Exploring Topics in the History and Philosophy of Logic

George Englebretsen

• Publisher: De Gruyter
• Year: 2015
• Print ISBN: 9783110442236
• eBook ISBN: 9783110433814
• Edition: 1
• Topic: Philosophy
• Subtopic: Ancient & Classical Philosophy

While post-Fregean logicians tend to ignore or even denigrate the traditional logic of Aristotle and the Scholastics, new work in recent years has shown the viability of a renewed, extended, and strengthened logic of terms that shares fundamental features of the old syllogistic. A number of logicians, following the lead of Fred Sommers, have built just such a term logic. It is a system of formal logic that not only matches the expressive and inferential powers of today's standard logic, but surpasses it and is far simpler and more natural. This book aims to substantiate this claim by exhibiting just how the term logic can shed need light on a variety of challenges that face any system of formal logic.

1 Liar’s Lookout

In the country of concepts only a series of successful and unsuccessful prosecutions for trespass suffices to determine the boundaries and rights of way.

Ryle

How wonderful that we have met with paradox. Now we have some hope of making progress.

Niels Bohr

These are old fond paradoxes to make fools laugh i’ the alehouses.

Shakespeare

1.1 Preliminaries: Our First Stop

A philosophical problem has the form: I don’t know my way about.

Wittgenstein

It is a wholesome plan, in thinking about logic, to stock the mind with as many puzzles as possible.

Russell

On his first expedition to Antarctica, Roald Amundsen and his crew were forced to spend the winter of 1898 in an ice field. Months of cold and darkness, coupled with inactivity, produced extreme physical and psychological disintegration in the crew. More food, less food, more exercise, less exercise, more sleep, less sleep – none of this had more than limited effect. The men had vitamin C deficiency, scurvy. As the British navy had learned a century earlier, citrus juices could alleviate the problem but until the 20th century no one could explain why. Amundsen and his men had no citrus fruit. Eventually the ship’s doctor noted that while well-cooked meat had no curative effect, lightly cooked, nearly raw meat (in this case penguin and seal) did. As it turns out, heat destroys vitamin C. (For much more on the expedition see Brown 2012, 28-32.)

Logicians and other philosophers so love the kinds of anomalies offered by paradoxes. The numbers of paradoxes, puzzles, and dilemmas is legion. In the country we are exploring, a good way to get name recognition is to dream up an example of a logical, mathematical, epistemic, moral, etc. paradox and attach one’s name to it. There are, for example, Moore’s, Russell’s, Curry’s, Yablo’s, Grelling’s, Richard’s, Cantor’s, and Berry’s. Other paradoxes have more colorful names. Unlike Elliot’s take on the naming of cats, the naming of paradoxes is not “a difficult matter” but more like “just one of your holiday games.” There are, for example, the Raven, Grue, No-No, the Ship of Theseus, the Hooded Man, the Truth-teller, the Arrow, the Racetrack, the ‘Heterological’, the Hangman, the Heap, the Prisoner’s Dilemma, Hilbert’s Hotel, not to mention what the Tortoise said to Achilles. Most famous of all is the ancient, but ever-popular Liar paradox.

Anomalies, of any kind, and wherever found, as the doctor on the Amundsen expedition surely knew, are instructive. They have causes. Until these are found, examined and understood, attempts to alleviate them or mitigate their unwelcome consequences will tend to multiply and yet be limited in their results. Logical anomalies, such as the Liar paradox (and its many cousins), have been subject to alleviate their symptoms. Moreover, many of these have involved theoretical explanations of the causes of such anomalies. Often these programs of logical preventative medicine are clever and sophisticated. Yet just as often they either address only a small range of the symptoms or require cures that may be worse than the disease. As with matters of health, even the seriousness of logical pathologies has been subject to dispute. Ancient Stoic logicians, especially Chrysippus, took Liar-type paradoxes to be lethal, threatening not only their logic but their epistemology and ethics as well (see Papazian 2012). By contrast, later medieval scholastic logicians saw such paradoxes to be little more than curiosities taking the Liar (the star among other paradoxical sentences – the so-called insolubilia) to be interesting but hardly dangerous (see Dutilh Novaes 2008). Contemporary mathematical logicians, especially after Tarski (Tarski 1935), look upon these paradoxes with something verging on horror. They are often seen as a frightening symptom of a fatal disease at the core of natural language, one that precludes the very possibility of any satisfactory definition or account of truth. Consequently, they have amassed a formidable arsenal to combat the ever-present threat of the Liar and its allies.

1.2 A Scenic (but “Risky”) View

Only paths that can be kept can be strayed from.

Ryle

First, a general statement of the paradox:

The Liar: A speaker says that she is lying (that what she is saying is false).

Needless to say, the speaker’s statement can be formulated in a number of ways (‘What I am now saying is false’, ‘I am now lying’, ‘This very statement is false’, etc.). The obvious question that confronts us is this: Is the Liar statement true or false? Neither answer will quite do. If it is true then it has the property ascribed to it – falsity; if it is false then is doesn’t have that property but rather the contrary – truth. It seems, per impossible, to be both true and false. Or else it seems, again per impossible, to be neither true nor false. As you would guess, some logicians are eager to accept the first choice (Liar-type statements are both true and false), others are happy to allow that such statements are neither true nor false. A very old, venerable principle of traditional logic is the Law of Bivalence, which holds that every statement is either true or false but not both. Both solutions above challenge the law. The first admits at least one truth value other than the standard truth values (true, false), namely both-true-and-false. This is often referred to as truth value glut. The second allows some statements (viz., paradoxical ones) to be neither true nor false, lacking a truth value, truth valueless. This is often referred to as a truth value gap. It goes without saying that gluts and gaps hardly exhaust the ways of dealing with Liars. Moreover, solutions for the Liar generate their own, further Liar-type paradox – the Strengthened Liar:

The Strengthened Liar: A speaker says (in effect) that what she is saying is false or both-true-and-false or neither-true-nor-false.

This is known as the Liar’s Revenge – if the Liar is defeated it will have its revenge via the Strengthened Liar (see Beall 2007). We will soon see that this revenge poses little more real danger than the Liar itself. For our present purposes we need only consider the classical Liar.

An immediately obvious and striking thing about the Liar is that it involves what some speaker says. Just what is it that a speaker says? Suppose someone, in an appropriate context, with an appropriate tone of voice, etc., utters the declarative sentence, “Je demeure à deux pas d’ici.” Now you ask me what the speaker said. I could accurately answer in several ways. Among them, I could

1. quote the speaker directly. She said, ‘Je demeure à deux pas d’ici’.

2. translate the sentence word for word. She said, ‘I live at two steps from here’.

3. translate the sentence into idiomatic English. She said, ‘I live nearby’.

4. quote her indirectly, using English. She said that she lives nearby.

5. quote her indirectly, using French. Elle a dit qu’elle demeure près d’ici.

This last answers your question by offering a reference to the proposition (that she lives nearby) which is expressed by any number of sentences (including the speakers original French sentence and its translations).

The salient distinction here is between sentences and propositions. Yet not all philosophers and logicians are sanguine about the prospects either of drawing this distinction or even countenancing the idea of propositions at all. For example, Saul Kripke, a formidable logician to say the least, has cast doubt on the utility of propositions (being “unsure that the apparatus of ‘propositions’ does not break down”) (Kripke 1972, 21). For him and many others, sentences are all the logician and philosopher of language needs. Propositions are unnecessary abstractions since sentences are the proper bearers of truth values. As Kripke says, “Sentences are the official truth vehicles” (Kripke 1975, 691, n.1). This may well be the “official” doctrine, but a case can be made that propositions are the proper bearers of truth and falsity.

Concepts and thoughts are prior (in every way, as Aristotle would say) to terms and sentences. The former are creatures of the intellect. They can be kept private, but often they are made public by expression. We humans use our language to express our concepts and thoughts (and we use it for much else besides). Whatever the ontological status of these two pairs might be, it is certain that the abstract products of conception and thought are profoundly different sorts of things from the perceptible bits of language used to express them. They are categorially distinct in the same way that prime numbers and prime ministers are categorially distinct. “There is a categorical difference between sentence and statement or proposition” (Goldstein 2006, 20). There are things we can sensibly (truly or falsely) say about the one kind of thing that we cannot say in the same sense about the other (unless we are speaking nonsense or using words nonliterally). Prime numbers are even or odd; prime ministers are competent or incompetent. Numbers aren’t the sort of things that can sensibly be said to be either competent or incompetent. Prime ministers are not the sort of things that can sensibly be said to be even, and though some are odd, they are not at all odd in the same way the 7 and 15 are odd. Sentences are the sort of things that can sensibly be said to be French or English or Greek; propositions are not. When I told you earlier what the French speaker said by quoting her indirectly, it didn’t matter which language her sentence was in; I could use any language to quote it indirectly. Sentences must be in some specified natural language, but the propositions they are used to express are not (though I must use some language to specify them). It makes no sense to say of a proposition that it is French or English or Greek.

Although they are originally products of language, propositions are not part of any language. Unlike the sentential utterance that brings a proposition into existence, the proposition itself is neither grammatical nor ungrammatical, neither English nor French and so forth. Strictly speaking, it is the proposition, not the sentence, that is true or false; when we call something that was said boring, original, false or plausible, we are characterizing a proposition and not the sentence that was used for saying it. (Sommers 1994)

Sentences are often used to express thoughts (propositions) with the implicit understanding that these thoughts are meant to be taken as true. Such sentences, when used this way, are statements (other kinds and uses of sentences are questions, commands, etc.). Since what is expressed (private, abstract concepts and thoughts) are categorially distinct from the vehicles of expression (publicly available terms and sentences), statements and propositions are categorially distinct. To understand an expression is to grasp the concept or proposition it is being used to express. If you understand French you know what our French speaker has said, otherwise you only hear her statement and are still in the dark about where she lives. I can express the same proposition using either an English or French expression, but the proposition is neither. Statements (again, sentences used to make a truth-claim) are the sort of things that can be written in ink, heard on a radio, typed in an email, etc. Propositions cannot sensibly be said to be written in ink, heard on a radio, or typed in an email. They can be said to be true, false, surprising, believed, doubted, well-understood, known for more than 2000 years, etc. Statements are said to have such attributes only secondarily, by being used to express propositions that have them primarily (again, they are prior in every way to statements). The proposition that the moon causes the tides can be sensibly said to have been well-understood 2000 years ago. Indeed, we could say that that proposition was true 2000 years ago. And this in spite of the obvious fact that the sentence ’The moon causes the tides’ could hardly be said to be well-understood or known 2000 years ago, there being no English sentences at that time.

Including ships, and shoes and sealing wax, cabbages and kings, we can talk of many things. Sentences are certainly among the things we talk about, and so are propositions. For example, we could say something about the sentence our French speaker used, saying of it that it is short, grammatical, and French. We could likewise say something about the proposition she expressed with her sentence, saying of it that it is true, profound, well-known. Given the fact that a speaker can talk just as well about sentences and propositions, a speaker could use a sentence to say something about any sentence, including that sentence itself. In that case we would say that the sentence is self-referential. Examples of self-referential sentences are ‘This sentence has five words’, ‘This sentence has 42 words’, ‘This sentence has the word “has” in it’, ‘This sentence is English’, ‘This sentence is French’. As well, a speaker could use a sentence to say something about any proposition, including the one that sentence is being used to express. For example: ‘The proposition expressed by this sentence is true’, ‘What I’m now ...

Table of contents

• Cover
• Title
• Copyright
• Contents
• Ded
• Getting Oriented
• 1 Liar’s Lookout
• 2 Ryle's Way With the Liar by Fred Sommers
• 3 The Logic Mountain Range
• 4 On the Term Functor Trail
• 5 The Four Corners
• 6 Referential Falls
• 7 Strawberry Fields
• 8 Into the Metaphysical Bogs
• 9 Ratiocination: An Empirical Account by Fred Sommers
• 10 Back to Logic Lodge, Base Camp
• Endnotes
• References
• Index