff the ordered triple <John’s Hand, John, Tuesday> is a member of a subset of DxDxD, where “D” denotes the domain. This ternary expression can be parametrized by introducing a binary relation with satisfaction conditions tied to the ternary relation. Parametrizing with respect to time (and replacing ‘at’ in the name for readability) we have:

(II) continuantPartOfTuesday(John’s Hand, John)

Satisfied iff the ordered pair <John’s Hand, John> is a member of a subset of DxD. We might continue parametrizing, this time with respect to John, resulting in:

(III) continuantPartOfTuesdayJohn(John’s Hand)

Satisfied iff <John’s Hand> is a member of a subset of D. Finally, we might complete the parametrization by introducing an individual:

(IV) continuantPartOfTuesdayJohnJohn’sHand

Satisfied iff the individual is a member of D. Parametrization resulting in an individual is called total.  Otherwise, the parametrization is called partial. (II) and (III) are thus partial parametrizations, and (IV) a total parametrization.
Our choice of example was only illustrative, of course, since rather than parametrizing ternary BFO-FOL relations, TQC replaces them. Nevertheless, temporally qualified continuants may be understood as the result of parametrizing the binary existsAt relation of BFO-FOL, with the domain understood as restricted to continuants (rather than applying to all entities), and the range restricted to temporal regions (and so unchanged). For example, in BFO-FOL the sentence “John exists during 2000 and 2085” might be characterized as (where “2000-2085” names a temporal region):

(V) existsAt(John,2000-2085)

Satisfied iff <John,2000-2085> is a member of CxT, where C and T are, respectively, continuant and temporal sorts of D. Total parametrization results in the individual:

(VI) existsAtJohn2000-2085

Satisfied iff the individual is a member of D. If we assume the existsAt relation is implicit for any individual, then we can, without loss, drop this portion of the name, resulting in the familiar:

(VII) John2000-2085

More generally we might represent individuals such as “John2000-2085” with the variable notation reflecting continuants existing at temporal regions as: xt, yt, zt, with “t” subscripts reflecting temporal indices. These observations suggest some reason to think individuals postulated by the TQC strategy may be translated into BFO-FOL. 
Additionally, commitment to (i) has a clear relationship with BFO-FOL, as straightforward translations for tensed binary relations in BFO-OWL follow the pattern:

R∀(x,y) =def ∀(t)(existsAt(x,t) & existsAt(y,t) & R(x,y,t))

R∃(x,y) =def ∃(t)(existsAt(x,t) & existsAt(y,t) & R(x,y,t))

In light of our observations concerning (ii), we might then characterize TQC’s commitment to (i) as accepting only the first definitional pattern, replacing the existsAt conjuncts with our temporally qualified continuant notation:

            R∀(x,y) =def ∀(t)(R(xt,yt,t))

Again, these observations provide some reason to think were BFO-OWL to accept (i) and (ii), the resulting implementation would be translatable into BFO-FOL.
The same cannot be said for commitment to (iii). To be sure, some proposed TQC classes and relations have apparently straightforward translations into BFO-FOL, e.g. the class Phase appears only terminologically distinct from a certain BFO-FOL class discussed below. Others, however, have neither obvious parallels in nor translations to BFO-FOL, e.g. the class Tqc, the binary relation tqcOf, etc. This is perhaps to be expected for computational artifacts, but it is a noteworthy cost that TQC commitment to (iii) appears to conflict with a major design principle adopted by BFO developers. Proponents of TQC will likely reply these computational artifacts are needed to address the particular challenge, and that the native machinery of BFO-FOL is inadequate for the job. Hence, they might continue, the cost is worth paying.
But such a response exaggerates the need for these computational artifacts. We demonstrate why in a follow-up post, where we introduce an alternative strategy with a clear translation into BFO-FOL.  

Trust Logic, Not Tortoises

Carroll's note What Achilles Said to the Tortoise holds many lessons, many of which related to putative justification - or lack of justification - for basic logical inferences. Recently, Romina Padro, pulling from coursework and discussions with Kripke, has argued one more lesson should be added to the list, namely, that under certain conditions adopting basic logical inferences is impossible. I've a few thoughts on this new lesson, in particular how it might play with the old lessons. Check it out a recent draft here!

A Version of the Anti-Realist Manifestation Argument

Suppose sentence U is such that relevant experts currently have no effective procedure for determining whether what the sentence expresses is true or false. Semantic realism is the thesis that understanding U consists in grasping U's truth-conditions, and these conditions may obtain or not regardless of whether relevant experts are in principle able to recognize it. Semantic anti-realists agree that understanding U consists in grasping U's truth-conditions, but claims these truth-conditions are constrained by speaker evidence. Why accept this latter thesis over realism? Consider the following so-called Manifestation Argument against semantic realism a central tenet of which if found in the supposition in line (1):

(1)   SUPPOSE: U has evidence-transcendent truth-conditions
(2)  Relevant experts understand U
(3)  If speaker S understands sentence P then S grasps P’s truth-conditions
(4)  Hence, relevant experts grasp U’s truth-conditions
(5)  If S grasps P’s truth-conditions, this manifests in S’s use of P
(6)  Hence, experts manifest grasp of U’s truth-conditions in use of U
(7)  Hence, relevant experts manifest grasp of U’s evidence-transcendent truth-conditions in use of U
(8)  It is not the case relevant experts manifest grasp of U’s evidence-transcendent truth-conditions in use of U
(9)  Hence, U does not have evidence-transcendent truth-conditions

Since (9) conflicts with semantic realism, if the argument is sound this thesis is false. Come explore whether it is sound or not, as well as the delicate dialectic between realists and anti-realists in a paper I'm working on here! 

Poisoned Wine Puzzle

Problem

An assassin has infiltrated your wine cellar where you keep 1000 pristine bottles of vintage. The assassin poisons one of the bottles before being startled and fleeing the cellar. You are left unsure which bottle of wine has been poisoned. However, you know the poison is powerful, and that even a diluted drop would be deadly. Moreover, you know the poison takes effect within 24 hours of imbibing. Once it takes effect, death is immediate.

Fortunately, you have a poison testing device. Less fortunately, you only have ten non-reusable testing cups for the device. So, for example, you might use a cup to test one bottle of wine for poison, but afterwards the cup cannot be used to test another. As with human ingestion, the poison can be detected when it takes effect, sometime within 24 hours of the test, and once it takes effect, it may be detected immediately.

Perhaps even less fortunately, you have intended to serve the 1000 bottles during a lavish party in exactly 24 hours. Clearly you cannot do so now. Always the optimist, you are sure there is a way to isolate the poisoned bottle using only the ten testing cups, allowing the remaining 999 bottles for the party.

Question

How do you find the poison?

Solution

Let's begin with some intuitive labels. Name the testing cups A, B, C…J. Name the bottles 1-1000.

Let's try on a few putative solutions, observing where they fail and why, and also where they succeed and why.

First Pass - Clearly, we might test ten bottles of wine, one for each cup. That is, we might take some small portion of wine from, say, bottle 1 and place it in testing cup A. Similarly, we might take some small portion of wine from bottle 2 and place it in testing cup B, etc. At some time within 24 hours we will then know if one of the tested bottles has been poisoned.
However, since the testing cups are non-reusable, this approach would leave us with 990 bottles untested. A more subtle approach is warranted.

Second Pass - We might instead divide the 1000 bottles into ten allotments of 100 bottles each, e.g. 1-100 would form an allotment, as would 101-200, and so on. We might then take, say, a small portion of wine from each of the 100 bottles in allotment 1-100, which we place in testing cup A. Similarly, a small portion of wine from each of 101-200 would be placed in testing cup B, and so on. At some time within 24 hours we will then know if one of the allotments contains the poisoned bottle of wine.    
This pass is a significant improvement over the first. To see why, assume the poison is in wine bottle 425. Then testing cup A-D and F-J will not indicate poison has been detected, while testing cup E will. Of course, that leaves us with poison somewhere among 100 bottles. We assumed the poison was in bottle 425, but we would've achieved the same result had we assumed the poison was in any of the 401-500 bottles. In the interest of having a stellar party rather than merely an amazing party, let's press on.

Third Pass - Keep the second pass attempt, but add that A also tests bottles with names ending in "1", B tests those ending in "2", C in "3" and so on until we reach J which tests, in addition to 901-1000, those bottles with names ending in "0".
The result seems to significantly narrow down the poison. For example, say the poison is in bottle 467. Then E will indicate poison since E tests 401-500. So will G, since G tests not only 601-700, but also every bottle ending in "7". Unfortunately, both E and G overlap in many other places, e.g. 407, 417, etc. This pass offers little advantage over the last. We must continue.

Final Pass - Each preceding pass was an attempt to uniquely identify each bottle by some distribution of samples over testing devices. You're an optimist, but the preceding failures might lead you to think the glass is half empty, i.e. there is no such arrangement. Note, however, there are 1000 bottles and 10 testing cups. Each testing cup either will or will not test a given bottle of wine. But then there are 2^10=1024 ways we might uniquely distribute the bottles along the cups. Since there are only 1000 bottles, we should be able to uniquely identify each bottle. We need only find the right arrangement.

And these observations reveal the lines along which we may find our solution. There are two 'values' with respect to each bottle of wine, that a testing cup might take. For the purposes of book-keeping, if a testing cup tests a bottle, say that cup is evaluated as 'True' at that bottle, or 'T'. Otherwise, say that cup is evaluated as 'False' at that bottle, or 'F'. If, for example, testing cup A tests bottle 456, then A will be evaluated as 'T' at that bottle. If not, then 'F'. I chose these values to better illustrate our unique identification using a feature of a commonly taught algorithm for evaluating sentences in propositional logic: truth tables. Appealing to truth table distributions of 'T' and 'F' will provide the needed distribution of 1000 bottles of wine (rows) over 10 testing cups (columns).

But working with a truth table with over 1000 rows is cumbersome. Let's illustrate instead with a smaller example and then generalize. Specifically, let's examine our solution with only 3 testing cups, resulting in 2^3=8 rows in the truth table, each of which will correspond to a bottle of wine:


         A         B        C   
1     T         T         T
2     T         T         F
3     T         F         T
4     T         F         F
5     F         T         T
6     F         T         F
7     F         F         T
8     F         F         F
 

Each row is distinct. Then according to the table every testing cup will test bottle 1, A and B will test 2, A and C will test 3, and only A will test 4. A will not, however, test any of the remaining bottles. Rather, B and C will test 5, B will test 6, C will test 7, and no cup will test 8. Then if no cup indicates poison has been detected, it is in bottle 8. If only C, then 7; only B, then 6; only A then 4. If both B and C, then 5; A and C, then 3; A and B, then 2. If all testing cups indicate poison has been detected, then the poison is in bottle 1.

It is easy to see how our illustration of 3 testing cups and 8 bottles generalizes to the case of 10 cups and 1000 bottles. In that case, as stated, we are able to uniquely identify 2^10=1024 bottles, and hence, 1000 bottles with some to spare.

So, retain your optimism. It looks like you'll have a party to remember (or not!) after all. Though, keep an eye on the guest list; there's an assassin out there looking for you.

Hat of a Different Color (Part II)

Review

In a previous post we began examining the Three Hats Puzzle. Here we complete our solution. So far, we've introduced axioms characterizing a domain of five hats (two blue and three red) and three individuals (Alex, Barbara, and Cherise). We've additionally asserted that Alex and Barbara may see the hats of other participants, but Cherise cannot, and that neither Alex nor Barbara know what color hat they are wearing, while Cherise knows her hat color. From these axioms we were able to infer if Alex (or Barbara) sees two hats, then they cannot both be blue. This should sound correct, since if, say, Alex saw two blue hats, and since there are only two blue hats, then Alex would know his hat color. Since he doesn't, he doesn't.

To solve the puzzle, however, we must infer that Cherise knows her hat color.

Informal Solution

We will ultimately introduce first-order axioms and infer the solution to the case, but first we should examine the case informally to see what axioms we might need. Let's think about Alex for a moment.

  • We know Alex may see two red hats. To see why, note that if Alex sees two red hats, then he does not know what color hat he is wearing, i.e. he could be wearing a blue hat or red hat. Moreover, Barbara gains no new information based on Alex's claim that he does now know what color hat he is wearing, as all three participants could be wearing red hats.
  • We know Alex may see a red hat and a blue hat, but here we must be careful. The distribution of the hats matters. It is permissible for Alex to see Barbara wearing a blue hat and Cherise wearing a red hat. Then Alex's hat may be blue or red. Moreover, Barbara gains little information from Alex's claim. If Alex's hat is, say, red and Cherise's hat is red, then Barbara's hat may be red or blue.
  • Note, however, Alex cannot see Barbara wearing a red hat and Cherise wearing a blue hat. This leads to contradiction. To see why, assume Cherise is wearing a blue hat. Then both Alex and Barbara see Cherise wearing a blue hat. Alex may speak truly when claiming he does not know what color hat he is wearing, as he may see Barbara wearing a red hat and Cherise wearing a blue hat. Nevertheless, this option leaves Barbara speaking falsely. For if Barbara sees Cherise wearing a blue hat and knows that Alex cannot see two blue hats and also knows that Alex does not know what color hat he is wearing, then Barbara can infer that her hat must be red. Clearly, if Barbara's hat were blue then Alex would know what color hat he's wearing, as there are only two blue hats.

Our informal solution results in only two options for Alex. Alex either sees two red hats or Barbara wearing a blue hat and Cherise wearing a red hat.

More importantly than all that though, is the fact that we've stumbled upon our solution to the problem! On either of these options it must be the case that Cherise is wearing a red hat. Indeed, this is information Cherise may infer from the constraints of the case. In other words, Cherise knows what color hat she is wearing without being able to see any hats at all.

Strengthening our Axioms

Our only task remaining is to formalize our informal solution, and verify the intended results follow from our formalization. To our axiom set we add the binary relation "W" with an intended reading that the first (individual) wears the second (hat).

1. ∀x∀y∀z((W(x,z) & W(y,z)) -> x=y)
Only one individual may wear a given hat
2. ∀x∀y∀z((W(x,y) & W(x,z)) -> y=z)
Every individual wears only one hat
3. ∀x∀y(W(x,y) -> (I(x) & H(y)))
Only individuals wear hats

We also require that seeing a hat entails the hat is being worn, but no one sees the hat they are wearing.

4. ∀x∀y(S(x,y) -> ∃z(W(z,y)))
If someone sees a hat then it's being worn by someone
5. ~(∃x∃y(S(x,y) & W(x,y)))
No one sees the hat they are wearing

Supplementing these general axioms are the following facts, reflecting that, say, anything Alex sees is either Barbara's or Cherise's hat (and mutatis mutandis for Barbara).

6. ∀x(S(a,x) -> (W(b,x) v W(c,x)))
Alex sees the hats Barbara and Cherise wear
7. ∀x(S(b,x) -> (W(a,x) v W(c,x)))
Barbara sees the hats Alex and Cherise wear

Finally, we add the fact that if Barbara sees Cherise wearing a blue hat, then Barbara knows what color hat she is wearing.

8. ∀x((S(b,x) & W(c,x) & B(x)) -> K(b,b))
If Barbara sees Cherise wearing a blue hat, Barbara knows what color hat she is wearing

These axioms and facts generate a class of models in which the following are permissible:

9. ∃x∃y(S(a,x) & S(a,y) & x≠y & R(x) & R(y))
Alex sees two red hats
10. ∃x∃y(S(b,x) & S(b,y) & x≠y & R(x) & R(y))
Barbara sees two red hats
11. ∃x∃y(S(a,x) & S(a,y) & x≠y & B(x) & R(y) & W(b,x) & W(c,y))
Alex sees Barbara wearing a blue and Cherise wearing a red hat

But, importantly, which rule out the following possibilities:

12. ∃x∃y(S(a,x) & S(a,y) & x≠y & R(x) & B(y) & W(b,x) & W(c,y))
Alex sees Barbara wearing a red hat and Cherise wearing a blue hat
13. ∃x(S(a,x) & B(x) & W(c,x))
Alex sees Cherise wearing a blue hat

Hence, we are able to infer that Cherise is wearing a red hat, i.e. the following is a theorem:

14. ∃x(W(c,x) & R(x))
Cherise is wearing a red hat

Which is essentially what we intended to show. The resulting axiom set thus far can be found here. (Exercise: Show Cherise knows the color of her own hat).

Checking Our Work

Proofs were generated with Prover9 and models were checked with Mace4. If you'd like to check the models yourself, I advise generating a model with Mace4, then looking at the 'cooked version'. You can make the model even more perspicuous by copying it into Notepad++, hitting ctrl+F, navigating to the "Mark" tab, then entering "-*" (no quotes) with "Bookmark Line" selected. This will bookmark each line that begins with "-" which, in Mace4 means the predicate or relation is not satisfied. You can then navigate to Search->Bookmark->Remove Bookmarked Lines, to remove all the unsatisfied predicates and relations. The result will be a small model that's easy to read.

Curtis & Robson on the Metaphysics of Time

Benjamin Curtis and Jon Robson, in A Critical Introduction to the Metaphysics of Time, provide an impressive overview of contemporary debates over the nature of time. Check out a draft of my review of the book here.

Of the wealth of material covered in this introductory text, I found the authors' discussion of future contingents fascinating, and yet perplexing. In particular, the authors claim the possibility of an (alethically) open future conflicts with the classical logic principle of bivalence. They then use deviation from classical logic to undermine the possibility of future contingents. I take issue with several claims made by the authors (you can see a few more in the review above). For one, bivalence is intuitive, but it's not limited to classical logic. Other non-classical logics incorporate this principle as well. Of course, this pushes the question back from logical principles to logical theories. To be fair, the authors claim classical logic is widely accepted due to its theoretical virtues. But there is no discussion of what theoretical virtues are desirable and why, or comparison against alternative logics.

One might think my complaint is unfair, since this is an introduction to the metaphysics of time and not a philosophical logic text. I would, however, relate this same claim to the authors. If you're going to appeal to classical logic to undermine metaphysical theses, more discussion of why philosophers have accepted classical logic over others is desirable. Otherwise, leave philosophical logic questions alone and focus on the metaphysics. 

Jason Turner's Factualism

Appearances to the contrary the world consists ultimately of facts, not of things. According to Factualism, familiar objects and properties of experience are mere abstractions from this single ontological category. Jason Turner’s recent defense of a version of this thesis in The Facts in Logical Space: A Tractarian Ontology, is precise, exhaustive, and persuasive. Those working on facts will find much of interest, as will those working at the intersection of formal logic and metaphysics...

...and those wanting to hear what I think of Turner's book an check out my review (in draft form) here before release.  

Content with Publicity

Check out my recent paper here (outline below)!

In Concepts: Where Cognitive Science Went Wrong (1998), Fodor provides a list of conditions he claims any adequate theory of concepts must meet. Among the entries is what is known as the publicity constraint - concepts must be shareable across distinct agents. In attached paper, I examine motivation for requiring theories of concepts meet the publicity constraint. I also extract, explain, and motivate four premises Fodor employs in arguing for this constraint. In passing, I outline aspects of Fodor’s Language of Thought Hypothesis, paying particular attention to the representational and computational theories of mind. Next, I formalize and defend Fodor’s argument that generalizable laws of psychology entail concepts must be public. I then evaluate Fodor’s argument, ultimately declaring it unsound given his commitment to an informational semantic account of mental state content coupled with his response to Frege Puzzles which plague such accounts. On Fodor’s behalf, I propose motivating the publicity constraint via argument to the best explanation, while noting such a tactic is an uphill battle.

Frede among the Skeptics

Check out my recent presentation on ancient Skepticism where I try to get clear on whether, according to Sextus Empiricus, Skeptics had beliefs. I engage with Michael Frede's two seminal papers The Skeptic's Beliefs (1979) and The Skeptic's Two Kinds of Assent and the Question of the Possibility of Knowledge. If the presentation piques your interest, check out the paper here (which is much better than the presentation imo).

Now That's a Hat of a Different Color...

Three Hats Puzzle
Three individuals, call them Alex, Barbara, and Cherise, enter a pitch black room, where they are led to a table on which rests five hats, 3 red hats and 2 blue hats. The hats are arranged in no obvious order, and no individual can discern the colors in the dark, but Alex, Barbara, and Charles know how many hats of each color there are. They each select a hat from the table, and wear that hat outside the room into a well-lit area. Alex looks at Barbara and Cherise, and says, “I don’t know what color my hat is.” Barbara looks at Alex and Cherise and says, “I don’t know what color my hat is.” Cherise does not look at anyone else, since Cherise is blind. Nevertheless, Cherise says “I know what color my hat is.” This is all that is said, and they each speak truly.

Challenge
Explain how Cherise knows her hat color.

Note: Like many puzzles, this has numerous lateral solutions, e.g. the red and blue hats are differently shaped, Cherise is colorblind but can see blue, etc. Lateral solutions are easily dismissed without affecting the details of the scenario, e.g. the hats share all properties save color, Cherise is not just colorblind, etc. The challenge is to find a logical solution. A logical solution will follow directly from the details of the scenario, and will not be easily dismissed since doing so will require changing the scenario.

Solution and Discussion
I pose this puzzle to students who are then encouraged to work in small groups (1-3 students) to find a solution. Once students have understood the puzzle and the distinction between lateral and logical solutions, groups are quick to reason in the following manner:

Since Alex speaks truly, she must not see two blue hats. If Alex did see two blue hats, then she would know her hat was red. Then Alex must see either two red hats or one red and one blue hat. Similarly for Barbara, who must see either two red hats or one red and one blue hat

This seems unsurprising; the reasoning involved is direct, it is an immediate consequence of understanding the details of the case. We may show this formally. First, we fix on our notation.

Our language is first order with identity. Our domain consists of eight objects, which we sort into individuals with the predicate “I” and hats with the predicate “H”. Let “a” denote Alex, “b” Barbara, “c”. Let “B” be the predicate applying to blue hats, and “R” the predicate applying to red hats. Sample axioms characterizing the domain include (see here for the full set):

1.      ∀x(Hx v Ix)
Everything in the domain is either a hat or an individual
2.      ~∃x(Hx & Ix)
Nothing in the domain is both a hat and an individual
3.      ∃x∃y∃z(x≠y v x≠z v y≠z & Ix & Iy & Iz & ∀w(Iw -> (x=w v y=w v z=w)))
There are exactly three individuals
4.      ∀x(Bx -> Hx)
All blue hats are hats
5.      ∀x(Rx -> Hx)
All red hats are hats
6.      ∀x(Hx -> (Bx v Rx))
Every hat is either red or blue
7.      ∃x∃y(x≠y & Bx & By & ∀z(Bz -> (x=z v y=z)))
There are exactly two blue hats

…And so on. We also introduce relations.  Let “S” stand for an irreflexive binary relation holding between an individual and a hat with the intended reading being that the individual sees the hat. Let binary “K” hold between individuals with the intended reading that the first individual knows the hat color of the second. Sample characterizing axioms include:

8.      ~∃x(Sxx)
The ‘sees’ relation is irreflexive
9.      ∀x∀y(Sxy -> (Ix & Hy))
Individuals see hats
10.     ∀x∀y(Kxy -> (Ix & Iy))
Only individuals know things
11.      ∀x∀y∀z((Sxy & Sxz & y≠z) -> ∀w(S(x,w) -> (w=y v w=z))
Individuals see at most two hats

…And so on. We also characterize the following facts concerning the case:

12.  ∃x∃y(x≠y & Sax & Say)
Alex sees two things
13.  ∃x∃y(x≠y & Sbx & Sby)
Barbara sees two things
14.  ~∃x(Scx)
Cherise sees nothing
15.  ~Kaa
Alex does not know what color hat she is wearing
16.  ~Kbb
Barbara does not know what color hat she is wearing

With these axioms in hand, we may infer the following additional facts as theorems simply based on the domain and relation constraints:

17.  ∃x∃y(Sax & Say & (Bx & By) v (Rx & Ry) v (Bx & Ry))
Alex sees either a blue/blue, red/red, or blue/red distribution
18.  ∃x∃y(Sbx & Sby & (Bx & By) v (Rx & Ry) v (Bx & Ry))
Barbara sees either a blue/blue, red/red, or blue/red distribution

We add two more plausible facts. Observe, there is a relationship between knowing one’s hat color and the possible hat distribution. Consider Alex. If Alex sees two blue hats she knows what color hat she is wearing, which we may formalize as (watch the scope; avoid the Drinker Paradox!):

19.  ∃x∃y(Sax & Say & x≠y & Bx & By) -> Kaa
If there are two blue hats Alex sees, then Alex knows her hat color

A similar fact pertains to Barbara, but we will leave that aside here. Importantly, given (15) the consequent of (19) is false. Hence, the following theorems can be inferred:

20.  ∀x∀y(Sax & Say & x≠y) -> ~(Bx & By))
If Alex sees two hats they are not both blue
21.  ∃x∃y∃z(Sxy & Sxz & y≠z & ((Ry & Rz) v (By & Rz)))
Someone sees either a red/red or blue/red distribution of hats

In other words, for Alex, Barbara, or Cherise, the only available distributions of colors are red and red, or blue and red. We have then matched the direct reasoning above with our axioms.

Of course, this is not the answer to the puzzle. To solve the puzzle we must infer Cherise knows what color hat she is wearing, i.e. Kcc (Exercise: Why won’t simply adding this fact to the axioms suffice?).

This step seems the trickiest for students. I suspect it is because moving forward in the solution requires indirect reasoning, i.e. assuming something for the sake of a contradiction. The stumbling block, however, often leaves them ready to abandon the puzzle. Don’t let the difficulty of the puzzle stand in the way…we’ll infer the solution next post. In the meantime, play around with the axiom set here. The syntax is readable by Prover9. All theorems were checked with this application. Models were checked with the bundled Mace4 finite model checker.

Square of Individuals

Suppose there are at least two distinct individuals, Alex and Bob, and that Alex is part of Bob.

Ground mereology has the two-place parthood relation as reflexive, i.e. everything is part of itself: the maximal part. The relation is not, however, symmetric as it is intuitively false that if x is part of y, then y is thereby part of x. There are at least two ways to reject symmetry: asymmetry or antisymmetry. On the former, any x part of y entails y is not part of x. On the latter, if x and y are parts of each other, they are identical. The first is too strong, since inconsistent in the presence of reflexivity:

1.      ∀x P(x,x)                                 Premise
2.      ∀x∀y P(x,y) -> ~P(y,x)          Premise
3.            SHOW   !                           DD
4.                  P(a,a)->~P(a,a)         2, ∀ Instantiation
5.                  P(a,a)                          1, ∀ Instantiation
6.                  ~P(a,a)                       4,5 MP
7.                    !                                 5,6 !

Ground mereology accepts instead the weaker antisymmetry. Additionally, parthood is taken to be transitive as it is plausible any part x of y which is part of z entails x is also part of z.

Useful definitions can be constructed from this characterization of parthood. Two individuals are said to overlap if they share a part in common, are discrete if they do not overlap, and an individual is said to overlap the complement of another, if the first shares a part with the complement of the second.

Remarks in hand, return to Alex and Bob, denoting the first with “a” and the second with “b”, and the parthood between them as “P(a,b)”. Observe, parthood entails overlap for these individuals:

1.      P(a,b)                                    Premise
2.      SHOW ∃x P(x,a) & P(x,b)     DD
3.             P(a,a)                             Reflexivity
4.             P(a,a) & P(a,b)               1,3, CI
5.             ∃x P(x,a) & P(x,b)          4, ∃ Introduction

Line 5 reflects that “a” and “b” share a part in common. Hence, if Alex is part of Bob, then Alex overlaps Bob. The converse does not hold (Exercise: Find a countermodel).

Observe next, to say Alex and Bob are discrete, is to deny they overlap. Equivalently, it is to claim they have no common parts. Moreover, we would be saying of, say, Alex, that Alex overlaps some part of the complement of Bob, and vice versa. For Alex to overlap Bob’s complement is for there to be a part of Alex that is not a part of Bob.

1.      ~∃x P(x,a) & P(x,b)                           Premise
2.           SHOW∃x P(x,a) & ~P(x,b)           DD
3.                 ∀x P(x,a) -> ~P(x,b)              1, Substitution
4.                 P(a,a)                                     Reflexivity
5.                 P(a,a) -> ~P(a,b)                   3, ∀ Instantiation
6.                 ~P(a,b)                                   4,5 MP
7.                 P(a,a) & ~P(a,b)                     4,6 CI
8.                 ∃x P(x,a) & ~P(x,b)                7, ∃ Introduction

Hence, if Alex is discrete from Bob, then Alex overlaps the complement of Bob. Since both overlap and complement overlap are symmetric, we can say the same for Bob (Exercise: Prove overlap and complement overlap are symmetric).

Our brief foray into the mereology wilderness permits, given the assumptions with which we began, a square of individuals (cp. Square of Opposition). As is well-known, blindly translating categorical sentences into first-order notation undermines logical relations of the traditional square, as classical logic permits conditionals which are vacuously true. We avoid the problem of existential import by sticking with individuals. Hence, our square parallels the tradition:

Untitled Diagram.jpg

Implication holds whenever the first is true then the second must be, and if the second is false so must the first be. Contraries are sentences which may both be false but which may not both be true. Contradictory sentences require that if one is true the other is false, and if one is false the other is true. Subcontraries are sentences which may both be true, but which may not both be false (Exercise: Verify the remaining corners).

There’s ∃x about Mary

PUZZLE:
Larry is married but Nick isn’t. Larry is looking at Mary, and Mary is looking at Nick.

QUESTION:
Is someone married looking at someone not married?

CHOICES:

A.     Yes
B.     No
C.     Not enough information to answer

This puzzle was posed to me by a student (thanks Richard!) after class one morning. A cursory google suggests 80% choose incorrectly. I am skeptical; I’ve found no empirical evidence supporting this claim (I’d be interested if anyone else has). Be warned, searching for the puzzle will likely turn up solutions, so if you’d like to solve it, best settle here and reflect for a bit. I’ll wait. Once you are finished, check your answer by clicking it above. Afterwards, scroll down for a solution and some discussion.

SOLUTION:
The hallmark of a logical solution to a puzzle is that once presented with the solution, nearly everyone agrees it is correct. The Wason Selection Test is an example. Many choose incorrectly. Bentham reports a psychologist once confessed to him that nearly everyone accepts the standard solution as correct once it is explained [1]. The puzzle under discussion strikes me as logical in this sense. I’m curious if you agree.

I’ve translated the puzzle into a standard classical first-order language, where the predicate and relations symbols are obvious. It is straightforward to prove the solution (I’m using Hardegree’s natural deduction system in Symbolic Logic: A First Course). In symbols:

1.      Ml & ~Mn                                            Premise
2.      Llm & Lmn                                          Premise
3.      SHOW∃x∃y(Mx & ~My & Lxy)          ID
4.                   ~∃x∃y(Mx & ~My & Lxy)      AID
5.                    ∀x∀y(~Mx v My v ~Lxy)      4, Substitution
6.                    ~Ml v Mm v ~Llm                  5, Universal Instantiation
7.                     Mm                                         1,2,6 DE
8.                     ~Mm v Mn v ~Lmn               5, Universal Instantiation
9.                     ~Mm                                      1,2,6, DE
10.                     !                                            7,9 Contradiction

I’ve assumed as premises the information provided in the puzzle. On the SHOW line you’ll find a symbolization of the follow-up question. On line 4, I assume the negation of the SHOW line. On line 5, I substitute negated existential quantifiers for universal quantifiers trailed by negation, which is then distributed via De Morgan application. Since both variables in line 5 are under universal scope, I instantiate without restriction; in particular, to the constants denoting Larry and Mary. The result is line 6, which says Larry is unmarried, Mary is, and Larry is not looking at Mary. However, on line 1 we assumed Larry was married. Similarly, on line 2 we assumed Larry was looking at Mary. Hence, by disjunctive syllogisms, we infer Mary must be married on line 7. Instantiating line 5 once more, this time with Mary unmarried, Nick married, and Mary not looking at Nick, results by similar reasoning in Mary not being married on line 9. Since we already inferred that Mary was married, we now find ourselves in a contradiction. Hence, we infer someone is looking at someone not married.

Another way to think about the solution is to observe (or suppose what is plausible) that Mary is either married or not. If married, then since Mary is looking at Nick who is unmarried, someone is looking at someone who is unmarried. If unmarried, then since married Larry is looking at Mary, someone married is looking at someone unmarried. Either way, the answer is "Yes."

DISCUSSION:
Several of my students, when initially posed with the puzzle, claim there is “Not enough information to answer”. I take this to suggest students have trouble thinking to make certain plausible suppositions. To be clear, I do not think students have trouble making plausible suppositions and reasoning from them in general. They do, after all, deliberate about the future. Rather, it just doesn’t naturally occur to them to make even plausible suppositions in certain contexts, such as the context of puzzles and the contexts of proofs.

Works Cited
[1] Bentham, J. (2008). Logic and Reasoning: Do the Facts Matter? Studia Logica. 88:67-84

Just Another Joke Page

Jokes and Jokes

"Your honor, I understand I'm on trial for a murder that happened 10 years ago. I'll admit, I have memories of committing the murder. Will you punish me though for something that happened so long ago? How can you be so sure I'm even the same guy? So much can change in 10 years. Look, when I was a teenager, 20 years ago, I put fireworks in a neighbor's mailbox. It exploded, and they never found out I was the culprit. Consequently, I never paid damages. But you wouldn't drag me into court today to charge me for the reckless behavior of a teenager, would you? I've changed so much since then! I mean, I killed a guy."

Funny Paper Titles

  • Schrodinger's Can't: What Quantum mechanics says about 'ought implies can'
  • Barcan up the Wrong Tree: Vindicating Quine's Objections to Ruth Barcan-Marcus's Quantified Modal Logic
  • Oh, the Humeanity! Justice as Bareness
  • Being Clever Only Gets You So Far: Notes on Zeno's Paradox(es)
  • The Anti-Disetablishmentanglement Problem: The Meta-Ethical Entanglements We Weave

Funny Instructor-Student Interactions

Instructor: "So, what are your thoughts?" *to student in last row*
Student: "Who? Me?"
Instructor: "No, no, the girl behind you."
Student: *turns only to be confronted with the wall*
Instructor: "Never mind, she looks busy. I'll let you answer for her."

Student: "But sarcasm is one of my many talents." *after being reprimanded for excessive sarcasm*
Instructor: "So you don't have any other talents then?"

Student: "Can you give an example of equivocation 'on the fly'?"                                                                                                                Instructor: "'On the fly' you say? Sure thing. (i) Mathematicians work with planes; (ii) Planes work with jet fuel; (iii) Therefore, Mathematicians work with jet fuel."