SpookyKitty 26 Report post Posted November 30, 2017 (edited) 1 hour ago, Grames said: The mathematical assertion of the equality of the two expressions on either side of the equal sign is tautological by definition. No it isn't. Nothing in the definition of either implies that the other is tautological. It's perfectly obvious you're just pulling this nonsense straight out of your ass. In fact, not only are not all statements of equality tautological, but there are obvious false ones such as 1 = 2. Quote The valid values y can take on for the independently variable value of x are limited by what makes the equality true. There is nothing "invalid" about substituting values for x or y which make the resulting statement false. In any case, you did not even remotely demonstrate that the given statement is a tautology because 1) you have not clearly stated the definition of a tautology and then 2) demonstrated how the given example meets all of the necessary conditions of being a tautology. Quote Truth tables simply map inputs and outputs and have little to do with truth in the epistemological sense relevant to philosophy. You really have no understanding of the subject at all if you think that's all truth tables are. 1 hour ago, dream_weaver said: Does a truth table for an engineering drawing count? If so, I've generated a few of them. Yes, that counts. Edited November 30, 2017 by SpookyKitty Quote Share this post Link to post Share on other sites
Grames 323 Report post Posted November 30, 2017 23 minutes ago, SpookyKitty said: In fact, not only are not all statements of equality tautological, but there are obvious false ones such as 1 = 2. And what makes the statement of equality 1 = 2 false? The two sides are not the same. This isn't hard. Quote Share this post Link to post Share on other sites
SpookyKitty 26 Report post Posted November 30, 2017 (edited) 4 minutes ago, Grames said: And what makes the statement of equality 1 = 2 false? The two sides are not the same. This isn't hard. Neither are the two sides of 2*3 + 1 = 7, and yet it is true. Edited November 30, 2017 by SpookyKitty Quote Share this post Link to post Share on other sites
gio 19 Report post Posted November 30, 2017 (edited) SpookyKitty, I advise you to watch the course on the History of Philosophy by Leonard Peikoff. This is not rationalism. Logic, like mathematics, ultimately comes from sensory experience, which is the way of grasping reality. There is no clash between percepts and concepts. Edited November 30, 2017 by gio Quote Share this post Link to post Share on other sites
Grames 323 Report post Posted December 1, 2017 35 minutes ago, SpookyKitty said: Neither are the two sides of 2*3 + 1 = 7, and yet it is true. Both sides are different ways of specifying a quantity, and the two quantities happen to be the same. 7 equals 7 and the tautology is complete. Quote Share this post Link to post Share on other sites
SpookyKitty 26 Report post Posted December 1, 2017 18 minutes ago, Grames said: Both sides are different ways of specifying a quantity, and the two quantities happen to be the same. 7 equals 7 and the tautology is complete. Your reasoning is completely circular. To say that 2*3 + 1 = 7 is true because both sides specify the same quantity is to assert precisely that 2*3 + 1 = 7. Quote Share this post Link to post Share on other sites
DavidOdden 92 Report post Posted December 1, 2017 Quote I don't understand what's so complicated about the concept of a tautology. Have any of you ever studied any formal logic? Who here knows what a truth table is? I’ve been there and done that, so maybe I’ll take a crack at explaining it to you. One of the basic concepts that an understanding of the term requires is that of “truth”, Quote the product of the recognition (i.e., identification) of the facts of reality. or, from How We Know, Quote truth actually pertains not to some match-up but to an awareness, a mental grasp, of the facts Cars exist, and they don’t have a truth value, in the parlance of philosophy. Propositions have truth value. At this point, we are somewhat stymied, without a definition of “proposition”. There is no controvery over the fact that propositions are not the same as sentences: Quote A sentence is a series of words; the proposition is the thought behind those words… The proposition is the cognitive content of the sentence, as distinguished from its linguistic form. Logic 150 deals with the topic entirely differently: it doesn’t have sentences, truth, of cognitive content. A truth table looks like this: P Q P ∧ Q T T T T F F F T F F F F There are also truth trees, and formal deductive systems like Kleene. “Truth tables” are about “T” and “F” – you can also use 1 and 0. “Propositions” in this approach are arbitrary atomic letters with no cognitive content. There remains a large unsolved problem of relating classical notions of proposition and truth to these kinds of tables. A formalist definition of “tautology” might be “a formula that always has the value T”, where “always” is a yet-to-be-defined weasel word. An example is P (P ∨ ¬P) T T F T With just bare-letter propositions and the three primitive Boolean operators, “always” refers to the fact that T is the value of the formula for any selection of T or F assigned to the primitive letters. Unfortunately, this cannot yet be connected to truth or cognitive content. The above is a canonical example of a formal tautology, though to compute that you have to go through a series of inferences (you refer to the reference tables for ∨, ¬, ∧ and supply the letter under the formula column that matches the given configuration of T and F for atomic letters on the left). We can call each such line an “imaginable world”: then a formula is a tautology if it has the value T in all “imaginable worlds”. A contradiction is a formula with the value F in all “imaginable worlds”. But again, this is strictly within the contentless formal approach. Things get more difficult when you move closer to real propositions, such as “Socrates drank”, “Aristotle rules” and “Plato blows”: here we have expressions with two words, and no logical connective. Truth tables require logical connectives. At the very least, we would like to be able to at least express these sentences, and their denials, the hope being that you might devise a definition of “tautology” and “contradiction” in the formal approach, once you venture into predicate calculus. The stumbling block is translating from such things with cognitive content (direct or indirect), to some content-free form. Objectivist epistemology prohibits context-dropping, while standard formalist approaches encourage it. Let’s say that we have a person Xocrates, and we know that he was born in Macedonia, he first went to Albania at age 18, he ate one fish in Albania, and he died the instant he ate that fish. In formalizing a proposition about Xocrates – “Xocrates went to Albania after his birth in Macedonia” – the formalist approach hopes to substitute some “definition” of terms into a formula. The proposition that he was born in Macedonia and that he first went to Albania at age 18 are clearly relevant to proving that Xocrates went to Albania after his birth in Macedonia, whereas one can intuit that the circumstances of his fish problem are not relevant, so formalists will drop that context. Formalists can even drop extremely relevant context, such as the fact that birth entails existence. It is just as contradictory to say that Xocrates was born and did not exist, as it is to say that being whipped and burned is the same thing as not being whipped and burned. The sentence “For all integers x, there exists an integer y such that 2x + 3 = y” can probably be translated into some symbolic form, and it would prove to be a tautology if you have the correct translation. In all imaginable worlds (‘for all imaginable values of x and y’), this formula (or would-be formula, if we could fully formalize it) has the value T. So one problem with “tautology” is that some people seem to think that “For all integers x, there exists an integer y such that 2x + 3 = y” is not a tautology (and I’ll assume that this is a tre statement and not an error). Thus there must be at least two definitions of tautology, which is a problem. Actually, it’s well-known that there are zillions of definitions of tautology: it’s not really a concept, it’s more a meme, a vague notion that lives somewhere. 1 William O reacted to this Quote Share this post Link to post Share on other sites
Grames 323 Report post Posted December 1, 2017 4 minutes ago, SpookyKitty said: Your reasoning is completely circular. To say that 2*3 + 1 = 7 is true because both sides specify the same quantity is to assert precisely that 2*3 + 1 = 7. Is it less circular if the equality were to be 2*3 + 1 = 5 + 2 ? Quote Share this post Link to post Share on other sites
SpookyKitty 26 Report post Posted December 1, 2017 @DavidOdden Yes, I'd say you understand the formal definitions of tautology in propositional logic perfectly. However, we have a substantive point of disagreement where you say that formal propositions involve context-dropping and weasel-wording. When we assert a formal proposition such as "p or not p" we are not context-dropping but rather abstracting over all possible meanings of the atomic proposition P. You are further incorrect in saying that this has no connection to any non-formal notion of truth. The connection is that it does not matter what actual proposition you substitute for p in the formal tautological statement. When the substitution is done, the resulting proposition must be true. That such results can be proved is precisely why formal languages are so useful and general. You are correct in saying that sometimes it is not clear how to formalize certain natural language expressions. However, this does not imply an ambiguity or any other kind of problem in the formalism, but rather a possible ambiguity or confusion in the natural language expression. When interpreting a formal language over some object domain, a correct interpretation requires that every symbol have exactly one meaning, so no confusion can possibly result. Natural languages do not have this requirement, which is why miscommunication is such a common problem with them. (Also why you would never want to use a natural language to deal with mathematical abstractions if it can be reasonably avoided). Quote The sentence “For all integers x, there exists an integer y such that 2x + 3 = y” can probably be translated into some symbolic form, and it would prove to be a tautology if you have the correct translation. In all imaginable worlds (‘for all imaginable values of x and y’), this formula (or would-be formula, if we could fully formalize it) has the value T. So one problem with “tautology” is that some people seem to think that “For all integers x, there exists an integer y such that 2x + 3 = y” is not a tautology (and I’ll assume that this is a tre statement and not an error). Thus there must be at least two definitions of tautology, which is a problem. This is an error in understanding in how to interpret statements in first-order logic. Interpreting a statement in first-order logic is a little bit more involved than interpreting statements in propositional logic. According to wikipedia, https://en.wikipedia.org/wiki/First-order_logic Quote An interpretation of a first-order language assigns a denotation to all non-logical constants in that language. It also determines a domain of discourse that specifies the range of the quantifiers. The result is that each term is assigned an object that it represents, and each sentence is assigned a truth value. In this way, an interpretation provides semantic meaning to the terms and formulas of the language. The study of the interpretations of formal languages is called formal semantics. What follows is a description of the standard or Tarskian semantics for first-order logic. (It is also possible to define game semantics for first-order logic, but aside from requiring the axiom of choice, game semantics agree with Tarskian semantics for first-order logic, so game semantics will not be elaborated herein.) The domain of discourse D is a nonempty set of "objects" of some kind. Intuitively, a first-order formula is a statement about these objects; for example, ∃xP(x){\displaystyle \exists xP(x)} states the existence of an object x such that the predicate P is true where referred to it. The domain of discourse is the set of considered objects. For example, one can take D{\displaystyle D} to be the set of integer numbers. The interpretation of a function symbol is a function. For example, if the domain of discourse consists of integers, a function symbol f of arity 2 can be interpreted as the function that gives the sum of its arguments. In other words, the symbol f is associated with the function I(f) which, in this interpretation, is addition. The interpretation of a constant symbol is a function from the one-element set D^{0} to D, which can be simply identified with an object in D. For example, an interpretation may assign the value I(c)=10{\displaystyle I(c)=10} to the constant symbol c{\displaystyle c}. The interpretation of an n-ary predicate symbol is a set of n-tuples of elements of the domain of discourse. This means that, given an interpretation, a predicate symbol, and n elements of the domain of discourse, one can tell whether the predicate is true of those elements according to the given interpretation. For example, an interpretation I(P) of a binary predicate symbol P may be the set of pairs of integers such that the first one is less than the second. According to this interpretation, the predicate P would be true if its first argument is less than the second. If we restrict the domain of discourse to the integers {0,1,2,3}, then the statement “For all integers x, there exists an integer y such that 2x + 3 = y” is false, and is therefore not a tautology. Quote So one problem with “tautology” is that some people seem to think that “For all integers x, there exists an integer y such that 2x + 3 = y” is not a tautology (and I’ll assume that this is a tre statement and not an error). This is completely wrong. So wrong that there in fact exists an effective method for determining whether a given first-order statement is a tautology, i.e., it's so easy that a computer can do it. Quote Thus there must be at least two definitions of tautology, which is a problem. Actually, it’s well-known that there are zillions of definitions of tautology: it’s not really a concept, it’s more a meme, a vague notion that lives somewhere. Even if this is true, this only suggests a problem with the terminology and not the concept. in a similar fashion, just because the English language uses the same word to refer to both a kind of fish and a stroke of luck (fluke) that in no way implies a problem with either of the two concepts in and of themselves. 35 minutes ago, Grames said: Is it less circular if the equality were to be 2*3 + 1 = 5 + 2 ? No. Your circularity now just involves two statements instead of one. Quote Share this post Link to post Share on other sites
dream_weaver 472 Report post Posted December 1, 2017 3 hours ago, SpookyKitty said: 4 hours ago, dream_weaver said: Does a truth table for an engineering drawing count? If so, I've generated a few of them. Yes, that counts. The more difficult element to the truth table on the engineering document is validating that it is, indeed, true, before sending it to the production floor. Gio suggested Leonard Peikoff's History of Philosophy courses (available for free in the course section at the Ayn Rand Institute web site). Leonard Peikoff also does an Introduction to Logic course, if it is logic, and how it is derived from the evidence of the sense that is being sought. It is available at the e-store. Spoiler If it is the mathematical angle desired, consider proofing this breakdown of this old conjecture here, or the core of the breakdown here, here, and here. It would be nice to have a relevant angle pointed out that hasn't been pursued on it yet. The key term is relevant, if it can be persuasively pointed out or indicated. There is a thread here pertaining to it. It is hyperlinked from the thread there. Post to either, as both are currently being regularly monitored. Quote Share this post Link to post Share on other sites