by E. John Winner
Most of what I know about logic I learned studying Aristotle, Kant, and Peirce. However, I did take an introductory undergraduate course in symbolic logic. The surprising take-away was that with a properly formed compound sentence, one could assert just about anything, and still hold the assertion, as a whole, to be true. In short, the standard symbolic logic course of the 1970’s was an indoctrination in epistemic relativism, and in a certain kind of rhetoric. A simple course in truth tables can contribute to public skepticism, which can then be manipulated rhetorically. So much for truth! 
In 1963, Edmund Gettier caused a minor controversy with a brief paper  demonstrating that certain rules governing the truth values of conjunctive propositions and disjunctive propositions effectively undermined the understanding of knowledge as “justified true belief,” for which he provides the generally accepted formula:
S knows that P IFF
P is true,
S believes that P, and
S is justified in believing that P.
“Suppose that Smith and Jones have applied for a certain job. And suppose that Smith has strong evidence for the following conjunctive proposition:
(d) Jones is the man who will get the job, and Jones has ten coins in his pocket.
(…) Proposition (d) entails:
(e) The man who will get the job has ten coins in his pocket.
Let us suppose that Smith sees the entailment from (d) to (e), and accepts (e) on the grounds of (d), for which he has strong evidence. In this case, Smith is clearly justified in believing that (e) is true.
The evidence is that the prospective employer has told Smith that Jones would be hired, and for reasons unknown, Smith has counted the change in Jones’ pocket. Unfortunately, the employer changes his mind and hires Smith, who has apparently forgotten that he himself has ten cents in his pocket. The proposition he believes before the hiring is announced is true, but not based on the logical justification he presumes for it. And if he doesn’t even know that he has ten cents in his pocket, he can no longer believe (e), so obviously this true proposition cannot be knowledge.
In the present case (known as “Gettier 1”), we immediately notice the clear lack of any sort of temporal verification process that we would expect in the real world. E.g., in the real world, at time X-1 (prior to the hire being announced), Smith would not know that “the man hired will have ten coins in his pocket.” He only has a hypothesis that this will be the case. Even if at X+1 (after the hire has been announced) he discovers the ten cents in his pocket, and thus may rightly claim that “the man hired has ten coins in his pocket,” he may also be aware that this had no dependence on his prior beliefs that Jones would get hired and Jones has ten coins in his pocket. The logical structure, reassuring Smith of his claim to knowledge, has completely broken down. But it was wholly artificial anyway, so such was inevitable.
The world of which Gettier writes is not the real world but a possible world, governed by the rules of a given logic. A real world Smith would not be too troubled about whether he holds a “justified true belief.” He’s just happy to get a job! And that he has ten cents in his pocket … well, there’s nothing much one can buy with ten cents these days. Maybe he’ll feel charitable and give it to Jones.
But in the logically possible world of the example, none of this matters. Motivation, emotional response, charitable behavior…all of these count for nothing. (Any method of verification or falsification would matter IFF we accept that Smith begins with a hypothesis, not a knowledge claim, and this isn’t allowed in terms of the given example.) Of course, Gettier doesn’t have to explain all of this in the context of his assumed readership.
The community of epistemologists for which Gettier was writing, trained in formal logic and having inherited many of their questions from the project of Logical Positivism, would have understood that Gettier’s cases are not problematic because of their “real world” application, but because of the formal-logic problems they pose. We can see this when Gettier remarks that knowledge is understood as “someone’s knowing a given proposition.” Note that what is known (or not) is not a thing, nor an idea, nor a theoretical model, nor anything other than a statement. This means in the case given that what Smith claims to know is not that the man getting hired has ten coins in his pocket, but the proposition “The man who will get the job has ten coins in his pocket,” which turns out to be true; but since it is derived from false inferences, Smith’s initial belief has no logical ground, and cannot be knowledge.
We can see this problem better in Gettier’s second example (“Gettier 2”):
“(f) Jones owns a Ford.
Smith’s evidence might be that Jones has at all times in the past within Smith’s memory owned a car and always a Ford and that Jones has just offered Smith a ride while driving a Ford. Let us imagine, now, that Smith has another friend, Brown, of whose whereabouts he is totally ignorant. Smith selects three place names quite at random and constructs the following three propositions:
(g) Either Jones owns a Ford, or Brown is in Boston.
(h) Either Jones owns a Ford, or Brown is in Barcelona.
(i) Either Jones owns a Ford, or Brown is in Brest-Litovsk.
Each of these propositions is entailed by (f).”
Here Gettier gives an empirical justification for (f); but the only justification for (g), (h), or (i) is the logical entailment made viable by its disjunctive truth-value:
A B A or B
T T T
T F T
F T T
F F F
(As long as one statement is true, the proposition is true.)
“But imagine now that two further conditions hold. First Jones does not own a Ford, but is at present driving a rented car. And secondly, by the sheerest coincidence, and entirely unknown to Smith, the place mentioned in proposition (h) happens really to be the place where Brown is. If these two conditions hold, then Smith does not know that (h) is true, even though (i) (h) is true, (ii) Smith does believe that (h) is true, and (iii) Smith is justified in believing that (h) is true.”
What Smith asserts is not who owns a Ford, and who is in Boston/Barcelona/Brest-Litovsk; he asserts the proposition “Jones owns a Ford OR Brown is in Barcelona,” which just happens to be true, and is empirically justifiable. That the first statement is false and any justification of the second statement are both unknown to him, however, so his assertion cannot constitute knowledge.
Notice that this also has little real world applicability. Why would a real world Smith be doing this? Is he making a bet? No, but logically-possible-world-Smith is testing the relationship between what logic says should be the case, and what the logical principle of knowledge says cannot be the case. The problem has to do with the formal structure of knowledge claims.
All of these problems can be traced back through the tradition that – thanks to promotions by Bertrand Russell and others – finds its main progenitor in Gottlob Frege. One of Frege’s key texts influencing that tradition was “On Sense and Reference” , and one finds in it several strategic missteps that played a significant role in the ultimate failure of the Logical Positivist project. Indeed, the very project itself was misguided. The Logical Positivists seemed bent on developing a “logically perfect language,” through which mathematics and the natural sciences could establish their claims: mathematics through a purified, deductive logic and the natural sciences through theories composed of true sentences concerning empirical reality. It never occurred to the Positivists (as it did to the Pragmatists) that mathematicians and scientists could develop their own logics and methodologies and their own languages for communicating their discoveries and inventions. When physicist Richard Feynman famously dismissed philosophers’ claims to provide explanations for the physics of the post-Einstein universe, he may have been thinking of some phenomenologist’s obscure ontological ramblings, but his remarks apply equally to Russell and Carnap. 
But this project, embedded implicitly in Frege’s text, is not its worst strategic error. That would be the decision to treat the sentential proposition as the principle bearer of truth, and the necessary object of logical analysis. In the development of logic after Frege, this decision has had beneficial consequences, including the development of symbolic logic and formal logic and computing languages. But this is a completely wrongheaded way to think of any language, including artificial or formal ones. A sentence has little meaning outside the context of a paragraph, which may itself have meaning, but more often requires the even larger context of the whole text. And in any case, it will require the context of a community of readers and writers to even make sense of it (even using Frege’s sense of ‘sense’).
This is made evident in Frege’s decision to draw many of his sentences for analysis from the texts of history. His remarks on the sentence “Napoleon, who recognized the danger to his right flank, himself led his guards against the enemy position,” is particularly embarrassing. It takes Frege some three paragraphs to admit that there is an implication in the sentence that Napoleon’s recognition of the danger to his right flank motivates his decision to lead his guards against the enemy position. Yet any reader of a history text in which such a sentence like this would occur would recognize it immediately and in passing. The sentence does make a truth claim, but it is structured rhetorically. Because of his refusal to treat the sentence structure as such, Frege’s discussion of it is hopelessly misguided. The principle structure of historical texts is narrational and cannot be reduced to the sentences used to construct such narratives. “Napoleon, who recognized the danger to his right flank, himself led his guards against the enemy position,” taken in and of itself, without the contextual narrative and without a proper accounting of its enthymemic structure, communicates little useful information.
Frege’s other major strategic error was to insist that propositions had a double referentiality: the “proper name” elements referring to entities and the proposition as a whole referring to a truth value. If correct, this would mean that every true propositions refers to “the True.” Rereading “On Sense and Reference” specifically to discover Frege’s working definition of truth or “the True,” I find direct reference to either, so all we’re left with is the sense. Given his discussion of the conditions determining the truth of the conditional “If the sun has risen, the sky is cloudy,” it seems clear he’s operating with the assumption of a correspondence theory of truth. What “the True” might be can only be surmised as a world described entirely with true propositions. Despite the very different logical approach, this would appear to be yet another attempt to achieve what Hegel termed Absolute Knowledge. But whereas for Hegel this would require an elaborate encyclopedia in order to account for the historical development of this knowledge, for Frege the final achievement would be a precise and logically ordered dictionary, and a thin manual on the logically proper grammar with which to use it.
I’m not going into an elaborate discussion of the technical arguments that have buzzed for years around the topics raised so far.  After all, I’m not a professional epistemologist, and I don’t come to these problems to try to make right of either Gettier or Frege or the justified true belief principle. I really want to say that they are all wrongheaded, and even if they aren’t, I don’t see what use they can be to anyone without an interest in professional epistemology. First, they are moot as far as our daily experience is concerned. No one goes around popping out propositions in order to test their truth value! No one cares if a sentence in a biography of Napoleon is logically well-ordered, merely that it is well written and persuasive, with enough documentation cited to be considered a feasible interpretation of events. Second, as far as their implied usage is concerned – purified deductive processes for mathematics, logically precise language for use in scientific inquiry – I have already remarked that this implication is also mooted, by the evident fact that most mathematicians and scientists have achieved success in their inquiries without recourse to such devices. 
But what if we were to develop understandings of logic, of language; of inquiry and invention; even of mathematics, that were not intended to provide foundation to the sciences, but instead derived from the logic, methodologies and languages that the sciences have already themselves developed, in order to explain how they developed, and what their commonalities are; and how they might intersect with common languages so that they could be better expounded to the non-specialist?
[A]ll the followers of science are fully persuaded that the processes of investigation, if only pushed far enough, will give one certain solution to every question to which they can be applied. They may at first obtain different results, but, as each perfects his method and his processes, the results will move steadily together toward a destined center. So with all scientific research. Different minds may set out with the most antagonistic views, but the progress of investigation carries them by a force outside of themselves to one and the same conclusion. This activity of thought by which we are carried, not where we wish, but to a foreordained goal, is like the operation of destiny. No modification of the point of view taken, no selection of other facts for study, no natural bent of mind even, can enable a man to escape the predestinate opinion. This great law is embodied in the conception of truth and reality. The opinion which is fated to be ultimately agreed to by all who investigate, is what we mean by the truth, and the object represented in this opinion is the real.
We shouldn’t get hung up on the “predestinate” character of the truth. Peirce only means that reality is waiting to be discovered, given the proper investigation. Or rather, given the convergence of investigations within the community of investigators.
For Peirce, knowledge comes as the result of inquiry. Truth is a product of the exhaustion of lines of inquiry that converge to the satisfaction of all those engaged in it. These inquiries themselves effectively establish the logics of their methodologies, which can then be summarized through logical statements. However, for the classically trained Peirce, these statements, as propositions and premises, are not the most basic material requiring analysis. Rather, the argument is. How sentences weave together convincingly was more important to him than their ability to stand up to logical parsing. A good methodology, well explained, effectively constitutes a good argument.
Peirce was a quirky character. He was very much an intellect of the 19th century. He was known and respected by professional logicians (much more so than his contemporary, Frege), and he hoped to produce a “systematic philosophy,” much like that of Hegel’s Encyclopedia. But his commitment to a socially contextualized theory of truth (which, as social, inevitably instigates further inquiry) and to the Pragmatic Maxim (which defines a concept by the consequences and actions that it would impel or necessitate) and finally his development of semiotics (dependent on the contingency of context) as the ground of logic, left him spending much of his later writing working and reworking his ideas and their terminology. Ultimately the implications of his Pragmatic theory of truth – and the logic and theory of knowledge it implied – had to be picked up by William James , and finally taken to the goal by John Dewey:
The present-day mathematical logician may present the structure of mathematics as if it had sprung all at once from the brain of a Zeus whose anatomy is that of pure logic. But, nevertheless, this very structure is a product of long historic growth, in which all kinds of experiments have been tried, in which some men have struck out in this direction and some in that, and in which some exercises and operations have resulted in confusion and others in triumphant clarifications and fruitful growths; a history in which matter and methods have been constantly selected and worked over on the basis of empirical success and failure.
The structure of alleged normative a priori mathematics is in truth the crowned result of ages of toilsome experience. The metallurgist who should write on the most highly developed method of dealing with ores would not, in truth, proceed any differently. He too selects, refines, and organizes the methods which in the past have been found to yield the maximum of achievement. Logic is a matter of profound human importance precisely because it is empirically founded and experimentally applied.” 
There is certainly a meaningful expression, ‘the truth’, but its meaning is probably found in the common language with all its rich history, rather than in epistemology. That we ask witnesses in law courts to swear or affirm to tell the truth (as they best understand it) in the recollection of their memories (with real world legal consequences), tells us more about the meaning of truth than any mathematical, logical, or epistemological insistence that there is something that we can call “the True,” which our mathematics or sciences should ultimately refer to or produce. 
 ‘Immigrants cross the border illegally and immigrants are employed instead of citizens’ is a truth-table verified true sentence; it just doesn’t remark immigrants who arrived legally, better qualified for the given jobs than citizens.
 See, for instance: https://www.youtube.com/watch?v=X8aWBcPVPMo
 For a brief survey, I suggest the article “The Analysis of Knowledge:” Ichikawa, Jonathan Jenkins and Steup, Matthias; Stanford Encyclopedia of Philosophy (Summer 2018 Edition), Edward N. Zalta (ed.), https://plato.stanford.edu/entries/knowledge-analysis/
 I’m aware of the deep relationship between formal logic and mathematics developed over the past century. But once formal systems achieve studies of inquiry in their own right, they no longer require the kind of epistemological justification for them Logical Positivism promised.
 Charles S. Peirce, “How to Make Our Ideas Clear” (1878)
 “Any idea that helps us to deal, whether practically or intellectually, with either the reality or its belongings, that doesn’t entangle our progress in frustrations, that FITS, in fact, and adapts our life to the reality’s whole setting, will agree sufficiently to meet the requirement. It will be true of that reality” James, The Meaning of Truth, 1909: http://www.authorama.com/meaning-of-truth-1.html
 John Dewey, Reconstruction in Philosophy (1920; revised 1948), pages 137-138
 Worth listening to here are discussions between Richard Rorty and Hilary Putnam. Rorty would rather that philosophers stop using the word ‘true’ all together, while Putnam replies (correctly, I think) that it is too meaningful to be abandoned, but admits that “theories of truth,” even the Pragmatists’, are largely unsatisfactory. https://www.youtube.com/watch?v=RnaaZOt78Ys