Logic

To write natural deduction proofs in LaTeX, I use a package called fitch.sty. The package was written by Johan W. Klüwer and offers a nice clean way to typeset Fitch-style proofs. He provides a nice example:

Lovely. However, in some of my proofs, I wanted to have lines without numbers because they featured information that was not strictly part of the proof. For instance, like others, I commonly add a line that indicates the formula we’re out to prove after the list of premises. This is especially useful in teaching proofs. That line, I don’t want numbered — instead I want the counter to skip that line and continue after it, like so:

I had to dig around in the fitch.sty file itself to figure out how to do this, since there’s not really any documentation outside of it. I figured I’d share what I did for anyone facing the same issue.

Here’s what you do. Instead of beginning a line with “\fa” or something like that, add a line like this:

\ftag{~}{\vline\hspace{\fitchindent} CONTENT } \\

Where CONTENT is replaced by whatever you want to have on that line. The exact code for the my ‘∴ B’ line, for example, is:

\ftag{~}{\vline\hspace{\fitchindent} \fbox{$\therefore~ B$}} \\

And that’s all there is to it. I hope this helps someone looking to do the same thing as I was.

Happy typesetting!

Classics, Ethics

In Book 6 of his Meditations, Marcus Aurelius makes some remarks about the character of Antoninus Pius, his predecessor as Emperor of Rome, and his adoptive father. Marcus warns himself against being seen as like Julius Caesar, and says to avoid this he must live a good and humble life and conduct his duties in such a manner, as a follower of philosophy. Such a proper lifestyle can be seen in the life of Antoninus.

Marcus describes Antoninus as having a “keenness for logical action” and an “equable temper” (54). This is to say that his adoptive father maintained a calm disposition in his duties and private life. He was not rash in his behaviour, but rather thought his actions through with great care. Here the contrast with Caesar can be seen. Whereas Caesar was power-hungry and with a great ego, Antoninus was much more reasoned in his actions and had a “lack of vainglory” (54), a formidable and unusual trait for an emperor to have.

That “keenness” for a reasoned approach is again evident as Marcus describes Antonius’ “ambition to understand affairs” (54). Again, Antoninus is not seen to be acting on his intuition but to carefully take the time to understand what is going on around him, and apply careful thought prior to action. Rather than merely rely on others for information, Antoninus wanted not only to know the truth of the matter, but to “understand”, as Marcus says. All possible choices before Antoninus were carefully and completely examined before one was chosen. “He never rushed things.” (54).

Marcus also praises Antoninus’ disposition towards other people, especially in his public life. When he was criticized publicly, he not only “tolerated” challenges to his views, but he “was glad” when presented with a more favourable view to his own position. To those who were wrong, or offered only negative commentary without foundation or suggestion for improvement, Antoninus “endured” their scorn. Neither slander nor rumour had any effect on him. He also maintained strong, “unchanging” friendships (54). That is, he did not abandon his friends, nor was he quick to end a friendship over petty matters.

In his duties of public office, Antoninus is said to have enjoyed his work to the point of being completely focused on it for long periods of time. He was “energetic” when working and continued to work until late, taking no breaks at all. His duties as a statesman were guided by justice and a goal of protecting the people, as Marcus lists among the virtues one must seek in his position.

Marcus also comments on Antonius’ spiritual lifestyle. Antoninus is described as pious, having respect for the gods and acting in accordance with their laws. Marcus also makes a curious remark, that Antoninus “was religious but free from superstition” (55). In this sense, Antoninus would have characterized the humbleness and reverence that come with piety and the proper following of a religion, along with its moral code. However, more petty or sensational aspects that often accompany religious belief, such as certain types of ritual and accompanying fears of retribution, would not have formed a part of Antoninus’ thought. Thus, he was able to obtain virtue from his beliefs without hinderance.

In all other areas of life, Antoninus demonstrated this same humility. Happy to live simply, he did not need elaborate housing or clothing. His diet was “scanty” (54), which both showed this humility and allowed him to to focus on work for longer periods of time. He also required “little in the way of . . . servants” (54). These aspects of his life again contrast him with Caesar, who was known to enjoy a lavish lifestyle and demand much of those around him.

Marcus’ description here follows one earlier in Book 1 of his text, where he tells of his adoptive father’s advice to follow such a lifestyle, and to “honor genuine philosophers” (7). Both there and in this section, what is good is to put proper philosophical and moral thought and action above one’s selfish, personal inclinations. Marcus tells himself that being Antoninus’ “disciple” (55) in all of these traits will leave him with a clear conscious at death, knowing that he had lived a good life.

Source:
Marcus Aurelius. The Meditations. Trans. G. M. A. Grube. Indianapolis: Hackett, 1983.

Classics, Epistemology

In Book I of his traditionally-titled Against the Logicians (Book VII of Adversus Mathematicos), Sextus Empiricus presents a critique of how past philosophers have attempted to define the human being. He notes that these philosophers would pass off a definition as an “explanation of man” (132), a strategy that, for Sextus, would surely fail.

First is the account that goes back to Aristotle, though Sextus does not cite him directly. He remarks that some philosophers define man as a “‘rational, mortal animal receptive of thought and knowledge’” (132). Sextus points out that this does not conjure up man himself, but rather lists of some of his attributes.

With respect to attributes, Sextus says that there are two distinct types, those which are “inseparable” (132) from the thing which has them and others which “can be separated” (132) from the attributed object. Those which cannot be separated from the object include its extended dimensions, for an extended body is inconceivable without such attributes. Separated attributes, for Sextus, include activities or potential activities which an individual can take part in. For instance, speaking and sitting are attributes of the human being, since these are things which “happen to” people, “but not continuously” (132).

However, as Sextus states, neither the separable nor the inseparable attributes are identical to the thing that bears them. Hence, the Aristotelian definition fails to define the human being, since it only recounts some of humankind’s attributes: that human beings are animals, an inseparable attribute, and that human beings possess the separable attributes of “‘reasoning’ and ‘possessing knowledge’” (133).

For Sextus, the other component of the given definition, that human beings are mortal, is not an attribute at all. He remarks that mortality is not something that is ever an attribute of a human being, since what makes a human seemingly mortal, death, is never a part of the human but something that happens after the human ceases to exist. One is never at the same time human and dead, hence, humans do not have mortality as an attribute in Sextus’ language.

Sextus also makes the point that none of these attributes uniquely give us the idea of the human being. For instance, being rational or possessing knowledge are also attributes of the gods, while many animals exist besides human beings. Of course, such a definition does not mean to imply that each of these attributes are uniquely true of humans, rather, it is “all of them in conjunction” that present the human being.

To this, Sextus asks how these various attributes can be united to produce the human without also “exceeding” or “falling short” (134) of what human beings are. After all, human beings are not at all times rational or in possession of knowledge, nor do they possess mortality as an attribute because “death is not yet present” (134). So in this case, listing all of these attributes at any time may exclude some human beings, namely those who are not currently reasoning, possessing knowledge, or even simply still living. Hence, this definition exceeds that of human beings. Similarly, these are not the only attributes of human beings, so the definition in another sense falls short of producing the human.

The definition of man from Platonic sources also falls under Sextus’ scrutiny. He says that this definition describes man as a “‘featherless, two-footed, flat-nailed animal, receptive of political science’” (134). Such a definition fails to capture the human being for the same reasons as the Aristotelian one, namely that it gives a list of attributes, some of which are not constant in humans. Further, Sextus calls this definition “even worse than the others” (134) because it contains some negative attributes, such as ‘featherless’, along with positive ones, such as ‘twofooted’. That is, some of what is in this definitions are not attributes of humans, but an account of what are not human attributes.

Since no definition like those given can accurately capture the idea of human beings, or produce the “conception of particular men” (132), Sextus concludes that it is impossible to properly define the human being. Humans are, hence, left without a proper conception of themselves.

Source:
Sextus Empricius. Selections from the Major Writings on Scepticism, Man, & God. Ed. Philip P. Hallie. Trans. Sanford G. Etheridge. Indianapolis, IN: Hackett, 1985. pp. 132-134.

Metaphysics

In his third letter of correspondence with Samuel Clarke, Leibniz outlines his position on the nature of space. Contrary to Newton, and Clarke who defends him, Leibniz holds that space is “something purely relative” (14), that space is not a substance or a thing that exists, but rather the relationship between the objects it seems to contain.

In order to support this claim, Leibniz relies on his principle of sufficient reason, which  states “that nothing happens without a sufficient reason why it should be so rather than otherwise” (14). His axiom is essentially that nothing is arbitrary, and for Leibniz, this means that all things are the product of God’s reason. Leibniz’s argument proceeds by reductio ad absurdum:

  1. Suppose that there is absolute space.
  2. Whence, when God placed objects within space, there was no reason for him to place objects in a particular area of space and not in another, or in a different orientation. Hence, the placement of objects in the universe is arbitrary.
  3. But, according to the principle of sufficient reason, nothing is arbitrary.
  4. But this is absurd, so (1) must be false, and there is no absolute space.

If one accepts the principle of sufficient reason, it seems that one is committed to accepting the argument. Leibniz argues that as long as space is an absolute entity, it is impossible to imagine that the placement of objects within it is anything but arbitrary. The contents of the universe could be rotated by infinitely many degrees inconsequentially, and the reason that it takes any particular orientation cannot be explained causally. Notably, this does not entirely depend on Leibniz’s metaphysical claims about God. Although, for Leibniz, it is through God’s reason that things happen, the principle of sufficient reason can also be interpreted as referring to causality, claiming that nothing is uncaused, entirely spontaneous or arbitrary.

Hence, if space is absolute then the orientation of matter within it is arbitrary. This directly contradicts the accepted principle, and so space must not be absolute.

The fact that such rotation in the universe’s orientation would not cause changes to the physical universe is also mentioned by Leibniz, who asserts that the difference between orientations would be indiscernible (14). As Leibniz points out in a later letter, to “suppose two things indiscernible is to suppose the same thing under two names” (22). If all predicates corresponding to one object also correspond to another (that is, two objects share identical properties), then those objects are identical. This principle is one of second-order logic and can be symbolically represented as:

  • xy[∀P(Px ↔ Py) → x = y]

Since there could be nothing said about one orientation that differs from another, where a and b are any two universe orientations,

  • ∀P(Pa ↔ Pb),

hence,

  • (a = b).

The identity of the different universe orientations prompts Leibniz to write that “there is no room to inquire after a reason for the preference of one to the other” (15). This can be taken to mean that the very notion of absolute space is therefore meaningless, since we are left with infinitely many configurations and no reason to prefer any one.

It seems to me that there’s a refutation to this last point. If all of the possible orientations of matter in the universe are in fact identical, then there is in fact reason to for the selection of the current orientation of the universe. For if all universe orientations are identical, then there is exactly one universe orientation – that is, the orientation of the universe is the necessary one. The choice of orientation is necessary rather than arbitrary and therefore it is not in conflict with the principle of sufficient reason.

Source: Leibniz, G. W. and Samuel Clarke. Correspondence. Ed. Roger Ariew. Indianapolis: Hackett, 2000.

 

Epistemology, Metaphysics

In Book I of On the Nature of Things[1], Lucretius provides arguments for the existence of two main things that exist: matter and the void. Further, he argues that there are no other types of things besides these.

First, Lucretius asserts that matter is known to exist by means of sensory perception. We see matter, we interact with it, and thus it must exist. He argues immediately for empiricism by saying that sensory perception must be the “unshakable foundation” (423) from which an epistemology begins. Otherwise, he says, any talk of what is unseen is meaningless, for it is only in contrast with sensory perception that we can reason about anything that is outside of our view. That is, we cannot make any sense of talk about what is hidden from view without having a view in the first place, and thus what we see in our view must be the case. So, matter exists.

Secondly, we observe some important traits about the matter we see. First, it is always situated somewhere. Second, it can be moved from one location to another. Both of these require the existence of space, what he calls the void, for things always move or are located in something. Lucretius refers back to a few lines before this passage[2], in which he has argued that without space existing between material objects, their force of obstruction would constantly be acting upon each other and there would never be movement at all. He also, in these earlier lines, anticipates an objection on the grounds that fish swim through water seemingly unobstructed. Here, Lucretius argues that water must give way and create space for the fish, thus in order for there to be movement the space must be ontologically present, in modern terms.

Next, Lucretius argues that there must be nothing else besides matter and void. His reasoning is simply that anything else that one may describe, in an effort to posit a third constituent of being, can be shown to be either matter or void. This is done by Lucretius by asking whether or not this new entity is tangible. If it is, and is “susceptible of even the lightest and faintest touch” (433), then it must add to the aggregate matter of the universe and thus count as matter itself. If it is not tangible, however, then it is nothing but “that empty space which we call void” (438). This amounts to the claim that everything exists is either tangible, and is therefore matter, for everything that is tangible is matter, or is intangible, and is therefore void, for everything that is intangible is empty space, that is, void.

Lucretius’ argument for the existence of matter is an ontological an epistemological argument, stating that matter must exist because we cannot meaningfully talk about anything besides matter without first admitting matter into our ontology. However, it is somewhat anthropocentric in its approach, since it relies on what human beings, or perhaps any sentient being, can talk about or count as existing, not what fundamentally exists independently. In this sense, Lucretius is merely categorizing entities into two groups, based on whether or not they are tangible. Since everything is either tangible or not tangible, due to the laws of non-contradiction and the excluded middle, everything can rightfully be said to fall into one of these two categories.

One can rightfully be sceptical of Lucretius’ claim that everything that is intangible is empty space. The tradition of philosophy before and after him posit many so-called intangible objects which are thought to be more than the void. Among them are the forms, or universals, which are thought by Plato to have greater reality than material things. Meanwhile, talk of God or gods both before and after Lucretius often involves an immaterial but certainly active being. Lucretius goes on, beyond this brief passage, to deny that anything can act upon matter without being corporeal[3], but more work must be done to overthrow alternative thought on this matter.

Yet, we can still praise Lucretius for a simple but elegant approach to two formidable views. The simple approach to an epistemological foundation predates the rampant empiricism of the modern era, and even reminds one of Quine‘s naturalized epistemology in its reductionist (and dismissive) approach. The view meanwhile that space counts as some entity in itself, and is needed for the situation and locomotion of matter is one common in modern science, if still debated. One can easily read contemporary developments into Lucretius’ ancient words.

 


[1] T. Lucretius Carus. On the Nature of Things. Trans. Martin Ferguson Smith. Indianapolis: Hackett, 2001. Lines 419-440.

[2] Lines 335-345 and 370-383.

[3] Lines 440-444.

Metaphysics

Leibniz: The Best of All Possible Worlds

According to Leibniz, the actual world is the best of all possible worlds. He outlines a simple argument for this conclusion in The Monadology, §§53-55. The argument proceeds as follows:

  1. God has the idea of infinitely many universes.
  2. Only one of these universes can actually exist.
  3. God’s choices are subject to the principle of sufficient reason, that is, God has reason to choose one thing or another.
  4. God is good.
  5. Therefore, the universe that God chose to exist is the best of all possible worlds.

The argument can be broken down into three parts. Premises (1) and (2) say that God has a choice to make, for there are infinitely many possible worlds, and only one can be chosen. Premises (3) and (4) state that God chooses things with reason (this is the principle of sufficient reason, that is that all things that occur occur for sufficient reasons which fully explain them), and is good – hence, when God makes choices, God chooses that which is most good, or most perfect. Finally, the conclusion puts these two ideas, concluding that in this choice, as with all other choices, God chose that which is best, the best of all possible worlds which could have existed.

The argument is valid, for if there is are infinitely-many choices, and a good, reasonable God has made a choice of one, then that universe is the best. For, if God chose another less-good universe, then God would not be good, God would lack the idea of some universe, or God would not be reasonable, and some premises would be false. (Arguably, one could claim that the argument is invalid on the grounds that its premises may yield a false conclusion if the goodness of universes were identical, and that God did not choose to make any world actual. Clearly, since we know of an existing universe, the claim that none exists would be false, but it is not logically false and hence the argument could have weakness here. However, we will grant Leibniz the tacit assumption that “a universe exists” is a premise.)

Leibniz explains that the reason for the choice of  a world is in its “fitness, or in the degree of perfection that these worlds contain” (76). Hence, the suitability for a world for selection is a function of its level of perfection, and God, being perfect, would choose that world which is most perfect.

Premise (1) follows from the notion of God. God is, by definition all-knowing – a view defended in §48 – and, hence, has the knowledge of what any configuration of the universe would be. So, if God exists, premise (1) is true. Leibniz argues that God must exist in §§44-45 with the argument that “he must exist if he is possible. And since nothing can prevent the possibility of what is without limits, without negation and consequently without contradiction” (74) then God exists. The argument can be symbolized by modal logic as (◊G → □G),  ¬□¬G ∴ G, where G means “God exists”, and is valid.

One may hope to attack premise (2) with the claim that there could be multiple worlds. For if God is a transcendent being, beyond this world, it would seem that he could create all of his possible worlds independently from one another. It may be argued that God would never create anything imperfect, as the other worlds are said to be, but imperfections exist within the actual world even if they are in order with universal harmony. It may be the case that the entire collection of worlds, including the most perfect one, is itself perfect. It is not clear why Leibniz would deny this possibility, though it should also be noted that in this case, we may extend “world” to mean the set of all universes which God is again supreme over, and say that this set is the best of all possible worlds, which would have to be the case since it is the only set in existence.

Premises (3) and (4) seem less controversial. God is defined as being the “ultimate reason” (73) in §38, and hence God is the sufficient reason for things existing. God’s goodness is explained by his perfection in §41.

Hence, it follows from these premises that the actual world is the best of all possible worlds.

 

Source: Leibniz, G. W. Discourse on Metaphysics and Other Essays. Trans. Daniel Garber and Roger Ariew. Indianapolis: Hackett, 1991.