Proofs and Demonstrations

There seems to be widespread misunderstanding about what proofs are, or as I prefer to call them, demonstrations. The Greek word is ἀπόδειξις, which means “to make something manifest, to show something, to place it before one’s view as evident.”

For something to be evident (as you can surmise from the vid– root) is for it to be “fully seen, completely in view, manifest to one’s eyes”.

A demonstration aims to make the truth of something evident.  “Evidentness” is the aim or end of a proof or demonstration.  This is important, because “evidentness” is also involved in every demonstration, and not simply as its end.

For a demonstration to succeed in making something evident, it must be both logically valid and have true premises.  And the validity of the demonstration as well as the truth of its premises must themselves be evident.  Sometimes this requires further argument or demonstration, but sometimes it does not.

Demonstrations are arguments, and they take place at the level of λόγος or discursive reasoning.  The level of λόγος is also the level of language, so all arguments, proofs, and demonstrations (I am using these as synonyms now) are presented in language, which can be informal or highly formalized.

The dependence of demonstrations on evidentness is shown in that

  1. Demonstrations aim at making the truth of a proposition, the conclusion, evident.
  2. Demonstrations require evident propositions, i.e. true premises, to succeed.
  3. Demonstrations require evidently valid reasoning whereby the conclusion is reached.

What this shows is that there is a power of the soul upon which discursive reasoning, λόγος, depends, and to which it is inferior. In Greek, this power is called νόησις [noēsis].  In Latin, it is called intellectus.  It comes uncertainly into English, sometimes as “the intellect,” sometimes as “intelligence” sometimes as “understanding” or “the understanding,” and sometimes as “intuition.”  Modern English-speaking analytic philosophers sometimes call it “a priori intuition.”

Personally, I loathe to word “intuition,” since it connotes, to me and many people, something like “a vague hunch or feeling about something.” And νόησις is the opposite of vague.  It is, in fact, the very touchstone of certain knowledge.

Let me take an example I’m fond of, given by Charles Dodgson (aka Lewis Carroll of Alice in Wonderland fame).  One of the simplest logical forms is modus ponens, which has the following schema:

  1. P ⇒ Q
  2. P
  3. ∴ Q


  1. If P is true then Q is true,
  2. P is true.
  3. So Q is true.

Well and good. How do we know this is valid? How do we know it is the case that “If P is true then Q is true” taken together with “P is true” allows us to know that Q is true?

The answer is, we see that it is so. That is to say, it is evident. Or if you prefer, noetically evident.

There is no possible way in which modus ponens can be demonstrated to be valid. As with many other basic logical truths and axioms, it is “too simple” to be demonstrated. It is, rather, one of the basic elements any demonstration always and necessarily relies on.

Aristotle famously emphasizes this, that it is an error (and he adds, a sign of lack of sufficient education) to think that everything requires a demonstration. If this were the case, then every demonstration would require a demonstration, and that too another demonstration, resulting in an infinite regress, such that nothing would ever be demonstrated.  Demonstrations require that we can “go back” to evident starting points. If there were nothing that was evident without demonstration, that is, self-evident, there would be no demonstrations—remember, the very idea of a demonstration involves the idea of the evident.  Their purpose is to make something evident.

The reason one gives a λόγος is to make something evident, manifest to νόησις.  Two more Greek words are useful here. One of Aristotle’s favorite words is δήλον [dēlon], which means “clear, evident, manifest.”  Anywhere I’ve written evident in this post, you could substitute δήλον.  Another very important Greek word is ἀληθής, which means “true.” The noun form, for ‘truth’ is ἀλήθεια.  Note the initial alpha privative α-.  The Greek word for “truth” is a-lēthē-ia where λήθη (lēthē) like the river and the goddess Λήθη means “hidden, concealed, covered.” So the Greek word for truth means, literally, “unconcealment” and thus shows its connection to what is evident or δήλον.  What is true, ἀληθής, is what is evidentδήλον.

What is crucial to emphasize is that being evident/δήλον is something that relates to our power of intellectual sight, to the νοῦς and νόησις. νοῦς is the power or faculty of the soul. νόησις is the activity. νοῦςνόησις :: power of sight : activity of seeing. νόησις is the being-at-work or actuality or ἐνέργεια of νοῦς.

One consequence of this that Aristotle (and others after him, such as St. Thomas Aquinas) draws is that God is pure νόησις.  The divine knowledge, divine omniscience, does not need λόγος, because λόγος is both subordinate to νόησις and inferior to it. When Aristotle defines the human being as “the living being with λόγος, he is distinguishing us from both lower animals and higher divine beings, gods, and very emphatically from the Unmoved Mover, Aristotle’s term for God.

Unlike νόησις, which can only fail to happen, λόγος can be false as well as true, it can be ψεῦδος as well as ἀληθής.  Plato explores the ψεῦδης λόγος [pseudēs logos] at length in The Sophist. In that dialogue, the Eleatic Stranger and Theaetetus are attempting to answer the question posed by Socrates (who is present, but silent save at the beginning) “What is the sophist?” They arrive at what seems to be the correct definition, namely, that “the sophist is the one who gives the ψεῦδης λόγος,” when all of a sudden, an imaginary sophist (voiced by the Eleatic Stranger) pops up and declares this to be impossible. “To speak falsely,” he argues, “is to say that which is not. But that which is not—is nothing.  As nothing at all, that which is not can neither be said nor thought. Parmenides himself has shown than nonbeing or nothing or ‘that which in no way is’ cannot be thought or said, and therefore, since ‘that which is not’ cannot be said, there is no one who says ‘that which is not,’ and therefore, there are no sophists, and therefore,” the sophist concludes with a self-satisfied smirk, “I am not a sophist!”

I won’t fully go into Plato’s analysis of how it is possible to say that which is not (the short answer is what Hegel will later call determinate negation—one cannot say what “in no way is” or absolute nonbeing—but one can say what something is not, by saying it is something other than it is; the ψεῦδης λόγος is a kind of covering over or obscuring of something by means of something else, rather than making it manifest as what it is).

For our purposes here it is enough to note that the realm of discursivity/λόγος is the realm of both truth and falsehood, unconcealment and concealment, making evident and obscuring from view.

Now, every argument will have certain necessary components.  Arguments are not things that happen “all at once” (like νόησις) but rather step by step.  This means they have parts.  Arguments have a form; this form may be valid or invalid. Arguments are composed of propositions, the premises, which can be true or false.  And propositions in turn are composed of terms, which may be clear or unclear.  A sound argument will have clear terms, true premises, and a valid form.

This is not hard to understand. But because all arguments have this structure, all arguments may be challenged on any of these three grounds: the validity of the argument may be called into question, the truth of the premises may be called into question, and the clarity or meaning of the terms may be called into question.

What I want to emphasize is that this is always possible, for any argument, by the very nature of an argument.  It is therefore also subject to sophistical abuse.  The philosopher Peter Geach gives two examples of this:



It is trivially easy to use the sophistical tricks of

  1. demand terms used in a demonstration be defined. When they are defined, as they must be, in other terms, demand that the new terms be defined in turn. Repeat infinitely.
  2. demand that the premises of a demonstration be demonstrated. If a new demonstration is given, demand that its premises be demonstrated in turn. Repeat infinitely.
  3. demand that the validity of a demonstration be demonstrated. If a new demonstration is given, demand that its validity be demonstrated in turn. Repeat infinitely.

The trouble is that some of the ancient sophists and some people today believe that these sophistical tricks constitute actual refutations of arguments or demonstrations.

But they obviously do not, since they can be applied to any proof, argument, or demonstration whatever, regardless of what it actually says.

I bring this up because these are favorite techniques used by many modern atheists.  Frequently, one hears them say “There is no evidence for God.” If one gives a demonstration, they demand that the demonstration be demonstrated, and so on, ad infinitum, and when one fails to meet this impossible demand, they smugly conclude that one has failed to demonstrate the existence of God. This is sophistry.  One could ask them to demonstrate that one’s demonstration has failed, and if they answer, demand that they demonstrate the demonstration, etc.  Anyone can play sophistical games.

I think that a very common error today is that people misunderstand the nature of proof or demonstration.  They seem to believe proof is something that COMPELS ASSENT.  But that isn’t what proof does. The task of a demonstration is to make something evident, that is, to place it before one’s eyes as clearly true. No one, however, and certainly not a demonstration, can compel anyone to actually look at what is placed in front of them.  Whether or not to actually look at or follow a demonstration is a decision of the will.  And so is the act of  ASSENTING to the truth of a proposition. If I “know in advance” that a given conclusion is wrong, I need not pay any attention to any argument given for that conclusion—other than, as Peter Geach once put it, to locate the fallacy.

What can we take away from this?

One can willfully and sophistically reject a perfectly cogent demonstration.  The fact that one is unconvinced by a demonstration is not a refutation of that demonstration.  The mother of a criminal might refuse to be convinced that “her baby boy” committed the crime, regardless of any amount of argument or evidence presented to her. This is not, however, a refutation of the case for her son’s guilt.

One way to short-circuit at least some kinds of sophistical regress tricks is to invoke Socrates’ principle of fair play in dialogues, which boils down to taking turns:


In other words, in a dialogue, which is a back and forth, you get one.  If you ask me to demonstrate something once, that’s fair enough (assuming your are asking in good faith). If I comply, and you then ask me to demonstrate my demonstration, the proper response is “No. I gave you your one. Now it’s your turn to show there’s something wrong with my argument, if you can.  Until you do this, I have no more responsibility here.  But if you do give a refutation of my demonstration, then it will be my turn to show why your refutation fails.”

I’m constantly amazed that so many people think “I’m not convinced” is a refutation. It isn’t. It’s not even a statement about the demonstration, but about one’s own psychological state.  The mere fact that someone is unconvinced may be because the demonstration is defective; but it may also be because the person in question is stupid, or ignorant, or failed to follow the demonstration, or is willfully set against being convinced for some extraneous reason (like the mother of the criminal’s love for her son).

As I said, the purpose of a proof or demonstration is to make something evident as true. But “evident” means “can be seen to be true” not “must be seen to be true.”  A matter may be evident in itself, but not evident to some people. Nothing can be made evident to those who will not look, or who refuse to accept the testimony of their eyes. You can reasonably be said to have “fed” someone if you set food before them to eat.  If they refuse to eat it, that isn’t your responsibility, but theirs. There was food there to be eaten.

This is why it is almost always pointless to argue with the completely convinced ideologue.  It doesn’t matter what you say. He will refuse to hear you, or to look at anything that would contradict his ideology. Ideologues are often recognizable by their win/win stance: for example, many feminists hold that arguments for feminism are strong and sound, and also hold that any arguments against feminism are instances of “misogyny” and therefore are also arguments for feminism.  Marxists hold that arguments for Marxism are strong and sound, and arguments against Marxism show the ideological prejudices of the bourgeoisie, and are therefore arguments for Marxism. Freudians hold their arguments for sexual repression and neurosis to be strong and sound, and hold any arguments against sexual repression and neurosis to be evidence of sexual repression and neurosis and therefore as evidence for psychoanalysis.  And if you think it matters that strict Freudian psychoanalysis is not longer that popular, consider the way the pseudo-Freudian word “phobia” has invaded our modern political discourse, e.g. an argument for gay marriage is strong and sound; an argument against gay marriage is “homophobic” and therefore really an argument in favor of gay marriage. Any praise of Islam is justified; any criticism of Islam is unjustified, because it is “Islamophobia” and therefore indicative of mental illness, rather than reason.

“Racism” and “misogyny” in our day have also essentially become “-phobia” words.

Positions and arguments are dismissed as supposedly being signs of psychological and/or moral properties, e.g. “hate” and “fear.”

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