Essay presenting the logic to be used within this work. Written 8-24-23.
Before we learn what to think, we must learn how to think. This means that the first topic of our exploration of a philosophy of magick is logic, which may be defined as the science of reason. But before we explore logic properly, we must ground logic as a natural progression of the Xeno Position.
During our initial exploration of the Xeno Position, I observed—or rather, the alien observed—that in the “something” that is happening (which we will call “the world”), there are regularities. Patterns. The alien also observed that there appear to be two basic divisions to the “something” that is happening, which from convention the alien learned to call the physical and the mental. The search for knowledge about these two divisions of the world motivates the following thought:
“Since I am using the mental realm as a means to understand both the physical and mental realm, it behooves me to first learn about and understand the patterns in the mental realm so that I may better use those patterns to my advantage”.
It is clear from the beginning that the alien has certain logical intuitions built into their phenomenological experience of the mental realm that provide the basis for these observed patterns in the realm of the mental. These intuitions, such as the notions of equality/sameness, inequality/difference, truth, and falseness form the basis of a logic, a science of reason. This is not the place for a course in remedial logic, of course, but I will state some of these basic intuitions as the axioms of a simple logic that we will use throughout this work. Students of logic will note that these are basic axioms and derivation rules of two-value (meaning a proposition may have two possible truth values, “True” or “False”) propositional (meaning it attempts to capture the meaning of propositions) logic. This logic will then be expanded in various ways.
“P”—no matter the form it takes—always equals “P”. (Identity)
If “P” is false, “Not P” is true. If “Not P” is true, “P” is false. Similarly, if “Not Not P” is true, “P” is true. (Negation)
If “P” is true and “Q” is true, then “P and Q” are true. (Conjunction)
If “Not P or Q” is true then “if P then Q” is true. (Implication)
If “P” is true or “Q” is true, then “P or Q” is true and is true regardless of the truth value of the added premise. Note that this axiom can be used to introduce arbitrary premises into your argument, and is thus considered controversial by some logicians because one can go from “Not P” to “Not P or Q” (addition), which is logically equivalent to “If P then Q” (implication)…which clearly presents a problem, because I can always say an arbitrary proposition is not true (“it is not Saturday here”, for example) and then logically go from that to stating that “it is not Saturday here or pigs can fly” (addition, the rule here defined), then equate that with “If it is Saturday here, then pigs can fly” (implication). Which just can’t be right. It seems like we need a rule about the relevancy of contingencies, which is why some logicians prefer to use what are called “relevance” logics, which add such rules. For the purpose of this work, I will use a logic that includes the rule of Addition, but explain and argue for relevancy as necessary when using this rule. In other words, in our logic addition will never be arbitrary. (Addition)
If “P and “Q” is true, then both “P” is true and “Q” is true. (Simplification)
If “P” is true but “Q” is false, “P or Q” is true. Conversely, if “P” is false but “Q” is true, then “P or Q” is still true. And of course, if “P” and “Q” are true, then “P or Q” is also true. (Disjunction)
Both conjunction and disjunction are commutative, thus “P and Q” = “Q and P” and “P or Q” = “Q or P”. (Commutativity)
If “P or Q” is true and “Not P” is true, then “Q” is true. Conversely, if “P or Q” is true and “Not Q” is true, then “P” is true. (Disjunctive Syllogism)
If “If P then Q” is true and “P” is true, then “Q” is true. (Modus Ponens)
If “If P then Q” is true and “Q” is false, then “P” is false. (Modus Tollens)
If “if P then Q” is true and “if Q then R” is true, then “if P then R” is true. (Hypothetical Syllogism)
If “P or Q” is true, “if P then R” is true, and “if Q then S” is true, then “R or S” is true. (Constructive Dilemma)
With these basic rules, many thoughts can be formalized and their truth values deduced mechanically. However, despite the fact that the alien finds much of use in these elementary axioms and rules of inference, they just don’t tell the full story. They can’t formalize every possible meaningful proposition the alien finds in their world. Why is this? Because two-value sentential logic is only useful within certain contexts. The alien observes early on that some propositions vary between true and false based on various contextual conditions that might apply, some seem to be both true and false at the same time, while yet others seem to be meaningful yet bear no real relations to truth values whatsoever. The alien also observes that there are many propositions that add new logical elements that aren’t accounted for in this set of simple axioms and inference rules. Clearly we need something more. We need strategies for expanding our logic so that it can say all the things we need it to be able to say.
For an example, consider the proposition “It is raining.” Clearly this statement’s truth value depends on context, because it might be raining in San Francisco but dry as a bone in Las Vegas, and it may be either at various times. So we see that the temporal facts of the alien’s existence require them to include other intuitions, such as those involving time and space, into their ideas of logic. Thus we come to require framing devices called indexicals in our logic, such as “at time T and place P, Q is R”.
Now consider the proposition “Radiohead is a great band.” While syntactically it looks like a perfectly straightforward sentence that would have a definite truth value, semantically it is a statement of opinion and thus has no truth value at all. It is neither true nor false. This classification also applies to creative products that express propositions that have no truth value, such as poetry and fiction (however, see below for a framing strategy for fictional contexts).
Now consider the proposition “This statement is false.” (Liar Paradox). If it is true then it is false and if it is false then it is true. Yet it is a perfectly good sentence otherwise. So it seems that there are some propositions that are simultaneously true and false, which means that there really are—despite the arguments of classical logicians—true contradictions that don’t somehow destroy all logic (see below). Some logicians consider these propositions (called “paradoxes”) places where logic “breaks down”, but I disagree. I think the logic is working perfectly well, it’s just telling us something that those of us who are limited to two-value logic don’t like to accept: that some statements really are both true and false. It is my contention that the existence of paradoxes presents a brute fact about the mental portion of the alien’s world, and thus we must deal with that fact in our logic.
Now let us think more deeply about fictional or conjectural propositions. Consider the proposition “Vulcans have pointed ears.” Fans of the “Star Trek” franchise will be tempted to say that the truth value of this proposition is “true”, yet the fact of the matter is that Vulcans are fictional, thus the fact about the world that makes the proposition true (what I will henceforth call a “truth-maker”) does not actually exist except inside of a fictional narrative. If there is no truth-maker for a proposition, no fact about the world or state of affairs in the world (either physical or mental) that makes a particular proposition true, then in what sense can it be said to be true? Yet the proposition “Vulcans have pointed ears.” seems very much to be about something. It’s a meaningful sentence that seems to be either true or false…within a fictional context. So we are able to use another type of contextual framing device in our logic, where we say instead “In the world of ‘Star Trek’, Vulcans have pointed ears.” Suddenly the sentence has a truth-maker again, and thus has a clear and distinct truth value.
Now consider the following propositions: “It is possible that a dog will be born with spots.”, “It is necessarily the case that 3 is more than 2.” and “It is impossible to have a square circle.”. These propositions capture a new element of the alien’s new world…modality. Thus we need a new sort of operator for our logical system, one that allows us to say things like “Possibly A is true.”, “Necessarily B is false.”, and so on. Propositions using these modal operators may be seen as having truth values in either fictional/hypothetical contexts (rendering the operators a type of framing device) or we may use talk of “possible worlds” where to say “It is possible that it will rain tomorrow.” is to say “There exists a logically possible world where it rains tomorrow.” and to say “it is impossible for 2 to equal 3.” is to say “There exists no logically possible world in which 2=3.”
Let’s expand on the last example. Consider a new proposition: “Some dogs have spots.” This sort of sentence captures a new fact about the alien’s world…relative quantities. So again we have to add a new framing device to our logic and two new ways of talking: the universal quantifier, which typifies sentences of the type “All A’s are P”, and the existential quantifier, which typifies sentences of the form “there exists an A such that A is P” (which is logically equivalent to “some A’s are P”). To formalize our example statement we would then say “There exists at least one dog that has spots and not all dogs have spots.”.
Another important pattern about the two divisions to the world that the alien encounters is causation. This is an extremely difficult subject to deal with logically because we—by which I mean humans—don’t really understand causation yet. The same cause can have many effects, or one effect can have many causes. The same effect can be caused by many different possible causes, individually or severally, and the same cause (for example, flipping a coin) can lead to many different—even contradictory—effects, sometimes simultaneously. How then do we address this? The only real way I have discovered is through counterfactuals, sentences of the form “If A had not occurred, B would not have occurred.” or “If not A, then not B.”, but obviously this is hardly a satisfactory formalization of causation because while it will end up with the right truth values when we are analyzing events that have already occurred, it doesn’t help us predict or reason about what effects will in fact follow from what causes. It also must be noted that implication (the “if…then” logical construction) does not necessarily imply causation…all “if A then B” means is that “If we see an A, then we always see a B”, which does not fit our normal intuitions about causation. Counterfactuals are also notoriously difficult to deal with logically, so this doesn’t really help us as much as it seems. We will go into causation more deeply in the section of this work on metaphysics, but until then let us be provisionally content with using counterfactuals to talk about causality (and indeed provisional in all our talk about causality).
Now let us move on and examine the proposition “The engine is hot.” We may look at this claim as a binary issue…whether the temperature of the engine has crossed the red line on a dial, for example. But most of the time, we are looking at the whole temperature dial and estimating a degree to which the engine is considered “hot” based on the safe ranges of temperature for the engine. Thus this claim is partially true, partially false, because an engine is hot only to the degree that it isn’t and vice versa. Fuzzy logic attempts to capture our intuitions about such situations by using truth values of any real number between 0 and 1, similar to probability calculus. However, in fuzzy logic the number represents how true the proposition is, with 0 = false and 1 = true, as opposed to how likely some event is to happen. So in our example, we can say that the proposition “The engine is hot.” is .75 true when the temperature of the engine is measured at 75% of the way to the red line on the gauge.
Now let us consider the proposition “All the glasses of regular cow’s milk I have seen thus far have been white, therefore it is very likely that the next glass of regular cow’s milk I see will also be white.” This sentence captures a different kind of logic altogether: inductive logic. All of our logic thus far has been deductive…we move from a set of premises that we already know to be true to a conclusion that we (hopefully) also know to be true using trustworthy deductive rules. Induction is different. Induction reasons from a body of data and tries to estimate the likelihood of the next instance. All induction can do is point to probabilities, because the claim is merely “I have seen a correlation between A and B in more situations than I have not, therefore the likelihood these things will continue to be correlated is greater than it is not.” Inductive reasoning is often used in the physical sciences, because often all we have as evidence is a statistical fact that most of the times we have seen A so far we have also seen B. For example, deductive logic reasons from the laws of physics, the observed current relative positions of the Earth and Sun, and the current states of the Sun and Earth, that the Sun, in fact, will rise tomorrow. Inductive logic reasons from the fact that the Sun has risen every day in recorded history that it is most likely that the Sun will rise again tomorrow. Notoriously, as David Hume (1748) noted, induction can never get us from “most likely” to “is definitely the case”. This is the “Problem of Induction”: no matter how many times the Sun rises, we still can’t use induction to know for sure that it will rise again tomorrow. All we can know is that it most likely will rise tomorrow. This limitation of induction is what makes all reasoning from empirical sources vulnerable to skepticism, and is the scandal of the sciences.
Now consider the following argument: “This morning the streets were wet. I was awakened a couple of times last night by thunder. The dirt in my garden was muddy. Therefore, it is probable that it rained last night.” This is called abductive logic, also called “reasoning to the most likely explanation”. Like induction, this type of logic may come to wrong conclusions, as all it can say is what is probable, not what is true. For the purposes of this work, I will always hedge my bets when possible. If I am making an inductive or abductive argument, I will say so.
Now consider the propositions “John is the father of Amy”, “Monica is to the left of the building”, and “2 is between 1 and 3”. Each of these propositions talk about relations. We will discuss relations a great deal more in the section of this work on metaphysics, but for now I will merely mention this sort of logical operator as a fact of our logic. When defining a relation I will explain both the sorts of objects it relates (members of families, directionality, and order of numbers, as in my examples) and the sort of relation I am picking out for discussion (“father of”, “to the left of”, “between”).
Finally, consider the proposition “Amy believes that P.” This is a proposition about belief, which we will address more fully in the section of this work on epistemology, but for the moment I want to address this issue logically. It seems clear that there is a logic of belief, which in modern logic is called “doxastic logic”. For example, if I believe that “if P then Q” and I believe “P”, then I seem logically committed to believing “Q” by our normal rules of implication. This means that we need a new logical operator in our logic…one that indicates belief. However, this logic can be misleading if we make it too simple, because even if I believe “P” and “if P then Q”, if “Q” is obviously false I still should not believe it. So we instead need to make rules that take into account the whole system of beliefs held together and say “I should not hold a system of beliefs where I believe that ‘P’, ‘if P then Q’, and also do not believe that ‘Q’”. The scope of our logic is thus enlarged and the normative claim (the “ought”) of our doxastic logic includes the whole system of beliefs, any one of which can be interrogated separately or together for correspondence with our data. In this case, obviously if it is the case that “P” is true and “Q” is not true, then our belief that “if P then Q” needs to be interrogated and perhaps revised.
Now we have framed our basic intuitions as axiomatic rules, we have accounted for the particularity of the world through contextual and indexical framing devices, we have addressed the differences between truth-bearing sentences and other sorts of sentences such as opinions, we have addressed true contradictions, fictional worlds through narrative frames, possible worlds through modal operators, relative quantities through universal and existential operators, causality, fuzziness, induction, abduction, relations, and belief. What else needs to be accounted for in our logic?
Well, the Xeno Position is fundamentally an epistemological starting place, a “veil of ignorance” that I am using to explore what I understand to be the basic phenomenological human condition. Given that “veil of ignorance”, what other important fact about our epistemological situation needs to be able to be captured in our reasoning? In a word, inconsistency. One reality of the human condition is that sometimes it is rational to believe both “A” and “Not A”. Sometimes we may have incomplete or inconsistent information such that both “A” and “Not A” have equally overwhelming evidence. Sometimes our previous philosophical or personal commitments force us into a bind where we are committed to contradictory beliefs and we don’t have sufficient new information to revise our beliefs (we will explore belief revision in the section of this work on epistemology). Sometimes we simply can’t be sure which of two or more contradictory propositions are true and our circumstances force us to account for all the possibilities. And sometimes we have paradoxes or otherwise contradictory propositions that still seem to bear meaning. In any case, our logic, in order to be satisfactory, needs to be robust enough to deal with contradiction.
Why is this so important? In classical two-value logics, it is generally believed that from a contradiction one can then prove anything, rendering every possible proposition trivially true and ruining the most important feature of a logic: its ability to preserve truth values over the process of reasoning (i.e. from true premises, true conclusions necessarily follow). This issue of contradictions breaking logic is called the “Principle of Explosion” or “Ex Contradictione Quodlibet” (ECQ) in philosophy and is generally accepted by most logicians as a fact of “classical” western logics, such as Aristotelian logic and most modern mathematical logics. However, it is extremely notable that classical western logics are not the only logics. Eastern philosophy has detailed other logics that account for contradictions, presenting logics with four possible truth values instead of two. Arguably these four-valued logics preserve our logical intuitions and results more adequately than two-value logics because they account for propositions that are true, false, both true and false (paradoxes, certain mystical insights, etc), and neither true nor false (opinions, art, jokes, etc). Other western logics that do not “explode” have also been created in recent years (the 20th and 21st centuries CE), and in general the type of logic that accommodates contradictions is called “paraconsistent” logic. The only logical machinery such logics don’t provide that we have explored are the framing devices and the new operators we introduced to cope with issues of quantification, modality, fuzziness, and belief. So we can, with the other tools we have introduced, end up with a four-valued propositional logic that can be extended as needed to address various types of propositions using specialized framing devices and operators. Within this four-valued logic, we will consider propositions that have either the truth values of “true” or “true and false” as deductively true, and treat propositions with the truth values of either “false” or “true and false” as deductively false for the purpose of parsing arguments, but contradictions will be supported with additional arguments. This is the logic I shall use for the rest of this work.
Note that I am very specifically not claiming that this logical system the alien has built is somehow universal or final or handed down to us by divine fiat. The total set of all possible provable statements in a logical system are a function of the definitions, inference rules, and axioms that are chosen as the basis for that system. This means that a change in your axioms produces a different total set of provable statements in the resulting system. There is no way to avoid this limitation of logic, and there are many possible logical systems, especially when one includes paraconsistent logics, as I have here. All I am claiming here is that this logical system can be successfully used to talk reasonably about the subjects at hand. If someone can come up with a logic that better fits our purposes, I am more than happy to use it.
Thus the alien begins with simple and straightforward phenomenological intuitions that allow them to build axiomatic rules of logic which they must, upon examination and comparison with lived experience, expand using various strategies in order to fulfill two intentions: that their reasoning is truth-preserving (when given true premises, the conclusions derived by the rules are necessarily true) and empirically adequate (that their reasoning accurately and meaningfully represents the facts about the world, both physical and mental). Thus equipped, the alien turns to the next issue…the problem of knowledge itself.
Works Cited:
David Hume, An Enquiry Concerning Human Understanding, (F. Collier & Son, 1748, 1910)
Further Reading:
Graham Priest, Logic: A Very Short Introduction, (Oxford University Press, 2000)