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On “Is”

<<τὸ ὂν λέγεται πολλαχῶς…>> (Bill Clinton and Aristotle)

“Is” (to be) is a tricky word, and I think the ambiguous nature of this word has led to some misunderstandings of some of the arguments I present, which are typically written in Free Logic. “Is” has multiple meanings, and some of the meanings are more “ontologically committing” or “existentially loaded” than others. Some common logical notation that gets translated as “is” in ordinary language include: 1) “(∃x)”, 2) “=”, 3) “Px”, and 4) “≝”, and I would like to emphasize that they are not syntactically equivalent, and do not function in logical arguments in the same way.

1) The “is” of existential quantification: There is an x, e.g. there is something green, or (∃x)Gx,. This can be interpreted as a “particular quantifier” indicating that there is at least one individual x. Depending on the domain of discourse, the existential quantifier can be more or less ontological committing. One could say, there is a fictional detective that Arthur Conan Doyle wrote about, and use the existential quantifer, and one would not be committed to the reality of fictional beings, i.e. (∃x)(Fx & Wax) [read: there is an x such that x is fictional and Arthur Conan Doyle wrote about x], s = x satisfies the formula in this case, where “s” means “Sherlock Holmes”.

2) The “is” of identity: (x = y), e.g. Tully is Cicero, or (t = c). Sometimes the “is” of identity is combined with the existential quantifier to make strong existential claims, e.g. there is a planet named Venus: (∃x) [Px & (x = v)]. There are rules around identity that are, themselves, metaphysically complicated, and it is controversial how those rules should apply to logic. For instance, it is sometimes granted that (∀x)(x = x) can be introduced at any stage of an argument simply because everything is self identical. Also, if a = b, then b can be substituted for a in an argument in some, but not all, contexts. The contexts were such substitutions cannot occur are called “referentially opaque contexts”. For example, Clark = Superman. Lois believes Superman = Superman. But it doesn’t follow that Lois believes Superman = Clark.

3) The “is” of predication:  x is purple, or simply Px. This “is” is not very existentially committing, but merely ascribes properties to individuals, on could say Sherlock, where “s” is Sherlock, and “B” is the predicate “Brave”: Bs. In “Free Logic” to make strong “existentially loaded” or “existentially committing” claims, you might specify “Real Existence” as a kind of predicate said of an individual. This might run contrary to “Kant’s Dictum” that existence is not a real predicate, but alternative ways of forming existential claims about what exists in the world are problematic for other reasons. When I construct ontological arguments, I tend to use Free Logic. This is because free logic allows you to quantify over things that may or may not exist in reality, which is needed, if one is not to beg the question in ontological arguments.

4) The “is” of definition: for example, the name “God” is “the x such that x is perfect”, or g ≝ (ɿx)Px.  I might stipulate such a definition in an argument by writing “D1: God is perfect.” This is not an existentially committing sentence, but a stipulation of the meaning of a term. Definitions are not really propositions in the fullest sense, as they are not true or false, but merely what one means when one uses a term in a proposition. As such, a definition is usually assessed in terms of clarity and coherence rather then whether it is true.  The scholastics would make this point by saying that definitions pertain to the first act of the mind, not the second.  Explicitly adding predicates into a definition in order to prove that the thing defined has those predicates can be question-begging, this would include adding “real existence” as a predicate in the definition, e.g. A shmunicorn is a unicorn that exists, therefore shmunicorns exist would constitute a question-begging proof. Adding “existence” directly into a definition also entails that the thing defined would exist necessarily, since one can add necessity to any conclusion derived from zero premises. It would be unclear and possibly incoherent to say that shmunicorns exist of necessity, so such a proof should not command assent. My ontological arguments for God are never zero-premise, and always require one or more premises to reach the conclusion.

So this can help us to disambiguate.  Consider the following sentence: “There is an individual who is the author of this blog and who is Daniel, who is the only son of James and Kathy Vecchio, and who is.”

Axy ≝ x is the author of y
Sxyz ≝ x is the only son of y and z
d ≝ (ɿx)Sxjk
j ≝ James Vecchio
k ≝ Kathy Vecchio
b ≝ Vexing Questions blog

(∃x){[Axb ∧ (x = d)] ∧ E!x}

There are a lot of “ises” in that expression, but we can now see how each has its own function.

Self-Referential Unsound Modus Ponens 

[Image Source Credit: TeX]

An argument is sound if and only if it is valid and the premises are true. If those conditions are met, the conclusion must be true.

Consider the following argument:

P1. If God does not exists, this argument is unsound.
P2. God does not exist.
C. Therefore, this argument is unsound.

The argument is valid (Modus Ponens), so it is sound if the premises are true. But, if both premises are true, the conclusion is would have to be true, and the argument would both be sound and unsound. So consistency demands that we deny the soundness of the argument. At lease one of the premises must be false.  
Consider whether P1 is false. It is a material conditional, and so it is false when the antecedent is true (it is true that God does not exist) and when the consequent is false (it is false that this argument is unsound).[1] So P1 is false only if the argument is sound, which means that the falsity of P1 leads to a contradiction, since the soundness of the argument entails P1 is true. So, P1 cannot be false.  

P2 is the only premise that can be false. So given that the argument must be unsound, we must conclude that it is false that God does not exist.

So this unsound modus ponens proves the contradictory of the minor premise, whatever it might be!

I am probably not the first to note this, but it is new to me.

[1]The truth-table for the Material Conditional is as follows:

    p  q | p → q
1. T  T        T
2. T  F        F*
3. F  T        T
4. F  F        T
*The material conditional is only false on line 2.

Absolute and Relative Identity

I have seen some argue that any relative identity claim can be reduce to an absolute identity claim in the following manner:

1) x and y are the same F  ≝ is an Fis an F, and x = y.

However, I don’t think this works.  Part of the motivation for relative identity is that there may be circumstances like:

2) x and y are the same F, but x and y are not the same G.

But (1) and (2) are not compatible, since we would have to affirm and deny absolute identity between x and y.  So the relative identity theorist should reject (1) given his commitment to (2).

Relative identity is not just absolute identity, plus the idea that x and y fall under the same sortal.  Moreover, this would be to suggest that relative identity is derivative, and absolute identity is the more primitive notion.  I would argue that is it the other way around.  So I would define absolute identity in terms of relative identity in the following manner:

4) x = y ≝ for any sortal, S, if x is an S or y is an S, then x and are the same S.

In other words, the absolute identity between x and y is derived from the fact that for any sortal which belongs to either x or y, it is the case that x and y count as the same S.  I say “either x is an S, or y is an S” as opposed to “both x is an S and y is an S” to avoid situations where x can be counted as an S and some y cannot, but they are the same S for any sortal underwhich both can be counted.  For there to be absolute identity, it must be the case that all sortals that belong to x must also belong to y.  I believe (4) captures this.

So to say x and y are absolutely identical is to say that for any sortal underwhich x or y can be counted, x and y are the same sortal.


Domino Logic

Here are some really interesting videos in which dominos simulate logical reasoning and computing:


Vexing Links (8/5/2016)

Apologies for the hiatus. I am hoping to put some arguments out there soon. But in the meantime, here are some links of note:

1. My Ph.D. dissertation is now on ProQuest.

2. My review of Modality & Explanatory Reasoning by Boris Kment was recently published by the Polish Journal of Philosophy.

3. I’m currently reading Structure and the Metaphysics of Mind: How Hylomorphism Solves the Mind-Body Problem by William Jaworski.  I’m hoping to do a presentation on hylomorphism this fall, so this will really help.

4. Wisecrack has some great videos on the Philosophy of  Daredevil, and the Philosophy of the Joker.

5.  Looking forward to the Ultimate Game of Thrones and Philosophy, especially as it will feature contributions from Max Andrews and Tyler Dalton McNabb.

6. Dale Tuggy interview Timothy Pawl on Trinities Podcast: Pt 1 and P2.

7. Appropriate for our current political climate, the SEP has a new article out on the Ethics and Rationality of Voting.

8.  Illustrates the problem of semantics for AI: the Domino Computer.

9. The History of Philosophy without any Gaps has some great recent podcasts on the Trinity: Episodes 258 and  259.

10. Justin Brierley of the Unbelievable? Podcast explains the argument from Fine-Tuning.

Hebrews 3, Proving the Minor, and the Divinity of Christ

I was recently reading the Letter to the Hebrews and came upon an interesting passage:

Therefore, holy brethren, partakers of a heavenly calling, consider Jesus, the Apostle and High Priest of our confession; He was faithful to Him who appointed Him, as Moses also was in all His house. For He has been counted worthy of more glory than Moses, by just so much as the builder of the house has more honor than the house. For every house is built by someone, but the builder of all things is God. Now Moses was faithful in all His house as a servant, for a testimony of those things which were to be spoken later; but Christ was faithful as a Son over His house—whose house we are, if we hold fast our confidence and the boast of our hope firm until the end (Hebrews 3:1-6, NASB).

The logic of the passage jumped out at me, as I have been keen to find passages that affirm the divinity of Christ in light of my interactions with Biblical Unitarians.  This passage is concerned with demonstrating that Christ is worthy of more glory than Moses.  Thomas Aquinas dissects the passage in the following manner:

161. – But the Apostle’s reason is that more glory is due Him Who built the house, than to him that dwells in it. But Christ built the house: ‘You have made the morning light and the sun’ (Ps. 73:16); ‘Wisdom has built herself a house’, i.e., the Church (Pr. 9:1). For Christ by Whom grace and truth came, built the Church, as legislator; but Moses, as promulgator of the Law: therefore, it is only as promulgator that glory is due Moses. Hence, his face became bright: ‘So that the children of Israel could not steadfastly behold the face of Moses for the glory of his countenance’ (2 Cor. 3:7). Therefore, the sequence of thought is this: You say that Christ is faithful as Moses was. Why then overlook Him? Certainly this man was counted worthy of greater glory than Moses, by so much as he that has built the house has greater honor than the house. As if to say: Even though Moses deserves mention, Christ is more honorable, because He is the builder of the house and the chief lawgiver: ‘Behold, God is high in his strength, and none is like him among the lawgivers’ (Jb. 36:22). Therefore, if Moses is deserving of glory, Christ is more deserving: ‘For is the ministration of condemnation be in glory, much more the ministration of justice abounds in glory’ (2 Cor. 3:9).

162. – Then he proves the minor premise of his reason when he says: For every house is built by some man. But the minor is that Christ built that house. He proves this, first, because every house needs a builder; secondly, because the house of which he speaks was built by Christ, the builder of all things is God.

163. – First, therefore, he proves that this house, as any other, needs a builder, because its various parts are put together by someone. This is obvious in a structure in which the wood and stones, of which it is composed, are united by someone. But the assembly of the faithful, which is the Church and the house of God, is composed of various elements, namely, Jews and Gentiles, slaves and free. Therefore, the church, as any other house, is put together by someone. He gives only the conclusion of this syllogism, supposing the truth of the premises as evident: ‘Be you also as living stones built up, a spiritual house, a holy priesthood’ (1 Pt. 2:5); ‘Built upon the foundation of the apostles and prophets, Jesus Christ himself being the chief cornerstone’ (Eph. 2:20).

164. – Then (v. 4b) he proves that Christ is the builder of that house, for He is God, the builder of all things. And if this is understood of the whole world, it is plain: ‘He spoke and they were made; he commanded and they were created’ (Ps. 32:9) But there is another spiritual creation, which is made by the Spirit: ‘Send forth your spirit, and they shall be created, and you shall renew the face of the earth’ (Ps. 104:30). This is brought about by God through Christ: ‘Of his own will has he begotten us by the word of truth, that we might be some beginning of his creature’ (Jas. 1:18); ‘We are his workmanship, created in Christ Jesus in good works’ (Eph. 2:10). Therefore, God created that house, namely, the Church, from nothing, namely, from the state of sin to the state of grace. Therefore, Christ, by Whom He made all things, ‘by whom also he made the world’ (Heb. 1:2), is more excellent (since He has the power to make) than Moses, who was only the announcer (Thomas Aquinas, Commentary on Hebrews).

If I understand Aquinas’s analysis of the passage correctly, the author of Hebrews is trying to prove:

C1: Christ is worthy of more glory than Moses

And the premises that support this conclusion are:

P1: For all persons p1 and p2, if p1 is the builder of the house that p2 dwells in, then p1 is worthy of more glory than p2.1

P2. Christ is the builder of the house that Moses dwells in.

Now, C does follow reasonably well from P1 and P2 (see the footnote below). Aquinas notes that further support is provided in verse 4 for the truth of the minor premise, i.e. P2. This sub-argument has massive Christological significance, and the argument looks like this:

P3: For all x, if x is a house, then there is some person who built x.

P4: For all x, if there is some person who built x, the person who built x is God.

From (P3) and (P4), we can draw the conclusion that God is the builder of all houses, or:

C2: For all x, if x is a house, the person who built x is God.

So, given that there is some house that Moses dwells in:

P5: There exists some x such that x is a house and Moses dwells in x.

We can conclude:

C3: There exists some x such that x is a house and Moses dwells in x, and the person who built x is God.

Or in more readable English: God is the builder of the house that Moses dwells in.

But wait a minute! C3 doesn’t say anything like P2. The only way that C3 could be taken to support P2 is if we add a premise, which the author of the Letter to the Hebrews has suppressed, namely:

P6: Christ is God.

The author invites the reader to reason through his enthymeme, and keep in mind the truth that Christ is God, and so the creator of all things, including the Church and all of the houses of Israel, including that of Moses. So from C3 and P6, we can draw out:

C4: There exists some x such that x is a house and Moses dwells in x, and the person who built x is Christ.

And C4 just is P2.

Now, we are also told that Jesus is the Son over the house, but that it is His house. So, we get both the idea that Jesus is the Son of God and God, the creator of all things.

Suppose, for a moment, that the author did not intend such an argument.  Instead, he merely wanted to argue that Christ is the Son of the house, whereas Moses is the servant.  If so, then his entire point about builders being more deserving of glory than members of the house would be wasted ink.  For that entire passage would only prove that God is more worthy of glory than Moses, which is hardly in dispute.  The passage only makes sense if it can lend support to the authors actual conclusion, and the only way to validly reach that conclusion is if we identify Christ as God.

1To be more precise, we should say something like, P1′: For all persons p1 and p2, if p1 is the builder of the house that p2 dwells in, and p1 is not identical to p2, then p1 is worthy of more glory than p2. We would also need to then add P3′: Christ is not identical to Moses, which is a reasonable assumption given the Transfiguration, for instance.

Vexing Links (12/27/2015)

Happy Holidays, Merry Christmas, Happy New Year to Vexing Questions readers.  Here are some links of note:

  1. has released its latest video in its series on the existence of God: the Leibnizian Cosmological Argument (view the other videos in the series here)
  2. The Church of England released a beautiful ad featuring the Lord’s Prayer.  It was banned and created some controversy, but it is moving nonetheless.
  3. Dr. Lee Irons does a great job defending the Trinitarian perspective in a new book.  Here is an interview about his defense, hosted by Dale Tuggy.
  4. The SEP has some new articles and revisions of note: Thomas Williams revises an entry on St. Anselm, Olga Lizzini has a new article on Ibn Sina’s Metaphysics, and Jeffery Bower revises an entry on Medieval Theories of Relations.
  5. Some music I’ve been enjoying: Timothy Vajda’s As the Crow Flies, and Sigur Rós’s version of the Rains of Castamere.
  6. great philosophy website, with videos on logic.
  7. Brilliant physicist, George Ellis, is interviewed on Closer to Truth about What An Expanding Universe Means.
  8. Grasped in Thought blogs about Gaunilo’s failed objection to Anselm’s ontological argument.
  9. Maverick Philosopher has a beautiful Christmas reflection on the meaning of  the Incarnation and John 1:14.
  10. Dr. Alexander Pruss offers an interesting argument about physicalism and thinking about big numbers.

Colbert on Faith, Logic, Humor and Gratitude

In the video below, Stephen Colbert talks about faith, logic, and humor.  Even though Colbert says that the ontological argument is “logically perfect”, like Pascal, he does not think logic can lead to faith in God.  There must be a movement in the heart, which Colbert connects to gratitude, and which he lives out in his work as a comedian.  But it isn’t as though logic and emotion as opposed forces.  The feeling of gratitude makes sense within a worldview where there is a being than which none greater can be conceived.

When we reflect on our existence, the love we share, the struggles, the joys, the busy days, and the quiet nights, we feel we ought to give thanks.  This gratitude is not conditioned by the kind of life we have.  For we see that gratitude is often freely expressed by the most lowly among us, and we are irked when the richest and most powerful lack gratitude.  Such a duty to feel gratitude seems to exist for us all and it doesn’t matter who we are or the sort of life we have.

Now, if we ought to express an unconditioned gratitude, then we can do so.  But if we can express such gratitude, there must be at least possible that there is an object worthy of such gratitude.  It is, after all, impossible to express gratitude if there cannot be anyone to whom the gratitude is due.  So, we might say that our ability to express unconditioned gratitude is at least predicated on the possibility of there being someone worthy of such gratitude.  So, I think only a perfect being is worthy of unconditioned gratitude, and if is possible that there is such a being, such a being exists.  That is, for me, one way in which gratitude and logic connect to bolster faith.

Anyways, here is the Colbert video.  I love a comedian who can name drop Anselm and Aquinas!

Circular Reasoning, the Bible, and Atheism

I had a discussion the other day in which my interlocutor cited “reading the Bible” as the cause of his atheism. This perplexed me. And he is not the only one who has said this.  Here is a common meme expressing the same sentiment:

So, here is my response, in meme form:


If you are interested in how to approach scripture, I recommend reading Dei Verbum.

Beg Your Pardon, What is Begging the Question?

Here is my recent contribution to Attack of the P-Zombies. Enjoy!

Grasped in Thought

We’ve all met them. Usually they are fresh off of a critical thinking, or informal logic course. They are the fallacy mongers. Taught to identify informal fallacies in headlines and textbooks, they begin to “see” fallacies at every turn. And suffering them in any conversion is nearly intolerable. For those unfamiliar, I am talking about people who behave like this. Now, I am not saying that it isn’t important to be able to know and be able to identify informal fallacies. It is. But it can also become a hammer that turns all arguments into nails. This is especially dangerous because informal fallacies tend to be vaguely defined, and often resemble perfectly good methods of reasoning. Pro-tip: When you encounter such people, inform them that it is not sufficient to merely burp up fallacies at you. Ask them to explain to you what the fallacy means, and specifically how…

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