Category: Logica Docens

Syntactic versus Semantic Validity

Consider the argument:
 
   Bill is a brother
   —–
   Bill is a sibling.

Is this little argument valid or invalid?  It depends on what we mean by 'valid.' Intuitively, the argument is valid in the following sense:

D1. An argument is valid if and only if it is impossible that its premise(s) be true and its conclusion false.

(D1) may be glossed by saying that there are no possible circumstances in which the premises are true and the conclusion false. Equivalently, in every possible circumstance in which the premises are true, the conclusion is true.  Since it is impossible that Bill be a brother without his being a sibling, the opening argument is valid by (D1).

(D1), though correct as far as it goes, leaves unspecified the source or ground of a valid argument's validity. This is the philosophically interesting question. What makes a valid argument valid? What is the ground of the impossibility of the premises' being true and the conclusion's being false? One answer is that the source of validity is narrowly logical or purely syntactic: the validity of a valid argument derives from its instantiation of valid argument-forms.

Now it is obvious that the validity of the above argument does not derive from its logical form. The logical form is

   Fa
   —–
   Ga

where 'a' is an arbitrary individual constant and 'F' an arbitrary predicate constant. The above argument-form is invalid since it is easy to interpret the place-holders so as to make the premise true and the conclusion false: let 'a' stand for Al, 'F' for fat and 'G' for gay.

Valid arguments are either syntactically valid or semantically valid.  The opening argument is not syntactically valid but it is semantically valid.

D2. An argument is syntactically valid iff it is narrowly-logically impossible that there be an  argument of that form having true premises and a false conclusion.

According to (D2), a valid argument inherits its validity from the validity of its form, or logical syntax. So on (D2) it is primarily argument-forms that are valid or invalid; arguments are valid or invalid only in virtue of their instantiation of valid or invalid argument-forms. (D2) is thus a specification of the generic (D1).

But there is a second specification of (D1) according to which  validity/invalidity has its source in the constituent propositions of the arguments themselves and so depends on their extra-syntactic content:

D3. An argument is extra-syntactically valid iff (i) it is impossible  that its premises be true and its conclusion false;  (ii) this impossibility is grounded neither in any contingent matter of fact nor
in formal logic proper, but in some necessary connection between the senses or the referents of the extra-logical terms of the argument.

A specification of (D3) is

D4. An argument is semantically valid iff (i) if it is impossible that its premises be true and its conclusion false; and (ii) this impossibility is grounded in the senses of the extra-logical terms of
the argument.

Thus to explain the semantic validity of the opening argument we can say that the sense of 'brother' includes the sense of 'sibling.' There is a necessary connection between the two senses, one that does not rest  on any contingent matter of fact and is also not mediated by any purely formal law of logic. Note that logic allows (does not rule out) a brother who is not a sibling. Logic would rule out a non-sibling brother only if 'x is F & x is not G' had only false substitution-instances — which is not the case. To put it another way, a brother that is not a sibling is a narrowly-logical possibility. But it is not a broadly-logical possibility due to the necesssary connection of the two senses.

So it looks as if analytic entailments like Bill is a brother, ergo, Bill is a sibling show that subsumability under purely formal logical laws is not necessary for (generically) valid inference. Sufficient, but not necessary. Analytic   entailments appear to be counterexamples to the thesis that inferences in natural language can be validated only by subsumption under logical laws.

One might wonder  what philosophers typically have in mind when they speak of validity. I would say that most philosophers today have in mind (D1) as specified by (D2). Only a minority have in mind (D3) and its specification (D4).  I could easily be wrong about that.  Is there a sociologist of philosophers in the house?

Consider the Quineans and all who reject the analytic/synthetic distinction. They of course will have no truck with analytic  entailments and talk of semantic validity. Carnapians, on the other hand, will uphold the analytic/synthetic distinction but validate all  entailments in the standard (derivational) way by importing all analytic truths as meaning postulates into the widened category of L-truths.

Along broadly Carnapian lines one could argue that the above argument is an enthymeme which when spelled out is

   Every brother is a sibling
   Bill is a brother
   —–
   Bill is a sibling.

Since this expanded argument is
syntactically valid, the original argument — construed as an enthymeme — is also syntactically valid. When I say that it is syntactically valid I just mean that the
conclusion can be derived from the premises using the resources of standard logic, i.e. the Frege-inspired predicate calculus one finds in logic textbooks such as I. Copi's Symbolic Logic. In the above example, one uses two inference rules, Universal Instantiation and Modus Ponens, to derive the conclusion.

If this is right, then the source of the expanded argument's validity is not in a necessary connection between the senses of the 'brother' and 'sibling' but in logical laws. The question, however, was whether the opening argument as stated is valid or invaid. I say it is semantically, but not syntactically valid.

A juicier example is the Cartesian cogito:

I think
—–

I am. 

This looks to be  semantically valid and thus valid without the need of an auxiliary premise to mediate the inferential  transition from premise to conclusion.  It is valid absent an auxiliary major premise such as 'Whatever thinks, is.'

Validity and Anaphora

The following argument appears valid:

Some deity is called 'Zeus.'
Zeus is wise.
Therefore, some deity called 'Zeus' is wise. (D. E. Buckner, Reference and Identity, 118)

Now if an argument is valid, it is valid in virtue of its logical form.  What is the logical form of the above argument? The following argument-form, Buckner correctly states, is invalid:

Ex Fx
Ga
Ex (Fx & Gx)

So if the form just depicted is the only available form of the original argument, then the validity of the argument cannot be simply a matter of logical form. And this is what Buckner concludes: "It is clearly the anaphoric connection between the premisses that makes the argument valid, but no such connection exists in the formalized version of the argument. "(119)

Buckner seems to be arguing as follows:

a) The original argument is valid. 

b) The only form it could possibly have is the one depicted above.

c) The argument-form depicted is plainly invalid.

Therefore

d) The validity of the original argument cannot be due to its logical form, but must be due to the anaphoric connection between its premises.

I do not find this argument rationally compelling. (b) is rejectable.  I suggest that the original argument is an enthymeme the logical form of which is the following:

1) For some x, x is called 'a.'
2) For any x, if x is called 'a,' then x =a.
3) a is G.
Therefore
4) For some x, (x is called 'a' and x is G).

Logical Form, Equivocation, and Propositions

A re-post with minor edits and additions from 4 September 2017.

………………………………..

Ed Buckner wants to re-fight old battles. I'm game. The following post of his, reproduced verbatim, just appeared at Dale Tuggy's site:

The concept of logical form is essential to any discussion of identity, and hence to any discussion of the Trinity. Here is a puzzle I have been discussing with the famous Bill Vallicella for many years.

(Argument 1) ‘Cicero is a Roman, therefore Cicero is a Roman’

(Argument 2) ‘Cicero is a Roman, therefore Tully is a Roman’

My puzzle [is] that the first argument is clearly not valid if the first ‘Cicero’ means the Roman, the second the American town, yet the argument seems to instantiate a valid form. Bill objects that if there is equivocation, then the argument really has the form ‘a is F, therefore b is F’, which fails to instantiate a valid form.

I then ask what is the form of. Clearly not of the sentences, since the sentences do not include the meaning or the proposition. Is it the form of the proposition expressed by the sentences? But then we have the problem of the second argument, where both ‘Cicero’ and ‘Tully’ mean the same man. Then the man is contained in both propositions, and if the form is of the proposition, the argument has the true form ‘a is F, so a is F’, which is valid. But I think no one would agree that the second argument is valid.

So logical form does not belong to the sentences, nor to the propositions expressed by them. So what is it the form of?

Tully'sMy answer is that the logical form of the argument is the form of the Fregean propositions expressed by the sentences that make up the argument. Let me explain.

I agree with Ed that logical form is not the form of an array of sentence-tokens. It is rather the form of an array of propositions expressed by the sentences. (To be painfully precise: it is the form of an array of propositions expressed by the assertive utterance, and thus the tokening, of a series of sentence-types by a speaker or thinker on a given occasion. A sentence-token buried in a book does not express anything by itself!)

To solve Ed's puzzle we need to distinguish three views of propositions: the Aristotelian, the Fregean, and the Russellian. This would be a good topic for an extended post. Here I will be brief.  Brevity is the soul of blog.

An Aristotelian proposition is an assertively uttered meaningful sentence in the indicative mood that expresses a complete thought.  What makes such a proposition 'Aristotelian' as opposed to 'Platonic' is that the meaning of the sentence is not something that can subsist on its own apart from the assertive tokening of the sentence.  The meaning of the sentence depends on its being expressed, whether in overt speech or in thought, by someone. And this expression must be thoughtfully done and not mindlessly like a parrot or a voice synthesizer. If there were no minds there would be no Aristotelian propositions. And if there were no languages there would be no Aristotelian propositions. In this sense, Aristotelian propositions are linguistic entities.

In brief: An Aristotelian proposition is just a declarative sentence in use together with its dependent sense or meaning. Suppose I write a declarative sentence on a piece of paper. The Aristotelian proposition is not the string of physical marks on the paper, nor it is the producing of the marks; it is the marks as produced by a minded organism on a particular occasion together with the meaning those marks embody where meaning is first in the mind and only then embodied in the marks.

A Fregean proposition is a nonlinguistic entity that subsists independently of minds and language. It is the sense (Sinn) of a declarative sentence (Satz) from which indexical elements have been extruded. For example, 'I am blogging'  does not express a Fregean proposition because of the indexical 'I' and because of the present tense of the verb phrase.  But 'BV blogs at 10:50 AM PST on 4 September 2017' expresses a Fregean proposition.

Fregean senses are extralinguistic and extramental 'abstract' or 'Platonic' items. They are not in time or space even when the objects they are about are in time and space. This is what makes Fregean propositions 'Platonic' rather than 'Aristotelian.' Fregean propositions are the primary truth-bearers; the sentences that express them are derivatively true or false.  Likewise with the judgments whose content they are.

A Russellian proposition is a blurry, hybrid entity that combines some of the features of a Fregean truth-bearer and some of the features of a truth-maker. A Russellian proposition does not reside at the level of sense (Sinn) but at the level of reference (Bedeutung).  It is out there in the (natural) world. It is what some of us call a fact or 'concrete fact' (as in my existence book) and others, e.g. D. M. Armstrong,  a state of affairs.  

Now consider a singular sentence such as 'Ed is happy.'  For present purposes, the crucial difference between a Fregean proposition and a Russellian proposition is that, on the Fregean view, the subject constituent of Ed is happy is not Ed himself with skin and hair, but an abstract surrogate that represents him in the Fregean proposition, whereas in the Russellian proposition Ed himself is a constituent of the proposition!  

We needn't consider why so many distinguished philosophers have opted for this (monstrous) view.  But this is the view that seems to have Ed in its grip and that powers his puzzle above.

If we take the relatively saner (but nonetheless problematic) view that propositions are Fregean in nature, then the puzzle is easily solved.

Ed asks: What is the logical form the form of?  He maintains, rightly, that it cannot be the form of an array of sentences. So it must be the form of an array of propositions. Right again. But then he falls into puzzlement: 

. . . ‘Cicero’ and ‘Tully’ mean the same man. Then the man is contained in both propositions, and if the form is of the proposition, the argument has the true form ‘a is F, so a is F’, which is valid.

The puzzlement disappears if we reject the Russsellian theory of propositions. A man cannot be contained in a proposition, and so it cannot be the same man in both propositions.

‘Cicero is a Roman, therefore Tully is a Roman’ is plainly invalid. Its form is: Rc, ergo Rt, which is an invalid form. If we adopt  either an Aristotelian or a Fregean view of propositions we will not be tempted to think otherwise.

‘Cicero is a Roman, therefore Cicero is a Roman’ is plainly valid. ‘Cicero is a Roman, therefore Tully is a Roman’ is plainly invalid. The logical forms are different! If, on a Russellian theory of propositions, the forms are the same, then so much the worse for a Russellian theory of propositions!

Circular Definitions, Arguments, and Explanations

In the course of our discursive operations we often encounter circularity.  Clarity will be served if we distinguish different types of circularity.  I count three types.  We could label them definitional, argumentative, and explanatory.

A.  The life of the mind often includes the framing of definitions.  Now one constraint on a good definition is that it not be circular.  A circular definition is one in which the term to be defined (the definiendum) or a cognate thereof occurs in the defining phrase (the definiens).  'A triangle is a plane figure having a triangular shape,' though plainly true, is circular.  'The extension of a term is the set of items to which the term applies' is an example of a non-circular definition. 

Ibram X. Kendi, the race 'theorist' currently much-loved by the 'woke,' was recently asked to define 'racism.' He came out with this brilliancy:  “A collection of racist policies that lead to racial inequity that are substantiated by racist ideas." Video here.

B.  Sometimes we argue.  We attempt to support a proposition p by adducing other propositions as reasons for accepting p.  Now one constraint on a good argument is that it not be circular.  A circular argument in is one in which the conclusion appears among the premises, sometimes nakedly, other times clothed for decency's sake  in different verbal dress.  Supply your own examples.

C.  Sometimes we explain.  What is it for an individual x to exist?  Suppose you say that for x to exist is for some property to be instantiated.  One variation on this theme is to say that for Socrates to exist is for the haecceity property Socrateity to be instantiated.  This counts as a metaphysical explanation, and a circular one to boot.  For if Socrateity is instantiated, then it is is instantiated by Socrates who must exist to stand in the instantiation relation.  The account moves in a circle, an explanatory circle of embarrassingly short diameter.

Suppose someone says that for x to exist is for x to be identical to something or other.  They could mean this merely as an equivalence, in which case I have no objection.  But if they are shooting for a explanation of existence in terms of identity-with-something-or-other, then they move in an explanatory circle. For if x exists in virtue of its identity with some y, then y must exist, and you have moved in an explanatory circle.

Some philosophers argue that philosophers ought not be in the business of explanation.  I beg to differ.  But that is a large metaphilosophical topic unto itself.

A Reader Asks about Existence and Instantiation

My responses are in blue.

Hello, Dr. Vallicella. I am a reader of your blog. I just read your article "Existence: Two Dogmas of Analysis" in Neo-Aristotelian Perspectives in Metaphysics (eds. Novotny and Novak, Routledge, 2014, pp. 45-75 , and I thought it was fantastic. I will have to read it again at some point. There were some parts in it that I found very interesting, and I was hoping I could ask you about. I want to focus on what you said in section 6.6, page 57. You write: 

 
"It is clear that “Unicorns do not exist” cannot be about unicorns: There are none. So it is reasonably analysed in terms of “ The concept unicorn is not instantiated”. But then the concept must exist, and its existence cannot be its being instantiated." 
 
The question I wanted to ask you was specifically about the final part, "But then the concept must exist, and its existence cannot be its being instantiated". I will try to keep the questions as brief as possible, 
 
The thin theorist might not identify the existence of the concept of a unicorn with its being instantiated, but with the concept of the concept of the unicorn being instantiated, and so on . . .
 
1) If it were possible that there be an infinite number of concepts, would there be any problem with this view?
 
BV:  An infinite regress would arise.
 
2) Clearly, we have a regress here, but is it vicious?
 
BV: Yes, because there would be no explanation of  the existence of the first concept in the infinite series. You might reply by saying that the series is actually, as opposed to potentially, infinite.  If so, then every concept in the series would have an explanation of its existence.  To which my response would be:   what explains the existence of the entire actually infinite series of concepts?
 
(An analogous situation. Suppose the universe is a beginningless actual infinity of continuum-many states with each state caused by earlier ones. If so, every state would have a causal explanation. But if every state of the universe has a causal explanation, then one might plausibly suppose that the universe has a causal explanation, one that is internal to it. Some people have maintained this with an eye toward ruling out the need for a transcendent explainer such as God. "There is no need for God because a universe with an actually infinite past has the resources to explain itself."  My objection would be that this account leaves us with no explanation of why the entire series of states exists in the first place.  Given that the entire series is modally contingent, and thus possibly such as not to exist at all, then any explanation of it, assuming that there is an explanation of it, could only be external or transcendent. Now back to the main thread.)
 
One might also question whether the concept regress could even get started. You want to say that the concept unicorn exists in virtue of its being an instance of the concept concept unicorn. But these two concepts have exactly the same content. How then do they differ? The concept unicorn is an instance of the concept concept, but I fail to see any difference between the concept unicorn and the concept concept unicorn.
 
3) The overall worry is that if we define x's existence in terms of instantiation, and then ask 'what are "x's"', we say things in existence, and, this is circular, but, since we are simply dealing with the analysis of terms, aren't we only dealing with semantic circularity? I am not sure that there is any problem with this sort of circularity (if there is a problem, it would be with the informativeness with the analysis rather than the accuracy).
 
BV: But we are not merely dealing with the analysis of terms; we are seeking to understand what it is for an individual to exist, given that the existence of a thing is extra-linguistic.   Let's keep in mind what the question is.  The question is whether an adequate theory of existence could treat '. . . exist(s)' as a second-level or second-order predicate only, that is, a predicate of concepts, properties, propositional functions,  descriptions (definite or indefinite), or cognate items. That is the Frege-Russell theory that I have in my sights in the portion of text to which my reader refers.
 
Granted, it is  true that Fs exist iff the concept F is instantiated.  For example, it is  true that cats exist iff the concept cat is instantiated. (This assumes that there is the concept cat, which is certainly true in our world if not in all possible worlds: it depends on what we take concepts to be.) But the right-hand-side (RHS) of the biconditional merely specifies a truth-condition on the semantic plane: it does not take us beyond or beneath that plane to the plane of extra-linguistic reality.  The truth of the LHS requires an ontological ground, a truth-maker, not a truth-condition. For consider: if the concept cat is instantiated, then, since it is a first-order concept, and relational as opposed to monadic, it is instantiated by one or more individuals. Individuals by definition are impredicable and uninstantiable. My cat Max Black, for example, is categorially unfit to have any instances, and you can't predicate him of anything. The little rascal is unrepeatable and impredicable.
 
Now either the instantiating individuals exist or they do not. If they do not, then the truth of the biconditional above is not preserved. But if they do exist, then the sense in which the instances exist is toto caelo different from the sense specified by 'is instantiated.' To repeat, by definition, individuals cannot be instantiated; therefore, the existence of an individual –call it singular existence –  cannot be explicated in terms of instantiation.
 
The instantiation account of existence either changes the subject from singular existence to general existence (instantiation) or else it moves in a circle of embarrassingly short diameter.  We want to know what it is for individuals to exist, and we are told that for individuals to exist is for first-level concepts to be instantiated; but for these concepts to be instantiated, their instances must exist singularly and thus in a sense that cannot be explicated in terms of instantiation. To put it another way: the account presupposes what it is trying to get rid of. It wants to reduce singular existence to general existence, thereby eliminating singular existence, but it ends up presupposing singular existence. If you tell me that the instances neither exist nor do not exist and that this contrast first arises at the level of concepts , then I will point out that you are thereby committed to Meinongian objects, to pure Sosein without Dasein.
 
The circularity I allege is the circularity of ontological/metaphysical explanation.  Is 'Tom exists' true because Tom exists, or does Tom exist because 'Tom exists' is true?  If this question makes sense to you and you respond by opting for  the former, then you understand metaphysical explanation.  It is an explanation that is neither empirical nor narrowly logical. Somewhat murky it might be, but nonetheless indispensable for metaphysics.  Similarly with the question: does Tom exist because some concept C is instantiated, or is C instantiated because Tom exists? The question makes sense and the answer is the latter.
 
I want to note that these are questions someone asked me about this view, and I wasn't sure how to respond, even though I ultimately do agree with your analysis of the thin theory. For the third problem, I would have said that that sort of response would merely ignore the fact that the question 'what is existence?' has ontological consequences, and is not merely a question of semantics. [Right!] If that is all we are concerned with (semantics), then we are concerned with something different than what most classical philosophers are concerned with when they are talking about the question 'what is existence?', which is the ontological aspect of that question, and as such, the circularity issue is a real problem. [You got it!]
 
BV: The problem with Frege, Russell, Quine, van Inwagen, and the rest of the 'thin  crew' is that they try to reduce existence to a merely logical topic. An opposite or at least different mistake is made by the phenomenologists who (most of them, not all of them) try to reduce existence to a phenomenological topic.  Heidegger, near the beginning of Sein und Zeit, opines that "Ontology is only possible as phenomenology." 
 
So I got me a two-front war on my hands: against the nuts-and-bolts analysts to the West and against the febrile phenomenologists to the East.
 

Another Thought on Psychologism in Logic

Logic is prescriptive and proscriptive.  Logic prescribes how we ought to think if we would arrive at truth. It also proscribes those ways of thinking that lead to error.  But 'ought' implies 'can.' How we ought to think must be really possible, indeed really possible for us, where what is really possible for us is grounded in how we actually and contingently are. A real possibility of thinking this way or that must be based in actual abilities, actual abilities of real minds in the real order.  The logically normative must be psychologically implementable.  The ideal patterns residing in the ὑπερουράνιος τόπος of Plato must be realizable in enmattered minds.

There look to be the makings here of an argument for a defensible psychologism.  (Logic cannot be a part of empirical psychology, but how could it have nothing to do with the latter?)

The above train of thought is from a couple of years ago.  (Journal vol. XXXIII, pp. 22-23, entry of 4 January 2019) Now I find the following in the Martin Kusch SEP article on psychologism, referenced in the immediately preceding entry:

1. Normative-prescriptive disciplines — disciplines that tell us what we ought to do — must be based upon descriptive-explanatory sciences.
2. Logic is a normative-prescriptive discipline concerning human thinking.
3. There is only one science which qualifies as constituting the descriptive-explanatory foundation for logic: empirical psychology.
Ergo, logic must be based upon psychology.

The above is the second of five patterns of psychologistic reasoning that Kusch distinguishes.  He attributes it to Wilhelm Wundt.  My thought above runs along parallel rails.

Logic, prescribing as it does how we OUGHT to think, by the same stroke prescribes how we ought to THINK. The abstract patterns definitive of the oughts and ought nots of inference may reside in Plato's timeless heaven, but thinking and thus judging is in time and takes time.  Inference, in particular, takes time. Its analog up yonder is implication. And so the abstractly logical must touch ground in the matter of minds in time.  An abstract entity can't think.

But a concrete hunk of intracranial meat can't think either. And meat can't mean. Minds mean. If we were just meatheads we couldn't think or mean. Thinking is a psychic function.  Arguably, though, it is not the psyche as objectified and manifest to inner sense that thinks but the psyche as subject, the psyche as pre-objective, pre-mundane, and thus transcendental.  But from Descartes on it has proven to be a bear of a task to get a good solid grip on the transcendental. Husserl struggled with it life-long and  yet couldn't drag it out of the dreck into the clear light of day. And where the great Husserl failed we lesser luminaries and flickering lights are even less likely to succeed.

Must we regress to the spiritual? But how can we get a grip on it without objectifying it?  We cannot help but reify, but the Cogitans is not a res, not  spiritual substance.

The noetic as such embraces the logical, the psychological, the transcendental and the spiritual. 

On that gnomic note I end this meditation.

Related: Martin Kusch, Psychologism (from Ralph Dumain's Autodidact Project)

Has Any Philosophical Problem Been Solved? The case of psychologism in logic.

For Cyrus

……………

A reader is skeptical of my solubility skepticism. He adduces the problem of psychologism in logic which, he suggests, has been definitively settled in favor of the anti-psychologizers.  Here, then, is a problem that supposedly has been solved. There is progress in philosophy after all. My reader is joined by Robert Spaemann who, in his Persons, tr. O'Donovan, Oxford 2006, writes:

The refutation of psychologism in logic, with which Husserl and Frege are associated, is among the very few philosophical achievements that have brought an existing debate to a decisive close. (54)

Would that it were so! But alas it is not.  The existing debate rages on. Having been brought up on Husserl, and influenced by Frege, I was for a long time an opponent of psychologism in logic, and thought the issue resolved. Time to revaluate! Here is a post from August 2004 from my first blog:

ARE THE LAWS OF LOGIC EMPIRICAL GENERALIZATIONS?

Someone on a discussion list recently resurrected the old idea of John Stuart Mill and others that the laws of logic are empirical generalizations from what we do and do not perceive. Thus we never perceive rain and its absence in the same place and at the same time. The temptation is to construe such logic laws as the Law of Non-Contradiction — ~(p & ~p) — as generalizations from psychological facts like these. If this is right, then logical laws lack the a priori character and epistemic ‘dignity’ that some of us are wont to see in them. They rest on psychological facts that might have been otherwise.

But now consider this reductio ad absurdum:

1. The laws of logic are empirical generalizations. (Assumption for reductio)
2. Empirical generalizations, if true, are merely contingently true. (By definition of ‘empirical generalization’: empirical generalizations record what happens to be the case, but might not have been the case.) Therefore,
3. The laws of logic, if true, are merely contingently true. (From 1 and 2)
4. If proposition p is contingently true, then it is possible that p be false. (Def. of ‘contingently true.’)Therefore,
5. The laws of logic, if true, are possibly false. (From 3 and 4)Therefore,
6. LNC is possibly false: there are logically possible worlds in which ‘p&~p’ is true. (From 5 and the fact that LNC is a law of logic.)
7. But (6) is absurd (self-contradictory): it amounts to saying that it is logically possible that the very criterion of logical possibility, namely LNC, be false. Corollary: if laws of logic were empirical generalizations, we would be incapable of defining ‘empirical generalization’: this definition requires the notion of what is the case but (logically) might not have been the case.

The above is a good, but not a compelling, argument. For it presupposes the distinction between necessary and contingent propositions.  Is that distinction objectively self-evident? Martin Kusch, Psychologism, Stanford Encyclopedia of Philosophy:

Massey also invokes the stronger form of the claim that logical truths are not necessary (1991, 188). According to this criticism, the very notion of necessity which is presupposed in calling logical laws ‘necessary truths’, is beset with difficulties. The argument leading to this conclusion was developed in a series of well-known papers by Quine. Quine argued that the notions of analyticity, necessity and aprioricity stand or fall together and that the traditional distinction between analytic and synthetic truths is relative rather than absolute. But once this distinction becomes relative, necessity and aprioricity go by the board (Quine 1951, Engel 1991, 268–70). Massey summarises the implications of Quine’s arguments succinctly:

If we reject the concept of necessity … we also forego the concept of contingency. If it makes no sense to say that the truths of mathematics are necessary, it makes no better sense to say that those of psychology or any other so-called empirical science are contingent. But if we may not employ necessity and contingency to demarcate the deliverances of the empirical sciences from those of the formal sciences, how are we to distinguish them in any philosophically interesting way? (1991, 188).

Now I don't much cotton to Quine, but he is no slouch of a logician!  And he is certainly a looming presence in 20th century American philosophy.  So on the basis of his dissent alone, we ought to agree that the psychologism problem has not been solved.   I am assuming that a problem hasn't been solved unless it has been solved to the satisfaction of all competent practitioners.  It hasn't been solved until the debate about it has been brought to a decisive close. Kusch gives several reasons in addition to the one cited above why this is not the case with respect to the psychologism debate. 

First and Second Intentions: Buckner on Zabarella, Kant, Frege, and Wittgenstein

The following two quotations are from the Facebook Medieval Logic forum.
 
Giacomo Zabarella (1533 – 1589). “Now first intentions are names immediately signifying realities by means of a concept in the soul, for instance, animal and human being, or those concepts of which these names are signs. But second intentions are other names imposed on these names, for instance, genus, species, name, verb, proposition, syllogism, and others of that sort, or the concepts themselves that are signified through these names.”
Edward Buckner comments:
 
The distinction [between first and second intentions] is rediscovered in various ways by subsequent philosophers. I see something like it in Kant’s distinction between concepts which are ‘pure’, and concepts which are not, in Frege’s distinction between concept and object words, and possibly in Wittgenstein, who viewed logic as a sort of scaffolding through which we conceive the world, a scaffolding which cannot be described in words. (4121 “Propositions cannot represent logical form: it is mirrored in them”). If I understand Wittgenstein, it is that there can be no science of second intentions in Zabarella’s sense, for such a science would be a futile attempt to represent logical form. The Tractatus of course is such an attempt, which is why he says (654) his propositions, while nonsensical, can be used as steps [in a ladder] to climb up beyond them, then throw away the ladder.
 
Kant
 
I think Ed is wrong above about Kant.  For Kant, the pure is the opposite of the empirical. Every concept is either pure or empirical and no concept is both. A pure concept is one that is not drawn from experience, ein solcher der nicht von der Erfahrung abgezogen ist, but originates from the understanding in respect of both form and content, sondern auch dem Inhalte nach aus dem Verstande entspringt. The form of all concepts, including pure concepts, arises from reflexion Reflexion, and thus from the understanding. Empirical concepts arise from the senses, entspringen aus den Sinnen,  by comparison of the objects of experience. Their content comes from the senses, and their form of universality, Form der Allgemeinheit, alone from the understanding.
 
If Buckner is telling us that Kant's pure-empirical distinction runs parallel to Zabarella's first intention-second intention distinction, then that can't be right. For Zabarella's animal and human being, which are first intentions for him, count as empirical concepts for Kant. 
 
Any comparison of Zabarella (1533-1589) the Aristotelian and Kant is bound to be fraught with difficulty because of the transcendental-subjective turn of modern philosophy commencing with Descartes (1596-1650).  For Aristotle, the categories are categories of a real world independent of  our understanding; for Kant, the categories are precisely categories of the understanding (Verstandeskategorien) grounded in the understanding both in their form and in their content.  The categories of Aristotle are thus objective, categories belonging to a world to be understood, and not subjective, categories whereby a mind understands the world.
 
Pure Concepts of Reason as Limit Concepts
 
Kant also speaks in his Logic and elsewhere of Ideas which are pure concepts of reason, Vernunft, and not of understanding, Verstand. Die Idee ist ein Vernunftbegriff deren Gegenstand gar nicht in der Erfahrug kann angetroffen werden. (Logik, sec. 3)  The objects of these pure concepts of reason cannot be known by us because our form of intuition, Anschauung, is sensible, not intellectual. We can know only phenomena, not noumena. Among these Ideas, which are plainly limit concepts, are God, the soul, the world-whole, and freedom. And they are not merely negative limit concepts. Free will, for example, is objectively real despite its not being obejctively knowable. But more on this later.
 
Frege
 
I also think Ed is wrong about Frege.  But I'll leave that for later. Wifey wants to go out to dinner. Philosophy before bread, but happy wife, happy life!
 
As for Wittgenstein, I think Ed is on the right track. 
 
 

How Much Logic Do I Need?

A reader who reports that his main interest is in contemporary metaphysics inquires:

Should I learn as much logic as humanly possible during my PhD? Or should I learn only what I need along the way? I have a basic grasp of symbolic and predicate logic, but little meta-logic.

First of all, it makes no sense to oppose symbolic to predicate logic.  Modern symbolic logic includes both propositional logic and predicate logic.

Second, learn what you need as you go along.  For example, existence is one of the central topics in metaphysica generalis. To work on this topic in an informed way you have to understand the modern quantificational treatment of existence in mathematical logic. 

Here is the minimum required for doing metaphysics. First, a thorough grounding in traditional formal logic including the Aristotelian syllogistic. Second, modern symbolic logic including the propositional calculus and first-order predicate logic with identity.  Third, some familiarity with axiomatics and the concepts of metalogic including consistency and completeness of axiom systems.  Fourth, axiomatic set theory. Fifth, some (alethic) modal logic both propositional and quantified.

The best way to master these subjects, or at least the first two mentioned, is by teaching them to undergraduates.  

Truth and Falsity from a Deflationary Point of View

The following equivalence is taken by many to support the deflationary thesis that truth has no substantive nature, a nature that could justify a substantive theory along correspondentist, or coherentist, or pragmatic,  or other lines.  For example, someone who maintains that truth is rational acceptability at the ideal (Peircean) limit of inquiry is advancing a substantive theory of truth that purports to nail down the nature of truth.  Here is the equivalence:

1)  <p> is true iff p.

The angle brackets surrounding a declarative sentence make of it a name of the proposition the sentence expresses. For example, <snow is white> –  the proposition that snow is white — is true iff snow is white. (1) suggests that the predicate ' ___ is true' does not express a substantive property.  We can dispense with the predicate and say what we want without it. It suggests that there is no such legitimate metaphysical question as: What is the nature of truth?  Having gotten rid of truth, can we get rid of falsity as well?

A false proposition is one that is not true.  This suggests that 'false,' as a predicate applicable to propositions and truth-bearers generally, is definable in terms of 'true' and 'not.' Perhaps as follows:

2) <p> is false iff <p> is not true.

From (2) we may infer

2*) <p> is false iff ~(<p> is true)

and then, given (1),

2**) <p> is false iff ~p.

This suggests that if we are given the notions of 'proposition' and 'negation,' we can dispense with the supposed properties of truth and falsity. (1) shows us how to dispense with 'true' and (2**) show us how to dispense with 'false.'

But we hit a snag when we ask what 'not' means.  Now the standard way to explain the logical constants employs truth tables. Here is the truth table for the logician's 'not' which is symbolized by the tilde, '~'.

$$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & \cr & P & & $\lnot P$ & \cr height2pt & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & \cr & T & & F & \cr height2pt & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & \cr & F & & T & \cr height2pt & \omit & & \omit & \cr \noalign{\hrule} }} $$

But now we see that our explanation is circular. We set out to explain the meaning of 'false' in terms of 'not' only to find that 'not' cannot be explained except in terms of 'false.' We have moved in a circle.

The Ostrich has a response to this:

. . . we can define negation without reaching for the notions of truth and falsity. Assume that the notion of ‘all possible situations’ is coherent, and suppose it is coherent for any proposition ‘p’ to map onto a subset of that set. Then ‘not p’ maps onto the complement. The question is whether the very idea of a complement of a subset covertly appeals to the concept of negation. But then that suggests that negation is a primitive indefinable concept, rather than what you are claiming (namely that it is truth and falsity which are primitive).

So let's assume that there is a set S of possible worlds,and that every proposition (except impossible propositions)  maps onto to an improper or a proper subset of S. The necessary propositions map onto the improper subset of S, namely S itself. Each contingent proposition p maps onto a proper subset of S, but a different proper subset for different propositions. If so, ~p maps onto the complement of the proper subset that p maps onto.  And let's assume that negation can be understood in terms of complementation.

The most obvious problem with the Ostrich response is that it relies on the notion of a proposition. But this notion cannot be understood apart from the notions of truth and falsity.  Propositions are standardly introduced as the primary vehicles of the truth-values. They alone are the items appropriately characterizable as either true or false. Therefore, to understand what a proposition is one must have an antecedent grasp of the difference between truth and falsity. 

To understand the operation of negation we have to understand that upon which negation operates, namely, propositions, and to understand propositions, we need to understand truth and falsity.

A second problem is this. Suppose contingent p maps onto proper subset T of S.  Why that mapping rather than some other? Because T is the set of situations or worlds in which p is true . . . . The circularity again rears its ugly head.

The Ostrich, being a nominalist, might try to dispense with propositions in favor of declarative sentences. But when we learned our grammar back in grammar school we learned that a declarative sentence is one that expresses a complete thought, and a complete thought is — wait for it — a proposition or what Frege calls ein Gedanke: not a thinking, but the accusative of a thinking. 

Truth and falsity resist elimination.

Excluded Middle, Presentism, Truth-Maker: An Aporetic Triad

Suppose we acquiesce in the conflation of Excluded Middle and Bivalence.  The conflation is not unreasonable.  Now try this trio on for size:

Excluded Middle: Every proposition is either true, or if not true, then false.
Presentism: Only what exists at present, exists.
Truth-Maker: Every contingent truth has a truth-maker.

The limbs of the triad are individually plausible but collectively inconsistent. Why inconsistent?

I will die. This future-tensed sentence is true now. It is true that I will die. Is there something existing at present that could serve as truth-maker? Arguably yes, my being mortal. I am now mortal, and my present mortality suffices for the truth of 'I will die.' Something similar holds for my coat. It is true now that it will cease to exist.  While it is inevitable that I will die and that my coat will cease to exist, it is not inevitable that my coat will be burnt up (wholly consumed by fire).  For there are other ways for it to cease to exist, by being cut to pieces, for example, or by just wearing out.

By 'future contingent,' I mean a presently true future-tensed contingent proposition.  The following seems to be a clear example: BV's coat will sometime in the future cease to exist by being wholly consumed in a fire. To save keystrokes: My coat will be burnt up.

By Excluded Middle, either my coat will be burnt up or my coat will not be burnt up. One of these propositions must be true, and whichever one it is, it is true now. Suppose it is true now that my  coat will be burnt up. There  is nothing existing at present that could serve as truth-maker for this contingent truth.  And given Presentism, there is nothing existing at all that could serve as truth-maker.  For on Presentism, only what exists now, exists full stop.  The first two limbs, taken in conjunction, entail the negation of the third, Truth-Maker.  The triad is therefore inconsistent.

So one of the limbs must be rejected. Which one?

An Objection

You say that nothing that now exists could serve as the truth-maker of the presently true future-tensed  contingent proposition BV's coat will be burnt up. I disagree.  If determinism is true, then the present state of the world together with the laws of nature necessitates every later state.   Assuming the truth of the proposition in question, there is a later state of the world in which your shabby coat is burnt up. The truth-maker of the future contingent proposition would then be the present state of the world plus the laws of nature.  So if determinism is true, your triad is consistent, contrary to what you maintain, and we will not be forced to give up one of the very plausible constituent propositions.

Question: Is there a plausible reply to this objection? No. I'll explain why later.

Excluded Middle, Bivalence, and Disquotation

LEM: For every  p, p v ~p.

BV: Every proposition is either true or false.

These principles are obviously not identical.  Excluded Middle is syntactic principle, a law of logic, whereas Bivalence is a semantic principle. The first says nothing about truth or falsity. The second does. (See Michael Dummett, Truth and Other Enigmas, Harvard UP, 2nd ed. , 1980, p. xix; Paul Horwich, Truth, Oxford UP, 2nd ed., 1998, p. 79) Though not identical they might nonetheless be logically equivalent.  Two propositions are logically equivalent iff each entails the other.  Entailment is the necessitation of material implication. Can it be shown that (LEM) and (BV)  entail each other? Let's see.

The logical equivalence of the two principles can be demonstrated if we assume the disquotational schema:

DS: p is true iff p.

For example, snow is white is true iff snow is white. Or, if you insist, 'snow is white' is true iff snow is white. In the latter forrmulation, which does not involve reference to propositions, the truth predicate  — 'is true' — is merely a device of disquotation or of semantic descent. On either formulation, 'is true' adds no sentential/propositional content:  the sentential/propositional content is the same on both sides of the biconditional.  The content of my assertion is exactly the same whether I assert that snow is white or I assert that snow is white is true.  But if (DS) is granted, then so is:

DS-F: p is false iff ~p.

For example, snow is white is false iff ~(snow is white).    

Now if the disquotational schemata exhaust what it is to be true and what it is to be false, then (LEM) and (BV) are logically equivalent.

Given (DS) and (DS-F), we can rewrite (LEM) as

LEM-T: For every p, p is true v p is false.

Now (LEM-T) is simply a restatement of (BV). The principles are therefore logically equivalent given the disquotational schemata. 

But this works only if falsehood can be adequately explained in terms of the merely logical operation of negation.  This will NOT work if negation can only be explained in terms of falsehood.  For then we would enter  an explanatory circle of embarrassingly short diameter. 

Ask yourself: when is one proposition the negation of another? The negation of p is the proposition that is true iff p is false and false iff p is true.  To explain the logico-syntactic notion of negation we have to reach for the semantic notions of truth and falsehood.  But then falsehood cannot be exhaustively understood or reduced to negation.

It is telling that to explain negation and the other logical connectives we use TRUTH tables.  Such explanation is satisfactory.  But it would not be if the redundancy or disappearance or disquotational schemata gave the whole meaning of 'true' and 'false.'  (The point is made by M. Dummett, Truth and Other Enigmas, p. 7)

I take this explanatory circle to show that there is more to truth and falsehood than is captured in the above disquotational schemata.

Conclusion: if one's reason for accepting the logical equivalence of (LEM) and (BV) is (DS) then that is a bad reason.

Are there counterexamples to (DS)?  It seems to fail right-to-left if 'Sherlock Holmes is a detective' is plugged in for 'p' on the RHS of (DS).  Arguably, Holmes is a detective, but it is not true that Holmes is a detective.  For it to be true that Holmes is a detective, 'Holmes' would have to refer to something that exists.  But this requirement is not satisfied in the case of purely fictional items.  I am assuming that veritas sequitur esse, that truth 'follows' or supervenes upon being (existence):

VSE:  There are no true predications about what does not exist.

Since Holmes does not exist, 'Holmes is a detective' appears to express a proposition that is neither true nor false. Likewise for its negation, 'Holmes is not a detective.'  (LEM) is not violated since either Holmes is a detective or Holmes is not a detective. But (BV) is violated since the two Holmes propositions are neither true nor false.

It is worth noting that from 'Only propositions have truth-values' one cannot validly infer 'All propositions have truth-values.'  

Excluded Middle, Bivalence, and Tertium Non Datur

Dave Gudeman comments:

I was surprised to see you distinguishing between bivalence and the LEM. As far as I can tell, in the traditional and most common formulations, they are identical.

Here is the way I understand it.  They are not identical.  Excluded Middle is a law of logic, whereas Bivalence is a semantic principle. (See Michael Dummett, Truth and Other Enigmas, Harvard UP, 2nd ed. , 1980, p. xix; Paul Horwich, Truth, Oxford UP, 2nd ed., 1998, p. 79) If 'p' is a place-holder for a proposition, any proposition, then Excluded Middle is:

LEM. p v ~p.

If 'p' is a propositional variable, and we quantify over propositions, then we have the universal quantification

LEM*. For all p, p v ~p.

It is understood that the wedge in the above formulae signifies exclusive disjunction. Why is that understood? Because both p and not-p is excluded by the Law of Non-Contradiction:

LNC. ~(p & ~p).  

If I may be permitted parenthetically to wax poetic in these aseptic precincts, (LNC) possesses a 'dignity' in excess of that possessed by (LEM). What I mean is that there are some fairly plausible counterexamples to (LEM), but none that are very plausible to (LNC).  Few philosophers are dialetheists; many more accept truth-value gaps.

The laws of logic are purely formal: they abstract from content or meaning. They are syntactic principles. Bivalence, by contrast, is a semantic principle. It goes like this:

BV. Every proposition is either true or false.

Tertium non datur means that a third is not given: there is no third truth value.  (TND) is also a semantic principle:

TND. No proposition is neither true nor false.

So the difference between (LEM) and (BV) is that the first is a syntactic principle and the second a semantic principle. But is this a difference that makes a difference? Is there a conceivable case where (LEM) is true but (BV) false?  I don't know the answers to these questions. Either that or I forgot them.

But if you conflate the two principles,  then you are in good company. W. V. O. Quine, Mathematical Logic, Harvard UP, 8th ed., 1976, p. 51: ". . . the law of excluded middle, which is commonly phrased as saying that every statement is either true or false . . . ."

Atomic Sentences and Syncategorematic Elements

The Ostrich tells me that Frege has no copula. That's not wrong, but there is a nuance that muddies the waters. Suppose Al is fat. The symbolization as Fa suggests the absence of a copula and thus the absence of a syncategorematic element. There appears to be only two categorematic elements, a and F. Well, let's see.

………………………………..

According to Fred Sommers (The Logic of Natural Language, Oxford UP, 1982, 166), ". . . one way of saying what an atomic sentence is is to say that it is the kind of sentence that contains only categorematic expressions." Earlier in the same book, Sommers says this:

In Frege, the distinction between subjects and predicates is not due to any difference of syncategorematic elements since the basic subject-predicate propositions are devoid of such elements.  In Frege, the difference between subject and predicate is a primitive difference between two kinds of categorematic expressions. (p. 17)

Examples of categorematic (non-logical) expressions are 'Socrates' and 'mammal.'  Examples of syncategorematic (logical) expressions are 'not,' 'every,' and  'and.'  As 'syn' suggests, the latter expressions are not semantic stand-alones, but have their meaning only together with categorematic expressions.  Sommers puts it this way: "Categorematic expressions apply to things and states of affairs; syncategorematic expressions do not." (164) 

At first I found it perfectly obvious that atomic sentences have only categorematic elements, but now I have doubts.  Consider the atomic sentence  'Al is fat.' It is symbolized thusly: Fa.  'F' is a predicate expression the reference (Bedeutung) of which is a Fregean concept (Begriff) while 'a' is a subject-expression or name the reference of which is a Fregean object (Gegenstand).  Both expressions are categorematic or 'non-logical.'  Neither is syncategorematic.  And there are supposed to be no syncategorematic elements in the sentence:  there is just 'F' and 'a.'

But wait a minute!  What about the immediate juxtaposition of 'F' and 'a' in that order? That juxtaposition is not nothing.  It conveys something.  It conveys that the referent of 'a' falls under the referent of 'F'.  It conveys that the object a instantiates the concept F. I suggest that the juxtaposition of the two signs is a syncategorematic element.  If this is right, then it is false that atomic sentences lack all syncategorematic elements.

Of course, there is no special sign for the immediate juxtaposition of 'F' and 'a' in 'Fa.'  So I grant that there is no syncategorematic element if such an element must have its own separate and isolable sign. But there is no need for a separate sign; the immediate juxtaposition does the trick.  The syncategorematic element is precisely the juxtaposition.

Please note that if there were no syncategorematic element in 'Fa' there would not be any sentence at all.  A sentence is not a list.  The sentence 'Fa' is not the list 'F, a.'  A (declarative) sentence expresses a thought (Gedanke) which is its sense (Sinn).  And it has a reference (Bedeutung), namely a truth value (Wahrheitswert).  No list of words (or of anything else) expresses a thought or has a truth value.  So a sentence is not a list of its constituent words.  A sentence depends on its constituent words, but it is more than them.  It is their unity. 

We here touch upon the ancient problem of the unity of the proposition first descried by the immortal Plato.

So I say there must be a syncategorematic element in 'Fa' if it is to be a sentence.  There is need of a copulative element to tie together subject and predicate.  It follows that, pace Sommers, it is false that atomic sentences are devoid of syntagorematic elements.

Note what I am NOT saying.  I am not saying that the copulative element in a sentence must be a separate sign such as 'is.'  There is no need for the copulative  'is.'  In standard English we say 'The sea is blue' not 'The sea blue.' But in Turkish one can say Deniz mavi and it is correct and intelligible.  My point is not that we need the copulative 'is' as a separate sign but that we need a copulative element which, though it does not refer to anything, yet ties together subject and predicate.  There must be some feature of the atomic sentence that functions as the copulative element, if not immediate juxtaposition then something else such as a font difference or color difference.

At his point I will be reminded that Frege's concepts (Begriffe) are unsaturated (ungesaettigt).  They are 'gappy' or incomplete unlike objects.  The incompleteness of concepts is reflected in the incompleteness of predicate expressions.  Thus '. . . is fat' has a gap in it, a gap fit to accept a name such as 'Al' which has no gap.  We can thus say that for Frege the copula is imported into the predicate.  It might be thought that the gappiness of concepts and predicate expressions obviates the need for a copulative element in the sentence and in the corresponding Thought (Gedanke) or proposition.

But this would be a mistake.  For even if predicate expressions and concepts are unsaturated, there is still a difference between a list and a sentence.  The unsaturatedness of a concept merely means that it combines with an object without the need of a tertium quid.  (If there were a third thing, then Bradley's regress would be up and running.)  But to express that a concept is in fact instantiated by an object requires more than a listing of a concept-word (Begriffswort) and a name.  There is need of a syncategorematical element in the sentence.

So I conclude that if there are any atomic sentences, then they cannot contain only categorematic expressions.

Presupposition and Excluded Middle

If Socrates dies at time t, then Socrates was alive prior to t. If Socrates does not die at t, then Socrates was alive prior to t.  Since both 'Socrates dies at t' and 'Socrates does not die at t' entail 'Socrates was alive prior t,' we say that the latter is a semantic presupposition of 'Socrates dies at t.'

But wait a minute! Doesn't what I have written generate an inconsistent tetrad?

1) p entails q

2) Not-p entails q

3) Necessarily, for any p, either p or not-p (Law of Excluded Middle)

4) q is contingent.

The conjunction of the first three limbs entails the negation of the fourth. So something has to give.

It is a datum that q — 'Socrates was alive prior to t' — is contingent: true in some but not all possible worlds. So we either reject semantic presupposition (which requires the truth of both (1) and (2) ) or we reject Excluded Middle.

Why not reject Excluded Middle? Socrates dies at t and Socrates does not die at t are contradictories: each is the negation of the other.  There is no possible world in which both are true.  And yet there are possible worlds in which neither is true. Those are the worlds in which Socrates does not exist.