by Corry Shores
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[Logic and Semantics, entry directory]
[Graham Priest, entry directory]
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[The following collects the brief summaries for Priest’s book. The directory of entries without the summaries is found here:
http://piratesandrevolutionaries.blogspot.com/2015/07/entry-directory-priest-logic-very-short.html
]
Collected Brief Summaries for:
Graham Priest
Logic: A Very Short Introduction
Preface
Logic is an ancient discipline that was revolutionized in the 20th century with mathematical techniques and is currently very useful in information and computational sciences. This book will give a brief, broad, and non-technical overview.
Ch.1
Validity: What Follows from What?
“Logic is the study of what counts as a good reason for what, and why” (Priest, 1). An inference draws a conclusion from premisses (or from a premiss). It is valid if the conclusion follows from those premisses. It is deductively valid if it necessarily follows, that is, if no other conclusion could possibly follow, and it can be determined as such when “there is no situation in which all the premisses are true, but the conclusion is not.” An inductively valid inference is based on reasoning given in the premisses, yet other conclusions could also follow instead.
Ch.2
Truth Functions – Or Not?
Our intuitions about the validity of inferences are often correct, but sometimes they are misleading. One such case is the inference: q, ¬q / p, for example, “The Queen is rich,” “The Queen is not rich,” therefore “Pigs can fly”. Since the conclusion seems logically unrelated, we might erroneously think it is an invalid inference. By rendering these sentences into symbols and computing their truth values, we can see that there is no instance when the premisses are true and the conclusion not-true (false), and thus indeed it is valid. But since there is no situation where both the premisses can be true anyway, it is called vacuously valid. We also learn the truth tables for negation, disjunction, and conjunction, which are based on the truth conditions for these operations. If a sentence is true, then its negation is false, and vice versa. A disjunction is true only if at least one disjunct is true. And a conjunction is true only if both conjuncts are true. But conjunctions and disjunctions in English do not always map perfectly onto these truth tables.
Ch.3
Names and Quantifiers: Is Nothing Something?
When we speak of things, we might refer to some specific thing by name, like if we say, “Marcus came to the party”. In this case, what we are saying refers just to this one named person or thing. Or we might speak broadly and universally of all of a group of things, like if we said, “everyone came to the party”. In this case, what we say of the people or things applies to all of them. Or, we might refer to some thing, but without designating it specifically with a name, like when we say, “Someone came to the party”. Here we are saying something about a person or thing, but we are not specifying which one. When we want to speak of some thing or another, as in, “someone is happy,” we could use the existential quantifier and formulate this as, ∃x xH, meaning, there is some x such that x is happy. Or if we wanted to say, “Everyone is happy,” we could write ∀x xH, meaning, for all x, x is happy. Note that from just one quantified sentence an inference can be drawn. For example, if all people are happy, then there is some person who is happy. By using quantification, we can settle debates in mathematics and philosophy.
Ch.4
Descriptions and Existence: Did the Greeks Worship Zeus?
A definite description specifies a thing satisfying certain conditions, for example, “the man who first landed on the Moon”. Descriptions can be formulated symbolically by the use of variables that are predicated. The overall formulation takes the form ιxcx. Here, the ιx means, “the object x, such that…”, and the cx gives the conditions specifying the object. In our example we could write ιx(xM & xF) to mean, “the object x such that x is a man and x first landed on the Moon”. Furthermore, we may treat the whole description as something that can take predicates, and we can use Greek letters to stand for the whole description, thus possibly making the above formulation simply μ. This abbreviation will help us examine the validity of the Characterization Principle (CP), which is used in the Ontological Argument for God. We describe God as having a variety of properties that specify God, with the final one being “exists”: ιx(xP1 & … & xPn). The CP says that a thing characterized by certain properties in fact has those properties, and thus the whole described thing is predicated by the properties given in the description. Symbolically this involves substituting all cases of x in the description with that description itself. In this formulation we would get: ιx((xP1 & … & xPn)P1 & … & (xP1 & … & xPn)Pn), which in part says that the object that is omniscient etc., and exists, is in fact omniscient, etc., and does really exist. Using the Greek letters we can render the above substitution as: γP1 & … & γPn. But there is an important rule this argument breaks, namely that any predication to a non-existing entity is false. If there is a God, then the predication that God exists is true; but if there is no God in reality, then this predication is false. This means that for the argument to work, it must assume the truth of its conclusion at the outset, and is thus invalid. Yet there are cases where this rule does not apply, for example in instances of fictional entities like Greek gods whose properties can rightly be predicated to their description even though the thing described does not exist.
Ch.5
Self Reference: What is this Chapter About?
Paradoxical and otherwise problematic instances of self-reference lead us to suspect that we have more options than the following two: 1) a sentence can be just true, or 2) a sentence can be just false. Consider the “liar” sentence, ‘This sentence is false.’ If it is true, then it is false; but if it is false, then it is true. Either way, it’s truth-value will contradict what it says its truth-value is. So we have option 3) a sentence can be both true and false. Or consider the “liar cousin” sentence, ‘This sentence is true.’ Normally the terms in such a declarative sentence refer to things or situations by which we may determine the truth or falsity of the statement, that is to say, whether or not the indicated situation holds in reality or not. So if we say, “this chair is red,” we look to the indicated chair and its color, and we determine if the sentence is true or not. However, the terms in “this sentence is true” does not point us to such a determining situation, since we are only able to make two equally viable assumptions about its truth value, namely, that it is either true or that it is false; but, we have no way to make the determination one way or another, since it will always be consistent with what it says of itself under both assumptions. It would seem that we have no grounds that would allow us to determine whether it is true or false, and thus we have option 4) a sentence may be neither true nor false. The classical assumptions 1 and 2 lead us to conclude certain inferences are valid when our intuitions say otherwise. For example, “The Queen is rich,” “The Queen isn’t rich,” therefore, “Pigs can fly” (q, ¬q/p). Our intuitions tell us this seems invalid. But by just using assumptions 1 and 2, it is valid, since structurally speaking there is no situation where the premises are true and the conclusion is false. For, the premises can never all be true anyway. However, under the new assumptions, particularly that sentences can be both true and false, q, ¬q/p can be valid, if q is both true and false and p just false. For, q is at least true and ¬q is also at least true. However, our intuitions tell us that q∨p, ¬q/p is valid, but the new assumptions deem it invalid. Yet, perhaps it only seems intuitively valid if we forget that there are exceptional situations where sentences can be both true and false. There are other problems with the assumptions. When we assume that the liar cousin, “This sentence is true,” is neither true nor false, that means it cannot be true, but it says of itself that it is true. And while we might go along with saying that “This sentence is false” is both true and false, we might not feel the same way about “This sentence is not-true”. Here, we might conclude that it is both true and not-true (and not just true and false), which is a stronger contradiction that we may not want to accept.
Ch.6
Necessity and Possibility: What Will be Must be?
We can modify a statement of fact to indicate whether or not the referenced state of affairs is possibly the case or necessarily so. Modal logic allows us to deal with these modifications formally. Suppose “it will rain” is p. We write, “Possibly it will rain” as ⋄p, and we write “necessarily it will rain” as ◻p. Unlike truth-functional operators (like negation and conjunction), these modal operators do not alter the truth values of statements in a mechanically consistent way. To formally examine modally modified sentences, we think of there being other possible worlds about which we may make the same statements of fact, and these statements may be true or false depending on which alternate possible world it is in. In one possible world, it does rain tomorrow. But in another, it will not. We say something is possible when in at least one other world this state of affairs is false. However, no matter what possible world we conceive of, in all of them, if it rains, then fluid is falling. Such things which cannot be otherwise, when for example they are governed by fixed laws of physics, are considered necessarily true; for, in every other possible world they are true. We can diagram these possible world situations using boxes. In one box we give the statements of fact and their truth values for one situation or world (this world for example), and in other boxes we give the statements and their values for the other possible worlds. This helps us see which statements are necessarily true or false in one world and which are possibly so. This manner of formulation helps with certain debates, for example, it allows us to see that Aristotle’s argument for fatalism is fallacious. The argument makes us think that there is nothing we can do now to change the future, and also, that there is nothing in the past that we can regret or feel responsible for. The reasoning is as follows. If it is true that something will happen, then it will happen no matter what. But if it is false that something will happen, it will fail to happen no matter what. Either way, whatever happens occurs no matter what. By formulating this using modal logic, we see that it infers something incorrectly. There is a difference between the following two claims: 1) it is necessarily the case that if it is true that tomorrow I will get in an accident, then I will get in an accident, and 2) if it is true that if I will get in an accident, then I will necessarily get in an accident. If we just look at the semantic references, both formulations seem to have the same meaning. But on the level of their logical structure they are making different claims, and also structurally the second claim cannot be derived from the first, which is what is needed for the argument to hold. Aristotle’s fatalist argument would want you to believe that in every possible world you will get in an accident tomorrow, which is not so. It even acknowledges that the opposite could happen. However, there is a way to twist this fatalist argument a bit to remove that fallacy, and we may wonder whether or not this modification provides a valid argument for fatalism. We first say that there is nothing we can do now to change the past. This implies that states of affairs in the past are irrevocably true and statements about those situations are necessarily true. Now, suppose we do get in an accident tomorrow. This means it is true now if we say that we will. Suppose further that we said it yesterday also. We can say now that in the past it was true that we will get in an accident tomorrow. This means that it is irrevocably true that in the past we will get into an accident, and thus it is necessarily true that we will.
Ch.7
Conditionals: What’s in an If?
Conditionals are of the form, “if a then c,” or a→c. The first term is the antecedent, and the second, the consequent. Conditionals are false only if the antecedent is true and the consequent false, and they are true for all other value assignments. But there are many difficulties regarding conditionals, and some of which call into question the universal applicability of these value-assignments. For example, according to the truth table for conditionals, when the antecedent is false, then the whole conditional is true, regardless of whether or not the consequent is true. This means that the following two conditionals should both be true: “If Italy is part of France, Rome is in France” and “If Italy is part of France, Beijing is in France”. But intuitively, the second one seems false. So conditionals are not truth-functional, since a lot depends on the meanings of the terms. In order to evaluate them, we can use possible worlds, like with modal operators: “the conditional a→c is true in some situation, s, just if c is true in every one of the possible situations associated with s in which a is true; and it is false in s if c is false in some possible situation associated with s in which a is true.” Since Rome is by definition in Italy, that means in no possible world would it not be in France, were Italy to be in France. So that is why the first sentence is true. However, since Beijing is by definition a city in China and not a city of Italy, then in some possible worlds Beijing will not be in France, were Italy to be in France. And that is why the second sentence is false. Another problem with conditionals has to do with ¬(a&¬c), which has the same truth table as a→c, and in fact is called the material conditional and is symbolized as a⊃c. But although we might think that we can infer a→c from ¬(a&¬c), this is not in fact a valid inference, and we can show this using the possible worlds analysis. The important difference between a→c and ¬(a&¬c) is that a→c involves the relevance of a to c, where there is no such relevance implied in ¬(a&¬c). For this reason we can think of situations where a→c will be false but ¬(a&¬c) will technically true, thereby invalidating the inference. There are other cases too of inferences using conditionals that seem valid, and yet there are troubling counter-examples that call their validity into question.
Ch.8
The Future and the Past: Is Time Real?
We can use tense logic to analyze the validity of inferences that are based on statements referring to different moments in time. We first think of a one-dimensional series of situations arranged in their proper chronological sequence. We then think of statements of fact. They may or may not be true for one temporalized situation or another. Suppose a statement h is true only for the temporally situated moment s0. This statement refers to an instantaneous state of affairs, like the moment the first bullet entered Czar Nicolas’ heart. It will be false for all situations coming before and after that temporalized situation, since the event did not happen at those other moments. However, at a succeeding moment in the future, we can say truly that the event happened in the past. And likewise for a preceding moment in the past, we can say it will be happening in the future. We use the modifier P for past (“it was the case that”) and F for future (“it will be the case that”). So in moment s1, Ph is true, and for moment s-1, Fh is true. We can further designate temporal relations by compounding the modifiers. PPh would apply h to a situation coming before some other situation that is already in the past. FPh would apply h then to some situation coming after some other situation that is already in the past. Now, P and F refer to some determinate situation in the past or future. We can instead refer to all future situations with the modifier G (“it is always Going to be the case that”) and all past ones with the modifier H (“it Has always been the case that”). We can also make a model for this tense logic by arranging in sequence a number of s’s, placing s0 in the middle, and counting up and down the subscripts on both sides. This allows us to evaluate inferences based on tense modifiers. One example is McTaggart’s argument against the reality of time. If time is real, then the past and future are real, and thus they do not present logical contradictions. We then consider a sentence that is true just for the situation at one time-point. This means it did not happen in two temporally distinct time-points, and thus it did not happen both in the past and in the future: ¬(Ph&Fh). However, time flows, and so before it happened, it was in the future, and after it happened, it was in the past: Ph&Fh. The concepts of past and future present a contradiction, and thus time is unreal. One may object to the second formulation and say that it pretends that, for one situation that is located at one time point, the event can be both in the past and in the future. So to clarify the problem, we might then compound the modifiers and write ¬(PPh&FFh) to mean that the event did not happen at some determinate point coming before another in the past and at the same time happen at some determinate point coming after another in the future. Those following McTaggart’s reasoning can then say that still, because of the flow of time, PPh will be true and FFh was true, and thus, in contradiction with the prior, negated conjunction, PPh&FFh. But, by using the tense-logic model, we can display visually that the McTaggart argument is mistaken. There is never a singular temporalized situation where both terms in the past&future parings are true. Nonetheless, as this is a model that spatializes the flow of time, it might not be adequate for dealing with this argument about time’s non-spatial flow.
Ch.9
Identity and Change. Is Anything Ever the Same?
Over time, something’s properties might change. But it might either keep its identity or it might take on another one altogether. This presents a difficulty for philosophy and logic, especially since identity is a foundational concept in our thinking. We first distinguish objects and their properties, and we note that the properties may be variable while the objects remain constant. The ‘is’ of predication (x is red, or Rx) is different from the ‘is’ of identity (x is y, or x=y). However, Leibniz’s Law [of indiscernibles] uses properties to define identity. If two things share the same properties, then they are identical, and vice versa. This is a useful law in most applications, as for example when we use it for substituting terms in algebra. There are some other instances that at first seem to cast doubts on the applicability of the law, but these cases can be shown in the end to be mistaken for other reasons. However, there is one case that presents a big problem for the Law. We assume that identical things always were and always will be identical. When an amoeba A splits into amoebae B and C, then A has transformed into two other things in the sense of it having taken on new guises. This means that before the split, B and C were identical to A and thus were identical to each other. However, after the split they are non-identical. This contradicts the assumption that things that are identical always are so.
Ch.10
Vagueness: How Do You Stop Sliding Down a Slippery Slope?
A thing can change gradually over time. A true statement about that thing’s status at the beginning can later be false at the end of the development. But in many cases, it is not clear when exactly during that development the status changes without ambiguity. “Jack is a child” is true when Jack is very young and not true when Jack is old; but, when precisely in his young adult years does it cease being entirely true and instead “Jack is an adult” becomes entirely true? This issue is related to sorites paradoxes. Consider that “Jack is a child” is true at the beginning, and “If Jack is a child at the beginning, then he is still a child one second later” also is probably also true. That means by modus ponens, “Jack is a child one second later” is true. Using this same sort of reasoning, we can then conclude that Jack is a child two seconds later, and so on, meaning that he never ceases being a child. (We reiterate the structure, taking the affirmed prior conclusion that Jack is still a child in the succeeding second, and use it as a premise in an argument of the same structure, allowing us to conclude he is a child in yet the next succeeding second, and so on infinitely). One solution to these issues is to use fuzzy truth values. We can say for example that when he is 3 years old, the statement “Jack is a child” has a full truth value of 1. At 9 years “Jack is child” has a truth value of 0.75. At 14 years, 0.5. At 19 years, 0.25. And at 24 years, 0. And when we apply truth functional operators to statements with values between 1 and 0, we can determine the different resulting fuzzy values. Also, we can say that an inference is valid when both the conclusion and the premises meet a certain minimum level of truth value, which is determined by the actual context to which the statements apply. What we find then is that the sorites paradox does not hold when we use this fuzzy system. [For, in order for the modus ponens inference to work in all steps, we will need the minimum value to be 0 (in order to accommodate the final transitional step), which is too low to be meaningful.] Also, fuzzy values do not clear up the situation entirely, because we have the same problem when we need to determine precisely at what point the values change from 1 to something less than 1.
Priest, Graham. Logic: A Very Short Introduction. 1st ed. Oxford: Oxford University, 2000.
