Bob's article was written after I prodded him several times between 1974 and the late '90's to expand his brief seminar notes. Here are the quotes from his seminar hand-out notes.
(10) Many years ago, I was given a boost towards this formulation by Anatol Rapoport. See his "What is Semantics?" (sic.) in S. I. Hayakawa, ed., The Use and Misuse of Language. Greenwich CT: Fawcett Publications, 1962, pp. 11-25
- grammar, wherein word-to-word relations are examined; basic unit: sentence
- logic, wherein statement-to-statement relations are examined, basic unit: paragraph
- semantics, wherein word-or-statement-to-referent relations are examined
- general-semantics, wherein all the above and non-verbal world (physics, neurobiology, etc.)-human-evaluations/ behaviors relations are examined, evaluated, and, prophylactically and amelioratively taught about (10)
When Bob's work was not forthcoming, I decided to prepare my own version, which I sent to him in the mid 1990's. Bob never responded to my prodding nor to the article I sent him, except for a brief verbal reply to the effect that he should probably do so. I believe this took place at a Korzybski lecture in New York, but I don't remember which one. It's fair to conclude that Bob's paper was written partially as a response to my prodding and/or paper. Based on the content of his paper, I understand why it was not prepared earlier. Bob's understanding, even with my paper, was simply not up to the task. With regard to the distinction between "true" and "valid", Bob just didn't get it. The main problem seems to be his inability to keep the levels separate and the failure to discern specific multiordinal differences in the use of the word 'true' - "true" at language level 2 (logical) is not "true" at language level 3 (semantic), and Bob seems unable to distinguish between these two. At the logic level, the meaning of "true" is best illustrated by truth tables. A sentence (proposition) is "true", that is, it gets a truth value of "T". If the sentence is an axiom or a major premise, then the value of "T" may be just assumed. At this level there is no correspondence relation to any state of affairs in the world. The meaning of the word "true" at the level of logic is simply an arbitrary value. What is important is that there are methods of analyzing such a sentence in conjunction with other sentences that will allow this same value of "true" to be correctly applied to a sentence whose value is not assumed. Such sentences are "conclusions". These methods of analysis take into consideration certain rules of grammar (language level 1). A couple of statement which may reasonably be assumed to be "true" will be used as "premises", and, from those sentences, a conclusion will be "inferred". The method of inference is what is critical. Certain methods always produce "true" conclusions from "true" premises, but certain other methods do not.
-- grammar: word to word relationships (structure)
basic unit: sentence
-- logic: statement to statement relationships (structure)
basic unit: paragraph
-- semantics: word/statement to referent relationships (structures)
straddles the verbal/non-verbal domains; these relationships usually formulated in formal-technical, often mathematical, language
-- general-semantics: Language-nervous system (organism-as-a-whole-non-verbal levels-human behavior relationships (again, structures)
Each discipline listed above incorporates the concerns of its predecessor (as listed).*
Methods that always produce true results, given that the premises are true, are called "valid", and methods that do not are called "invalid" or fallacious.
The terms 'valid' and 'invalid' are specific technical terms that applies to a paragraph of three (or more) statements that form a specific example of a reasoning method or an argument that relates sentences at logic levels of language.
So, at logic levels of discourse, statements or sentences are described or evaluated as "true" or "false" and paragraphs or arguments are described or evaluated as "valid" or "invalid" (fallacious).
In Bob's article on page 85, Bob fails to understand this distinction when he writes
statements that may be evaluated as valid (related to what he called "consequence" and others refer to as "tarskian biconditionals")(4) and those that may be evaluated as true (or false).
Bob goes on to say,
In much philosophical writings (most that I have read), authors shuttlecock between true and valid, suggesting that for them the terms are interchangeable. (85)
In my experience with philosophical and mathematical writings, the distinction is clear. Statements are true or false, while arguments are valid or invalid, and every (well-formed) argument requires at least three statements. "Valid" applies to the relation among statements. An argument is valid if and only if it yields true conclusions from true premises. It is important to note that the validity of an argument does not depend on anything that the words in the sentences might refer to. Those are semantic considerations that are not relevant to logic, and this may be where part of the confusion comes from. The (logical level) part of the meaning of "true" depends only on the assumed (truth) value of the statement or if it is the conclusion of a valid argument using "true" premises. Only the grammatical form of certain well-formed sentences arranged in the correct form of a valid argument produce well-formed conclusion sentences, and their truth value depends only on the assumed truth values of the premise sentences.
Yet the bulk of Tarski's work is concerned with the "consistence theory of truth," with relations among statements such that the statements are 'true' if they are well formed and deductively (concludingly) coherent. Tarski's internally consistent 'true' is what I call valid. (86)
It looks like Bob has "almost got it" here, but the subsequent text shows that he does not.
I have plainly said that I offer the valid/true distinction as a simple, homemade formulation that can help in alerting you to when you've come to the boundary between observation as speculation; a most parlous crossing point if we are not sufficiently conscious of abstracting. (86)
It's clear from the above that Bob is applying "true" or "false" to sentences that represent observations - particularly at lower levels of abstraction. For Bob, a statement can be "true" or "false" if it represents an observation statement that corresponds in some structural way to the non-verbal observations, and that he applies the term valid to a higher level, "inference" statement. Bob is using both terms to apply to sentences, distinguishing them into categories related to the level of abstraction of the sentences in relation to the abstraction and semantic properties of the sentences.
The "correct" distinction between "truth" and "validity" is that one applies to single sentences, and the other applies to paragraphs containing three or more sentences. There is a lay tendency to use the phrase "valid conclusion" and this is a probable factor in the conclusion. Technically, "valid conclusion" means a "conclusion obtained using a valid argument and true premises".
That Bob is biased by this non-technical usage is illustrated by his next statement.
In distinguishing merely valid from true statements, we need to first consider the difference between statements of 'fact' and statements of inference. (86)
This is the difference between a premise and a conclusion, not the difference between valid and true.
Bob is nearly obsessive about formulations being seen as always having a semantic or referent aspect. He cannot take a view that looks at the structure and formal relations among sentences without regard for possible semantic considerations. This "bias" is understandable from the point of view of the general semantics idea of non-elementalism to split verbal words from their non-verbal referents or object level semantic reactions. To understand logic, however, this is exactly what must be done - but not to worry, we "re-assemble" the parts when we add semantic level structure. It should be noted that we are building a multi-level model one level at a time. See Levels or perspectives on the use of language.
I suggest a cautionary definition of a statement of fact, setting minimum requirements for accepting some statement as factual, while remaining formulationally dry under the Uncertainty Umbrella. (86)
Statements of fact and speculation both involve semantic level considerations, and neither of these are even relevant to the true/valid distinction. Once again, Bob is unable to understand the distinction between levels of abstraction that distinguishes logic from semantics. The entire discussion about statements of fact versus statements of inference is digressing into matters unrelated to the distinction between truth and validity.
This entire discussion in only relevant to a discussion of semantic issues where one is concerned about how representative statements are with respect to the non-verbal world they are taken to be about.
I reiterate, a statement's having met the minimum requirements won't guarantee that it's 'true'; we know that people can hallucinate in private and together in public. (87)
A hallucination is "seeing something that is not present", Bob is setting up a metaphor that suggests that a statement corresponds to its referent analogous to seeing corresponding to that which is seen, and in the case of the hallucination, the seeing occurs without that which is seen. The analog is that the statement that has no referent cannot be true. He is specifically invoking the correspondence theory of truth here.
That Bob is confusing the distinction between observation statements (fact) and conclusion statements (inferential) with the distinction between truth and validity can be seen again on page 87-88.
But as long as statements retain the inferential character, then can most strongly be evaluated as valid, i.e., verbally consistent with what has gone before. (87-8)
Bob goes on to relate statements whose inferential character is uncertain to a third truth value, "indeterminate in varying degrees", and he appears to be relating this directly to three-valued logics.
Thus the three-valued (and more) logics of Lukasiewicz and Tarski: statements may be true, false, or indeterminate in varying degrees. (88)
Unfortunately, these multivalued logics are not in the business of dealing with statements about the world as we know it. Three (and more) valued logics are strictly logic level constructs, and the "truth values" of the statements in these logics are arbitrary symbols from a set of three or more tokens which includes the conventional symbols "T" and "F". None of these "truth values" correspond to what we mean when we describe the relation between a statement about the world and the non-verbal conditions of that world.
We can see further evidence of Bob's confusion on page 88.
We can distinguish Tarski's logical 'truth,' expressing verbal/verbal relations (what I prefer to limit to validity) from Tarski's "semantic truth," expressing symbol/referent relations, both verbal/verbal and verbal/non-verbal, the level at which 'true' and 'false' may apply. For example, if John says "I am sick" and I, making what Tarski called a metastatement, say "John said 'I am sick'" (not "John is sick," which would be an inference on my part), John's statement serves as a referent for my statement, and my statement can be evaluated as true. If I report, "There's a rock on the road," the rock and the road, provided I'm not hallucinating, qualify as the referents for my statement. If I am hallucinating, then I as brain am the referent for my statement. (88)
Bob's examples are strictly natural reference examples that involve "real world" semantic relations. At logical levels of discourse, with which Tarski deals, "truth" expresses a property assigned to a statement, not a verbal/verbal relation, and it certainly does not express a verbal/non-verbal relation. "True" and "false" apply to statements. In what is now called "formal semantics" or "model theory", the structure consists of a language of tokens, a set of object tokens, a set of relations among those tokens, an assignment function from tokens in the language and object tokens and relations, and rules for forming well formed statements. A statement specifies a relation between language tokens. The statement is said to be "true" (in the model) if the objects assigned to the language tokens satisfy the appropriate relation on the objects.
We have an advantage over Tarski today; we have the benefit of decades of refinement and simplification of his early ideas. Tarski was developing these ideas, and, as we all know, the first formulations often need to be revised and improved as a result of effective time-binding. So, here is a very simple example that has all the minimum requirements.
|expanded explanation||compound statement truth value|
|P and Q are arbitrary well formed statements||P||Q||P&Q||P|Q||P>Q||P=Q|
|If statement P has the value "T" and||Q has T||T||T||T||T||T||T|
|Q has F||T||F||F||T||F||F|
|If statement P has the value "F" and||Q has T||F||T||F||T||T||F|
|Q has F||F||F||F||F||T||T|
To understand Tarski's "semantic" truth let's look at two simple statements X(A,A) and X(A,B)
X "refers to" relation x, and "A" refers to "a", so we must look at the relation x and see if x(a,a) is in the relation, and we see that it is. X(A,A) "is true" if and only if the relation x (referred to by X) is satisfied by the object tokens a and a, and since it is, then the statement X(A,A) is "true" in this model. Now, let's look at X(A,B). For this to be "true", the relation x must include (a,b), but we can see that it does not, so the sentence X(A,B) is "false" in this model.
Can you figure out what relations X, Y, and Z correspond to in our normal experiences? Click here for the answer.
Now that you understand the basic principle of Tarski's "semantic truth", what is now called model theory, we can continue looking at Bob's discussion.
On page 89 bob says,
So the conclusion, "Alphonse is a four-floogled hogwash," is valid but very probably not true.
Note that bob has used the word 'valid' in the lay sense of a "valid conclusion", which means a conclusion which is logically dependent upon the truth of the premises in an argument which has a valid structure. Here Bob has applied the word 'valid' to a statement, and that is not technically correct. The statement is "wrong" on two counts. The Argument is valid, so the "truth" of the premises carries through to the conclusion. Bob is unable to hold the notion of "true" as an arbitrary value being propagated from the assumptions to the conclusion. If premises are "true" and the argument is valid, then the conclusions are "true", and the meaning of "true" in these cases, is the "true" of logical levels. This "true" is NOT the same as it's multiordinal alternative "true" at semantic levels, where correspondence in structure is required.
Bob went on to deny that the conclusion is "true" in this strictly logical context. To do so is to confuse the non-logical context of factual type statements with the logical context. Can we go out and find a four-floogled hogwash? Most likely not, but this is an empirical question, not a logical one. Here again is evidence of Bob's obsession with finding physical referents to associate with statements; he is unable to understand the logic level of analysis taken in strict isolation from "facts". Bob's example argument is valid, and the "truth value" of the conclusion statement depends strictly on the assumed truth values for the premise statements. If they are assumed to be true, then the conclusion is equally true.
But he expressly denies this possibility:
Thus, while there's no particular advantage to be ungrammatical or illogical outside of poetry or play writing (writing a play), correct grammar and impeccable logic are not enough. (89)
Bob's view here is that semantic considerations must always be involved.
Further evidence that Bob is unable to separate statements of logical relations from statements with physical referents can be seen next.
At the level of semantic analysis, we first test for truth-value in Tarski's second sense. We first break out of the "neurolinguistic 'circle'" to check the relation between what we say and what we say refers to; eventually, what we can observe. We become extensional. (89)
Once again, Bob is harkening back to empirical observations in the "real world". At this point I can suggest another description of his difficulty. Consider that the word "true" is multiordinal; it "means" different things at different levels of abstraction, but those different things are related in a coordinated way. Tarski's insightful analysis provided a highly technical and formal model designed to shed light on the correspondence theory of truth. A statement is true if it corresponds to (has structural similarity to) the state of affairs in the world. Tarski built a special world and a special language, and he defined the correspondence relation between that language and that world precisely. He then show what he understood "true" to mean by defining it in terms of his special world and language. (You can review the simplified version above.).
For logic, "true" names the truth value "T", For model theory, "true" names the condition of a statement having objects that satisfy the relation specified in the model statement. For empirical science, "true" names the conditions of a statement not having been disconfirmed. In the former two cases the sense of "T" and "true" are absolute, but in the later case the sense is more probabilistic in that, according to the current modern philosophy of science, we can never know when a (theoretical) statement about the world is "true" in any absolute - can never be wrong - sense (the philosophers "strong" sense). Direct observation statements can perhaps be "true" in a somewhat "absolute" sense - subject to the interpretation and conventions behind the words used in the formulation of such statements. It is, however, somewhat pathological to attempt to "force" the later sense of true "backwards" onto the sense of "true" in model theory and the sense of "T" in logic truth values. Just because "true" has evolved with our understanding of how we relate to the world as hypothetical map makers and users does not mean that the earlier sense are completely obsolete; it just means that the context of the uses of the separate senses of "true" is more precisely determined.
On page 90, Bob says,
At the level of [T]arskian semantics, not unlike a chick breaking through its shell, we begin to partially 'break out' of the neurolinguistic 'circle'; we haltingly straddle the verbal-non-verbal nexus. Here we can check to see if our grammatically correct and logically valid formulating has resulted in 'true' statements.
This is a limited view of "true" only in relation to "actual" physical objects subject to investigation with respect to common usage. I regret to say that this shows a lack of understanding of Tarski. Tarski did not, at any time, present a language that "refers" to non-verbal levels. All the objects that Tarski's language "refers" to are linguistic verbal tokens. The objects are not anything that the lower case letters "refer" to, they are the lower case letters themselves. All the things that are mapped to by the assignment function from language tokens to object tokens and relations are strictly specified linguistic items. It is only an analogy suggesting that Tarski's language tokens correspond under the analogy to our words, that Tarski's object tokens correspond under the analogy to our non-verbal objects, and that Tarski's language relation tokens correspond to non-verbal structures in the world. But it is a failure to remember that this is only an analogy to say that Tarski is "straddling" the verbal-non-verbal nexus. Tarski intends to use the analogy in the reverse direction. He intends to say that this model he is showing us shows the structure of what we mean by the correspondence theory of truth in its simplest and purest form - where there are no uncertainties. It is a marvelous breakthrough in both philosophy and mathematics. It presents a clear and concise model. But the model suffers from one major inadequacy. In the empirical world we do not have any direct access to the "assignment function". Knowledge of it is only empirically gained, and what we do know is strictly the result of our abstraction process.
Bob is trying to reconcile the differences by understanding logic and Tarski only in terms of empirical uncertainty. Starting at the bottom of page 90 he says,
Achieving and maintaining a sharp awareness of the difference between validity and truth, inference-type and fact-type (descriptive-reportive) statements, lower order and higher order inferences ... can facilitate ...
Here he has again associated validity with inference-type and truth with fact-type (as he had done earlier), but he reverses his parallel construction with the next pairing, which should be higher-order and lower order inferences, in order to continue the hierarchy. While "valid" is a characteristic of a collection of sentences in a certain form and truth is a characteristic of single sentences, these in no way correspond to inference-type statements in general and fact-type statements in general. An "inference-type" statement would be a "conclusion" statement arrived at using valid reasoning methods, but it could also be a "conclusion" statement arrived at using invalid reasoning methods such as Korzybski pointed out and called "un-sane" behavior.
Here are two tables illustrating the different ways of putting these structures together.
For Bob, any "absolute" sense of "true" or "false" is completely rejected, so he aligns "truth" with "fact" and "valid" with conclusions of a certain type. In the more "correct" view shown on the right the two areas can be aligned with either the absolute logic on top or on the bottom. In theory and in language levels absolute logic comes first and provides the guidelines necessary for developing the probabilistic-empirical understanding. It also evolved this way. But is is also possible to see this as a higher level abstraction arrived at in our brain-mind-culture over millennia of examining our experiences in life. It's an over-simplified, high-level, abstract way of modeling relations between statements derived from millennia of analysis, so you could just as easily reverse the two major row sets in the right. The view presented here agrees with the four levels of language analysis presented by both Bob and myself.
If you take nothing else away from this discussion remember this.
|This page was updated by Ralph Kenyon on 2009/11/16 at 00:28 and has been accessed 27574 times at 357 hits per month.|