Bill Vallicella – Page 502 – Maverick Philosopher

Placeholders, Variables, and Logical Form

London Ed refers us to Understanding Arguments: an Introduction to Informal Logic, Robert Fogelin and Walter Sinnott-Armstrong, and provides this quotation:

Perhaps a bit more surprisingly, our definitions allow 'roses are red and roses are red' to be a substitution instance of 'p & q'. This example makes sense if you compare it to variables in mathematics. Using only positive integers, how many solutions are there to the equation 'x + y = 4'? There are three: 3+1, 1+3, and 2+2. The fact that '2+2' is a solution to 'x + y = 4' shows that '2' can be substituted for both 'x' and 'y' in the same solution. That's just like allowing 'roses are red' to be substituted for both 'p' and 'q', so that 'roses are red and roses are red' is a substitution instance of 'p & q' in propositional logic.

In general, then, we get a substitution instance of a propositional form by uniformly replacing the same variable with the same proposition throughout, but different variables do not have to be replaced with different propositions. The rule is this:

Different variables may be replaced with the same proposition [Ed: Let's call this the London rule], but different propositions may not be replaced with the same variable.

Suppose I am given the task of determining whether the conditional English sentence 'If roses are red, then roses are red' is a tautology, a contradiction, or a contingency. How do I proceed?

Step One is translation, or encoding. Let upper case letters serve as placeholders for propositions. Let '–>' denote the truth-functional connective known in the trade as the material or Philonian conditional. I write 'P –> P.'

Step Two is evaluation. Suppose for reductio that the truth value of 'P –>P' is false. Then, by the definition of the Philonian conditional, we know that the antecedent must be true, and the consequent false. But antecedent and consequent are the same proposition. Therefore, the same proposition is both true and false. This is a contradiction. Therefore, the assumption that conditional is false is itself false. Therefore the conditional is a tautology.

Now that obviously is the right answer since you don't need logic to know that 'If roses are red, then roses are red' is a tautology. (Assuming you know the definition of 'tautology.') But if if Fogelin & Co. are right, and the 'P –>Q' encoding is permitted, then we get the wrong answer, namely, that the English conditional is a contingency.

I am assuming that if 'P–>Q' is a logical form of 'If roses are red, then roses are red,' then 'P –>Q' is a legitimate translation of 'If roses are red, then roses are red.' As Heraclitus said, the way up and the way down are the same. The assumption seems correct.

If I am right, then there must be something wrong with the mathematical analogy. Now there is no doubt that Fogelin and his side kick are right when it comes to mathematics. And I allow that what they say is true about variables in general. Suppose I want to translate into first-order predicate logic with identity the sentence, 'There is exactly one wise man.' I would write, '[(Ex)Wx & (y)(Wy –> x = y)].' Suppose Siddartha is the unique wise man. Then Siddartha is both the value of 'x' and the value of 'y.'

So different variables can have the same value. And they can have the same substituend. In the example, Siddartha is the value and 'Siddartha' is the substituend. But is a placeholder the same as a variable? I don't think so. Here is a little argument:

No variable is a constant
Every placeholder is an arbitrary constant
Every arbitrary constant is a constant
——-
No placeholder is a variable.

A placeholder is neither an abbreviation, nor a variable. It is an arbitrary constant. Thus the logical form of 'Al is fat' is Fa, not Fx. Fa is a proposition, not a propositional function. 'F' is a predicate constant. 'a' is an individual constant. We cannot symbolize 'Al is fat' as Fx. For Fx is not a proposition but a propositional function. If 'a' were not an arbitrary constant, then Fa would not depict the logical form of 'Al is fat,' a form it shares with other atomic sentences.

Here is another argument:

Every variable is either free or bound by a quantifier
No placeholder is either free or bound by a quantifier
——-
No placeholder is a variable.

Here is a third argument:

Every variable has a domain over which it ranges
No placeholder has a domain over which it ranges
——-
No placeholder is a variable.

A fourth argument:

There is no quantification over propositions in the propositional calculus
——-
There are no propositional variables in the propositional calculus
If there are no propositional variables in the propositional calculus, then the placeholders in the propositional calculus cannot be variables
——-
The placeholders in the proposition calculus cannot be variables.

Punchline: because placeholders are not variables, the fact that the different variables can have the same value and the same substituend does not show that different placeholders can have the same substituend. 'If roses are red, then roses are red' does not have the logical form 'P –>Q' and the latter form does not have as a substitutution-instance 'If roses are red, then roses are red.'

As I have said many times already, one cannot abstract away from the fact that the same proposition is both antecedent and consequent.

What one could say, perhaps, is that 'P –> P' has the higher order form 'P –> Q.' But this latter form is not a form of the English sentence but a form of the form of the English sentence.

Ed can appeal to authority all he wants, but that is an unphilosophical move, indeed an informal fallacy. He needs to show where I am going wrong.

On Her Deathbed: “I Fear that There is Nothing on the Other Side”

This from a correspondent:

My grandmother is on her deathbed. My mother flew out to Boston to be there with her when she dies. Of course my grandmother is putting up a good fight; however, they expected her to die yesterday. My mother had a conversation with her while she was lucid. She asked her, “Why are you fighting so hard? Do you fear something?”

My grandmother’s reply, “I fear that there is nothing on the other side.” Here is a woman who has spent eighty nine years of her life devoting herself to the [Catholic] church and her family. Now, when it comes down to death she is clinging on because her entire life is behind her and the only thing that she faces in front of her is the uncertainty of whether there is a heaven awaiting her in the coming days.

If you were there at my grandmother’s deathbed and she would convey to you her fears, what would you tell her?

I'm a philosopher, not a pastor, and what a dying nonphilosopher needs is pastoral care, not philosophical dialog. But if I were to play the pastor I would say something along the following lines.

"You have lived your long life faithfully and devotedly in the embrace of Holy Mother the Church. She has presided over central events in your life, your baptism, first communion, confirmation, and your marriage. She has provided guidance, moral instruction, comfort, and community as you have navigated life's difficulties and disappointments. She provided meaning and solace when your parents died, and your husband, and your many friends and relatives. If your faith was a living faith and not a convenience or a matter of social conformity, then from time to time you had your doubts. But through prayer and reflection you have repeatedly reaffirmed your faith. You faith was made deeper and truer by those doubts and their overcoming."

"I ask you now to recall those moments of calm reflection and existential lucidity, those moments when you were at your best physically, mentally, and spiritually. I ask you to recall them, and above all I ask you not to betray them now when you are weak. Do not allow the decisions and resolutions of your finest and and clearest hours to be taken hostage by doubts and fears born of weakness. Your weakness has called forth the most vicious attacks of the Adversary and his agents. You have lived in the faith and now you must remain true to a course of life judged right at the height of your powers. Your doubts are of the devil and they must be put aside. Pray, and remain true to a course judged right."

So that is what I would say to the old Irish Catholic woman on her deathbed. I would exhort her to remain true to a course judged right in the moments of her highest existential lucidity and to bring her life to a successful completion. The hour of death is not the time to grapple with the devil of doubt!

To myself and the others for whom the hora mortis is still a ways off, to those in the sunshine of their strength, physical and mental, I say the following. Now is the time to wrestle with doubts and either defeat them or succumb to them. Now is the time to get serious about The Last Things. It is far better to get serious about them before they get serious about you. Now is the time to face the reality of death without evasion and to prepare for a happy death. Now is the time to realize that you don't have all the time in the world, that as the Zen Master Dogen says, "Impermanence is swift." Now is the time to stop fooling yourself about how you are going to live forever. For "What is your life? You are a mist that appears for a little while and then vanishes." (James 3, 14)

Related: Six Types of Death Fear

Three Possible Death Bed Thoughts

Boko Haram and the Kidnapped School Girls

An important piece by Ayaan Hirsi Ali.

On the Scientism Front

We defenders of the humanities need to do battle on three fronts against three enemies: scientism, leftism, Islamism. Each is represented by a disturbing number of crapweasels, individuals who won't own up to who and what they are. Thus prominent scientisticists — to give an ugly name to an ugly bunch — will deny that there there is any such thing as scientism. (See my Scientism category for documentation.) And the same goes, mutatis mutandis, for the Pee Cee crowd and the Islamists.

Here are two links so that you may know your enemies.

Thomas Nagel and Stephen C. Myer's Signature in the Cell

Why Neil de Grasse Tyson is a Philistine (HTs: Dave Lull, J. Orsak, W. Chambers, et al.)

Dallas Willard Remembered

Dallas Willard died on this date one year ago. Here is what I wrote at the time.

The Eliminative Materialist

The eliminative materialist is a bit like a man who blows his brains out to be rid of a headache. No head, no headache, no problem!

Are Rednecks People of Color?

If not, why not?

How Reasonable is it to Rely on Reason Alone?

Edith Stein, Finite and Eternal Being, tr. Reinhardt, ICS Publications, 2002, p. 22:

Reason would turn into unreason if it would stubbornly content itself with what it is able to discover with its own light, barring out everything which is made visible to it by a brighter and more sublime light.

Is it unreasonable to rely on reason alone, or is this exactly what reason demands? If the latter, how could reason validate its demand? Reason cannot validate itself by appeal to itself: A circular validation is no validation at all. So it is by a sort of transrational faith that reason relies on itself and accepts only what it can validate by its own lights. But if reason allows transrational faith in justification of itself, then it ought to be open to other transrational or suprarational sources of insight.

Solubility Skepticism, Religion, and Reason

Ruffin Crozat writes,

There is much depth in your short post on religion and reason from 6 May. Here are two points I often ponder about this topic:

First, I appreciate the difficulty of solving philosophical problems, but I wonder about the claim that they are insoluble (I suppose “insoluble” means “insoluble by humans alone”). If the problems are beyond mere human knowledge, how could we know this? One may inductively suspect insolubility by reflecting upon his experience of practicing philosophy, but how could he know the unknowable? If we can’t solve philosophical problems by philosophizing, then it seems we can’t conclude insolubility by philosophizing because this very conclusion would be a philosophical conclusion.

BV: I hold that the central problems of philosophy are most of them genuine, some of them humanly important, but all of them insoluble. And you are right, by 'insoluble' I mean insoluble by us or by beings of a similar cognitive architecture, ectypal intellects in Kant's jargon. Furthermore, pace Nicholas Rescher, I don't count a 'solution' that is relative to some set of background assumptions and cognitive values as a solution. Of course there are solutions in this sense. Nominalists solve the problem of univerals in one way, realists in another, conceptualists in a third, etc. But those are merely intramural solutions. What is wanted are solutions acceptable to all, solutions that hold ouside the walls of self-reinforcing enclaves of the like-minded.

You ask a very important question: How could one know that the central philosophical problems are insoluble? You yourself supplied the clue: by induction from philosophical experience. The best and the brightest have been at this game for thousands of years but not one single problem has been solved during this period, solved to the satisfaction of all competent practitioners. Everything is up for grabs, even the most elementary and picayune topics. Take a look at what is going one as we speak in the thread on logical form. Philosophers can't even agree on the most basic concepts of deductive logic. There is controversy everywhere. This is a plain fact.

The strife of systems and the ubiquity and longevity of controversy need explaining and I offer the insolubility thesis as the best explanation. Why haven't the problems been solved? Because they are insoluble. I agree with Benson Mates on this point. Of course, the following is an invalid argument form: Such-and-such has hitherto not been accomplished; ergo, such-and-such will never be accomplished. But then every inductive argument is invalid. Some inductive arguments, however, do quite reasonably support their conclusions.

But you can and should press your objection. If I maintain that the problems of philosophy are insoluble, then, given that the metaphilosophical problem of whether or not philosophical problems are soluble is a philosophical problem, it follows that the metaphilosophical problem is insoluble. Is this a difficult for my position? Not obviously. I simply 'bite the bullet' as they say. I accept that the meta problem is also insoluble.

In fact, the insolubility of the meta problem is further evidence of my thesis.

In other words, I am not dogmatizing. I am not claiming to know with certainty that the problems of philosophy are insoluble. I am not claiming to have solved the meta problem. I am merely claiming that the insolubility thesis is very reasonably maintained. Not every truth is such that we can know it to be true. With some truths the most we can expect here below is reasonable belief.

Compare God and the soul. I do not claim to know with certainty whether either exists. I claim merely that it it is reasonable to affirm both.

Second, I agree that it’s wise to intelligently practice religion and mysticism — which, by the way, rules out superstition and group-think! Take religion: religious practice does not exclude reason, as Mates’ quote implies. It is a false dilemma to say “One can seek truth either by reason or religion, but not both.” Why not both? If I try to lift a stone and realize I can’t manage alone, this would not entail that I can or should stop lifting. If a stronger person assists me, and I trust his assistance, I can still lift. He may request my help. He may even require that I give it my all, and I may grow from the effort. Likewise, intelligent religion requires reason.

Consider Christianity: The biblical conception of faith is “trust based on good reasons”. This point is clear in passages such as Hebrews 11:1 and 1 Peter 3:15. In the Gospels, Jesus himself reasons and encourages others to do the same. Christian faith calls for the whole self: heart, mind, soul, and strength.

I’d be interested in your thoughts on reason and intelligent religion.

BV: I basically agree with you. Reason in the end must confess its own infirmity. It cannot deliver on its promises. The truth-seeker must explore other avenues. Religion is one, mysticism is another.

Logical Form and the Symmetry Thesis

"The most conspicuous purpose of logic, in its applications to science and everyday discourse, is the justification and criticism of inference." (Emphasis added, Willard Van Orman Quine, Methods of Logic, 2nd revised ed., Holt, Rinehart & Winston, 1959, p. 33.

Perhaps the dispute in the earlier thread could be resolved if we all could agree on the following.

1. The most specific logical form of a deductive argument A is the form relevant for assessing whether the reasoning embodied in A is valid or invalid.

2. Every deductive argument has exactly one most specific form.

3. Symmetry Thesis: if the most specific form of A is valid, then A is valid; if the most specific form of A is invalid, then A is invalid.

In case 'most specific logical form' needs explanation, consider the difference between the following valid form from the predicate calculus and the following invalid form from the propositional calculus:

Fa
Ga
——-
(Ex)(Fx & Gx)

p
q
——-
r.

The former is the most specific logical form of 'Al is fat, Al is gay, ergo, something is both fat and gay.' The latter, if a form of the argument at all, is less specific: it abstracts from the internal subpropositional logical structure of the constituent propositions.

Now three examples in illustration of (1)-(3).

Example One. Call the following argument 'Charley':

Tom is tall
——-
Tom is tall.

Although the above display, which is a written expression of the argument and not the argument itself, shows two tokens of the sentence type 'Tom is tall,' the argument consists of exactly one proposition. Anyone who executes the reasoning displayed infers the proposition *Tom is tall* from itself. (I am using asterisks to mention propositions. So '*Tom is tall*' is an abbreviation of 'the proposition expressed by a tokening of the sentence type "Tom is tall".')

It is perfectly clear that the reasoning embodied by Charley is valid and that its form is 'P ergo P.' The reasoning is not from P to some proposition that may or may not be identical to P. Therefore the concrete episode of reasoning does not have the form 'P ergo Q.'

But let us irenically concede that if one wished, for whatever reason, to abstract not only from the content of the argument but also from the plain fact that the argument involves exactly one proposition, one could view the form 'P ergo P' as a special case of 'P ergo Q.' And I will also concede, to keep peace between Phoenix and London, that the argument instantiates the second invalid form, even though I don't believe that this is the case.

Either way, the Symmetry Thesis stands and the Asymmetry Thesis falls. For as G. Rodrigues in the earlier thread pointed out, 'P ergo P' is the most specific form of Charley.

Example Two. Call the following argument 'Kitty Kat.'

If cats like cream, then cats like cream
Cats like cream
——-
Cats like cream.

Please note that there is no equivocation in this example: 'Cats like cream' has the same sense in all four of its occurrences.

Kitty Kat's most specific form is 'P –> P, P, ergo P.' This form is valid. So Kitty Kat is valid, notwithstanding the fact, if it is a fact, that Kitty Kat also instantiates the formal fallacy, Affirming the Consequent: P –> Q, Q, ergo P. By (1) above, the fact, if it is a fact, that Kitty Kat instantiates Affirming the Consequent is irrelevant to the assessment of the validity/invalidty of the reasoning embodied in Kitty Kat.

Example Three. Call the following example 'Massey':

If God created something , then God created everything.
God created everything.
——-
God created something.

This argument fits the pattern of the formal fallacy, Affirming the Consequent:

If p then q
q
——-
p.

But the argument also has a valid form:

Every x is such that Cgx
——-
Some x is such that Cgx.

Please note that if an argument is valid, adding a premise can't make it invalid; this principle is what allows us to disregard the first line.

(Example adapted from Gerald J. Massey, "The Fallacy behind Fallacies," Midwest Studies in Philosophy VI (1981), pp. 489-500)

The most specific form of Massey is the predicate logic form above displayed. Since it is valid, Massey is valid.

Symmetry Thesis vindicatus est.

Is everybody happy now?

The Logic of Buddhist Philosophy

Beyond True and False, by Graham Priest. (HT: Allan Jackson)

A Question for Benson Mates

According to Benson Mates (1919-2009), all the major problems of philosophy are "insoluble though intelligible." (Skeptical Essays, U. of Chicago Press, 1981, p. 13) If true, this would explain why the problems of philosophy have not been solved. But "the rational minds among us are not inclined to give up the struggle, while the rest become religious mystics or philosophical obscurantists . . . ." (p. x)

But why continue to struggle with the problems of philosophy? To better appreciate the insolubility thesis? Apparently, Mates thinks that while the problems can't be solved or dissolved, one ought to keep trying to solve them anyway. But how rational is this? I should think that a "rational mind" should not attempt to do what he has already convinced himself cannot be done. Is it not more rational to seek a path to truth beyond philosophy?

How rational is it to place one's sole faith in reason when one has, by one's own lights, seen the infirmity of reason?

If a certain weight needs lifting, a weight beyond my ability to lift, and known to be such, does it make sense to struggle with it? Or is it more rational to seek assistance? By rejecting out of hand the assistance of religion and mysticism – which he foolishly conflates — Mates shows that his commitment to reason is irrational, as irrational as my pride-driven conceit that I am master of any difficulty that I should encounter.

The Liberal

A liberal is the kind of person who would extend the right to vote to felons but will not grant the right to life to the unborn. How 'liberal' is that? How 'inclusive'? How respectful of 'diversity'?

Should felons be allowed to vote? The conservative answers with alacrity. "Of course not. Why should those who cannot order their own lives prudently be allowed to have a say in the ordering of society?"

There is no wisdom on the left.

The Proctology of a Pessimist

Arthur Schopenhauer was a foe of noise in all its forms, as one can see from his delightful essay, On Noise. The “infernal cracking of whips” especially got on his nerves. (One wonders what he would say about the Beelzebubic booming of boom boxes.)

One day, a cleaning lady made what he considered to be an excessive racket outside his rooms. He asked her to quiet down, which led to an argument. Push came to shove, and the lady ended up at the foot of the stairs. The local court ruled in favor of the Putzfrau, and Schopenhauer was ordered to pay her a monthly sum of money for the rest of her long life. When at last she died, the philosopher opened his journal and penned what is arguably the greatest Latin pun of all time: Anus obit, onus abit.

What wit, what pith, what anagrammatical punsterism! All hail to Schopenhauer and his scowl of Minerva! Note first that the line is an anagram: there are two constructions, in this case two independent clauses, each of which represents a transposition of the letters of the other. A second example of an anagram: Democritus docet risum = Democritus teaches laughingly. The second thing to note is that ‘anus’ has two Latin meanings depending on whether the ‘a’ is short or long. Short, it means alte Frau, Greisin, old woman. (My Latin dictionary is Lateinisch-Deutsch.) Long, it means 1) Fussring, 2) (euphem.) After (= anus in the English sense).

Schopenhauer’s aphorism in English: The old woman/anus is dead; the burden is lifted. So Schopenhauer was not necessarily being crude, though of course he was punning.

Decadent Art, Buddhist Statuary, and the Taliban

Our Czech friend, Vlastimil Vohanka, writes:

A question: Do you remember the title of your blog post in which you argued, if I recall correctly, that the Taliban damage to the Buddha statues would be evil — or ought not to take place — even if nobody ever got to know about it? I also recall dimly that the post was a reply to Peter Lupu. Is the post still online, somewhere?

Vlasta, I believe you are referring to this post. It was a response, not to Peter Lupu, but to Mike Valle. (I had the pleasure of their company at Sunday breakfast yesterday.)

Here is how the post begins:

This by e-mail from a doctoral student in Canada:

I am writing to you because I have a couple of questions . . . about your recent (May 12) blog post, and I was curious to hear a bit more about your views. [. . .] My questions concern your assertion that "I also agree that if one is going to violate people's beliefs in the manner of that 'artist' Andres Serrano then one ought to do it on one's own time and with one's own dime, as the saying goes." I assume that you're referring to "Piss Christ" and the controversy that surrounded it.

That's right. Context is provided by Mike Valle's post to which I was responding.

1. Why do you feel that "Piss Christ" (or Serrano's other works–again, I assume you're referring here mostly to the religious icons and bodily fluids) is (are) a "[violation] of people's beliefs"? The claim that it "violates beliefs" is much stronger than simply saying that it is distasteful, since it ascribes an active quality to the work.

Of course, it is more than distasteful or disgusting, although it is that; it shows profound disrespect and contempt for Christianity. And it is not the work itself that violates the beliefs and sensibilities of Christians and plenty of non-Christians as well, but the work in the context of its production and public display. It should be offensive to any decent person, just as "Piss-Buddha," if there were such an 'art work,' would be offensive to me and other non-Buddhists. Buddha was a great teacher of humanity and should be honored as such. (That is why decent people were offended when the Taliban destroyed the ancient Buddhist statuary.) The same goes for Jesus and Socrates and so many others. Christians of course believe that Jesus was much more than a great teacher of humanity, but whether he was or not is immaterial to the point at issue. Or imagine "Piss-King" in which a figurine of Martin Luther King, Jr. is supended in urine. Everyone would take that, and rightly so, as expressive of contempt for the black American civil rights leader, as offensive as Southern racists' references to King back in the '60s as Martin Luther Coon.

The decadent art of the 20th century reflects not only the corruption of aesthetic sensibility but also a moral corruption. So my objection to Serrano is not merely aesthetic but moral. The purpose of art is not to debase but to elevate, refine, ennoble.

[. . .]