'Formal Logic' (of no use for philosophical work)

 

[under construction]

 

The term "logic" is often used for (the study and use of) what is commonly called "formal logic"--the vocables, rules and definitions of predicate logic, propositional logic, and some special supplements, "non-classical" logics, such as termporal logic and epistemic logic.

 

 

Purposes ascribed to 'formal logic'

 

These "logics" are not (directly) concerned with the study and application of the rules of rational/reasonable thinking and working. Rather, the provide those vocabularies, rules and definitions. What is the purpose these "logics" are supposed to serve?

 

(1) Formal logic is supposed to serve the precisification of philosophical theses and arguments 

(2) Formal logic is meant to serve the discovery of logical flaws in philosophical theses and arguments

(3) Learning and applying formal logic ensures logical thinking

 

The main problem with 'formal logic' is that in fact, none of these putative purposes is actually served by formal logic. Indeed, there seems to be no relevant purpose which 'formal logic' serves in fact.

 

 

The small expressive power of 'standard logic'; "seven signs" 

 

Let the "standard logic" encompass propositional logic and predicate logic. Standard logic offers a limited set of expressions which can be used for the representation, and (allegedly) for the precisification of statements. The expressive power of standard logic is small. Its vocabulary includes what we may call "the seven signs" (where "seven" is not meant quite literally: give or take a bit). These are, roughly, (1) negation, (2) conjunction, (3) adjunction, (4) material implication, (5) material equivalence, (6) the general quantifier, and (7) the existence quantifier.

 

 

The "formalization" of statements 

When you "formalise" a statement, what you do is: you search it for any instances of the seven signs. Whenever you find one, you replace that instance by the corresponding formal term. All remaining expressions are faded out: in their place, you insert a (meaningless) letter as a 'placeholder'. What you arrive at is a very sketchy 'propositional frame' of the statement, in which all instances of any of the seven signs is documented, all the rest being faded out.

 

 

#(1) Formal logic does not serve the purpose of precisification

 

Consider what we may call the "central philosophical terms"--terms such as "exist", "instantiate", "constitute", "colour", "sound", "thought", "live", "conscious", "mind", "state", "justice", "mean", "understand", "interpret", etc., which are required to express the content of philosophical statements. Formal logic could precisify philosophical statements only if its vocabulary included precise pendants to these terms. For example, when your aim is to precisify the statement "Consciousness supervenes on material states", you need precise terms expressing the notions 'consciousness', 'material', 'state' and 'supervene'. As a matter of fact, the vocabulary of standard formal logic does not include such terms. Its expressive power is, niceties aside, limited to the seven signs. Accordingly, aside from occurrences of the seven signs, philosophical statements cannot be represented, and hence not be precisified by standard logic. 

 

You may be tempted to ask: "What about special 'logics', such a 'modal logic', 'temporal logic', or the logic, say, 'of communication'? Don't these offer additional precisified terms, over and above the 'seven signs'?" The answer is: (1) Admittedly, these 'logics' enrich the 'logical' vocabulary--but only by a small handful of terms. Furthermore, (2) how these terms are eventually to be defined is a matter of dispute in more or less every case--which leads the purpose of precisification ad absurdum. So quite apparently, formal logic is not capable of serving the purpose at hand, the precisification of philosophical statements. Everything except of the seven signs, plus a handful of other notions, at most, is by no means precisified, but in fact just faded out.

 

 

#(2) Formal logic is not a suitable means for the detection of logical flaws

 

The idea that formal logic can disclose logical flaws in statements goes back to the view, popular among the "logical positivists", that many philosophical theses are (as they used to put it) "metaphysical", in the sense of being 'logically inconsistent', or (cognitively) 'meaningless', or 'senseless', or 'non[-]sense'.

 

o Carnap, "Überwindung der Metaphysik durch logische Analyse der Sprache" 1931

 

Auf dem Gebiet der Metaphysik (einschließlich aller Wertphilosophie und Normwissenschaft) führt die logische Analyse zu dem negativen Ergebnis, daß die vorgeblichen Sätze dieses Gebietes bietes gänzlich sinnlos sind. [...] Zwar finden sich verwandte Gedanken schon in manchen früheren Überlegungen [...]; aber die entscheidende Durchführung ist erst heute möglich, nachdem die Logik durch die Entwicklung, die sie in den letzten Jahrzehnten genommen hat, zu einem Werkzeug von hinreichender Schärfe geworden ist. " (Carnap 1931, 220)

 

By "Werkzeug von hinreichender Schärfe", Carnap seems to refer to formal logic. So the "logical analysis" he refers to seems to be the formalization of statements by means of formal language.

 

According to Carnap, there are two types of "metaphysical" statements:

(1) Sentences which contain words which have no meaning (or do not designate something). As examples he gives (1.a) 'Manche Dinge sind babig' ("Some things are 'babig'"), where the word "babig" does not exist, is undefined; and sentences which contain the word "principle" or the word "God", when these words do not designate something, such as (1.b) "x is a god" and (1.c) "x is the principle of y" (Carnap 1931, 224-227).

(2) Sentences whose elements actually are meaningful, but shose concatenation 'makes no sense' ("Sie bestehen aus Wörtern mit Bedeutung, sind aber aus diesen Wörtern so zusammengesetzt, dass sich doch kein Sinn ergibt."). Examples are (2.a) "Cäsar ist und" ("Caesar is and"), as well as (2.b) "Cäsar ist eine Primzahl" ("Caesar is a prime number") (Carnap 1931, 227)--because 'prime number' is a property of numbers, which cannot be ascribed to persons (Carnap 1931, 228).

 

Carnap never demonstrates how formal logic is supposed to achieve this task. the reason seems to be that it actually does not. The reason is that it lacks the means. Consider what the formalisation of Carnap's examples delivers. 

(1.a) "Manche Dinge sind babig" => [E]x F(x) [Be "[E]" a sign for existential quantification]

(1.b) "x is a god" => F(x)

(1.c) "x is the principle of y" => F(x,y)

(2.a) "Cäsar is und" => ??? 

(2.b) "Cäsar is a prime number" => F(x)

 

The proof of the pudding is in the eating. As soon as we execute the formalisation of "metaphysical" sentences in some cases, we see quite clearly that this does not lead to the disclosure of their logically problematic nature. In one case, (2.a), formalization does disclose the problem at hand, lack of grammaticity; yet in this case formalization makes no contribution, because the problem is perfectly obvious without formalization. In each of the other cases, formalization goes quite smoothly, which means: the logical problem (insofar as there is one) is not being disclosed. The disclosure of "metaphysical" sentences is not something formal logic can achieve. The reason is, roughly, that it lacks the means of representing the content of a statement: it merely represent their logical form.

 

 

#(3) Learning and applying formal logic does not ensure logical accuracy

 

The third purpose which is regularly ascribed to 'formal logic' is the improvement of logical thinking. (Actually, this is precisely what the study of 'real logic' is supposed to achieve. The belief that 'formal logic' achieves this effect surely goes, to a certain extent, back to fallacy, which is rather widespread, that 'formal logic' be (an important part of) 'real logic'.) 

 

In fact, the study and use of 'formal logic' does not seem to have any remarkable effect on the logical capacities of its users. Of course, it is obviously hard to show (as well as perhaps strictly speaking false) that it has really no effect on logical thinking at all. It is even hard to show that it has no 'relevant' effect--partly, because the term 'relevant effect' is stretchable. Yet in any case, the notion which the believers in 'formal logic' tend to have, that at least the champions of 'formal logic' are largely immune against the perpetration of crude logical flaws, or at least are largely immune against them in particularly relevant connections, can be exposed as an illusion. We show this now by demonstrating that even the most reputable of philosophers, who did learn and do apply 'formal logic', commit absolutely crude logical mistakes in most relevant connections. Here are some examples. [REPETITION! /against arbitrary definition]

 

 

 

Example (1): Quine's (1951) inductive argument against 'a priori truth' is strongly inexhaustive, hence perfectly unvconvincing

 

In his celebrated article "Two Dogmas of Empiricism", W.V.O. Quine wants to show that 'a priori' truth (sic; instead of "a priori", he mistakenly uses "analytic") does not exist. The reason, he argues, is that the distinction between 'a prioricity' and 'a posterioricity' cannot be defined.  Starting from a definition of 'a priori' as "true by virtue of meanings and independently of fact" (1951, 21), he justifies this assumption by arguing that no acceptable definition of 'meaning' can be given. And this, in turn, he tries to show by means of an inductive procedure, taking particular definitions of "meaning" one by one, in order to dispense of them all.

 

It is a simple logical requirement on such an inductive argument that in order to be convincing (ideally: conclusive), the induction must be executed exhaustively. In fact, however, Quine's sample of definitions of meaning is restricted to just three (1951, 21-2). Ogden & Richards' The Meaning of Meaning, one of the most-know's in the meaning theory of Quine's time, represents more than twenty--and there are some important additional theories which were in the debate right at the time when Quine wrot his article, including, for instance, C.L. Stevenson's 'emotivist' conception of meanings as dispositions. Quine seems to be perfectly unfamiliar with the state of the art in the meaning theory of his time; in any case, he disregards almost the entirety of definitions of meaning at hand. In any case, to construe an inductive argumentation on the basis of only a tiny sample of cases, as he does in "Two Dogmas", is nothing but a crude logical mistake. (For a more detailed exposition click here.)

 

 

Example (2): Lewis 1970, arbitrary re-definition of "true", causes invalid argument

 

In order to be able to say that non-declarative sentences (e.g., interrogative sentences, directive sentences) be "true", Lewis introduces a peculiar definition of the predicate "true", which is not in accordance with the real meaning of that word (a sentence is "true" in this peculiar sense when a certain kind of putative paraphrase of that sentence is true--in the normal sense of that word). Let us refer to Lewis' peculiar conception as"truthLEWIS". It is clearly inadequate: it does not conform to any common usage of the word "true".

Lewis introduces his peculiar definition in order to save the general assumption that all sentences have truth conditions. Against this view, it may be objected that non-declarative sentences (interrogative, directive sentences) cannot be true or false, and hence do not have any conditions of their truth.

Lewis' argument says, roughly, that since all sentences, even non-declarative ones, are trueLEWIS under certain conditions, all sentences have truth conditions, such that the objection which seems to threaten the truth-conditional approach fails.

However, Lewis' argument is fallacious, eventually due to the mistake of re-defining a term without regarding the requirement of adequacy.

Given that truthLEWIS is not truth (in the ordinary sense), the possibility of truthLEWIS does not imply the possibility of truth (in the ordinary sense). Due to the homonymy between the two notions, Lewis confuses them. His argument, being based on the mistaken equation, is invalid. For a more detailed exposition click here. 

 

 

 

o Davidson (1951), faulty interpretation of Frege's semantics

o Austin (1962), descriptive fallacy

o

 

 

#(4) Other purposes? A test question.

 

 

"What are the most impressing pieces of philosophical progress which were made possible by 'formal logic', and how exactly did 'formal logic' make them possible?"

 

 

 

 

 

Cf. the content of the "Logical Flaws in Literature" page.

o Quine's inexhaustive execution of an inductive procedure. This is a logical mistake because knowledge and awareness of how induction works would have sufficed to that the argument is invalid.

o Davidson's interpratory mistake of taking to be meaning what Frege calls (and just because Frege calls it) "Bedeutung". This is a logical mistake because logical thinking would have made clear to him that the word "Bedeutung", Frege does not refer to the meaning of an expression, but to what that expression's extension. 

o @category mistake: Austin, "constative utterance" being true/false AND happy/unhappy (THE POINT: unrecognizes by others!!)

 

 

 

[...] Reasons

 

o Use of formal means avoids real scrutiny. The application of formal languages is supposed to guarantee unequivocality, precision and clarity of philosophical writing. Yet in fact formal languages do not guarantee these values. In fact it reduces them. For as soon as you put something 'formally', you can lean back: nobody will check your stuff for category mistakes, bad definitions, or logical flaws any more; and even if they tried to, they could not: because the results of 'formalization' are too gross to actually represent category mistakes, bad definitions, etc. (TEST: Just 'formalize' "Utterances are performed sentences": Is there anything in the formula that uncovers the category mistake?)

 

o Expressive poorness of 'formal language'. What formal means offer are only a minimal range of expressions. That means, there actually are only a handful of words of normal language which 'formal' language actually offers replacements for. Ask yourself what the formalisation of the following terms is (this is an arbitrary choice of common-or-garden philosophical key terms): 'altruistic', 'belief', 'correspondence', 'definition', 'equal', 'formal', 'general', 'hypothesis', 'implication', 'justification', 'knowledge', 'logic', 'meaning', 'naturalistic', 'ontological', 'propositional', 'quantifier', 'reference', 'statement', 'time', 'use', 'vague', 'world', 'you', 'zombie'. When ambiguity, vagueness or lack of clarity infects philosophical work, then the reason lies in the vagueness, ambiguity or lack of clarity of the central terms.

Only rarely is the problem located in the lack of precision of the word "and", or difficulties concerning quantification and bracketing. 

 

o The symbols which 'formal language' provides are really dispensable. (And if they were useful, they could easily be introduced by means of common definition).

-- If you want to express quantification, try this: "There is at least one entity which ...", "There is no entity whatsoever which ...", "For all entities, ...".

-- The disambiguation of vel/aut is quite easy in normal language, too: "Either ... or ___ or both"; "Either ... or ___, but not both". You definitely do not need formal means for this.

! As a test, collect the most relevant three instances of theses/arguments which in your opinion can be captured only within 'formal language' -- I predict you' find not even one.

 

o It is quite an amazing fact that philosophers make use of 'formal language' when common language would perfectly do.

 

o It is another amazing fact that philosophers usually do not apply formal language exactly when clarity and precision would be needed the most. As an example, take Quine's "Two Dogmas of Empiricism": (a) The attack he drives against what he calls "analytic" truths (which strangely resemble Kant's 'a priori' truths ;-) ...), (b) the arguments he uses to support his attack, and (c) the relation between the first dogma and the second (which he says is close to identity): in none of these cases is Quines text even approximately clear. Why does Quine not apply formal means here?

 

...

 

o Many authors complement their formulae with natural language paraphrases. Why is the paraphrase needed? And what does the 'formal' statement achieve which the paraphrase, if it is good, does not?

 

o The following terms, for instance, are not available in 'formal' language: a priori, action, altruistic; belief, behaviour, bad; categorical, conditional, confirmation; definition, (etc.). 

 

o => partial formalisation (where the crucial parts are non-formal!)

 

o Formal means: treated like a 'magical' instrument; serve the implicit self-ascription of expertship.

 

 

 

 

 ... 

 

Yes, clarity and precision are indispensable, ... but: 'formal' languages actually do not further them!

 

Adherents of the study of formal languages defend them as an efficacious device in the service of aims like clarity and precision. When I argue that 'logic' courses should not concentrate on 'formal' languages, does this mean that in my view any of these requirements is debatable?--Of course, not: quite the opposite. The problem is, however, that the study and (slily selective) application of 'formal' means by real-life philosophers really does not advance these values. For in fact, only very few real-life philosophers are earnest about those 'formal' methods.

 

For example, real-life philosophers regularly use, within their 'formal' statements (!), terms quite alien to the 'formal' language they (seem to) apply ('semi-formal means' this is called), terms which though undefined in the formal language they do not bother to define in any acceptable way, and which regularly infect the matter by the full range of the well-known dangers of natural language use (vagueness, lack of clarity, etc.) which those 'formal means' are actually meant to avoid. Furthermore, real-life philosophers quite frankly omit quantification wherever they like to (i.e., in particular, when quantification would force them to make decisions they find inconvenient ...). Also, they regularly present the reader with semantically incomplete expressions (for about the same reasons). For some examples from the most prominent philosophers ckick here [sorry, now still a blind link; but contact me if you are particularly interested in this matter!].

 

 

... 

 

Why were 'formal' languages ever advanced to the extent they were?

 

@focus on matters related to mathematics (Frege, Whitehead, Russell)

@focus on matters concerning scientific research (Schlick, 'the early' Wittgenstein, Carnap) [@philosophers posing as scientists]

@projection of eschatological expectations to sth. which was then new, thrilling, and ...: mysterious

... and then ...

@study of 'f.' l. began to be / suddenly was perceived as an indicator of serious, hard, rational uncompromising philosophical work [@philosophers posing as scientists]

@study of 'f.' l. became a condition of recognition within the broad 'analytic' movement [@philosophers posing as scientists]

@the introduction of courses in 'f.' l. produced legions of philosophers who had invested much work in their FL skills, and now wanted and had to be employed--within FL courses ...

@those legions of FL skilled philosophers were/are not ready to admit (or even consider the idea) that the annoying stuff they have invested so much labour in is really just for the birds.