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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.

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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.

 

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. Reasonablefaith.org 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. Carneades.org 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:

IMG_0973.JPG

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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Paradox of Omnipotence

It is sometime alleged that the concept of omnipotence is logically self-defeating and therefore impossible.  If omnipotence is an essential attribute of God and impossible, then God cannot exist.  But are these paradoxes any real threat to orthodox theistic belief, or is the threat overblown?

Perhaps the most common example of the paradox of omnipotence is posed in the following dilemma: Either God can create a stone so heavy that he cannot lift, or God cannot create a stone so heavy that he cannot lift it.  If God can create such a stone, then there is something an omnipotent being cannot do—namely, lift the stone.  If God cannot create the stone, then once again an omnipotent being lacks the ability to do something.  So, no matter what, it seems omnipotence is impossible.

The traditional theistic response has been to embrace one of horns of the dilemma while denying that the horn leads one to the conclusion that omnipotence is impossible.  To understand this, we first must understand what “omnipotence” means.  Most traditional theists define “omnipotence” as the ability to do anything logically possible.  So an omnipotent being could create stars, black holes, and even unicorns.  But such a being could not draw a round square, since roundness and squareness cannot cohere in the same object at the same time and in the same way.  But what about a really heavy stone, what is so logically incoherent about it?  Notice that the stone is attributed not with merely being super heavy, but with being so heavy that an omnipotent being cannot lift it.  We might put it this way: a being that can create anything and lift anything is tasked with creating something that a being that can lift anything cannot lift.  If there is no weight that is too heavy for an omnipotent being to lift, then it is simply not possible for an omnipotent being to create such a stone.  No such stone can exist.

There have been some other interesting examples of omnipotence paradoxes.  For instance, it appears that an omnipotent being cannot create legal U.S. currency.  Why?  Because in order for currency to be legally made, it must be minted at one of the dozen or so authorized mints around the United States.  So, if God were to just bring a dollar into existence, having not been minted legally, God would be guilty of counterfeiting the bill.  A clever theist might come up with a way around this, i.e. God could incarnate himself as man and then apply for employment at the U.S. mint.  Then he could be said to have at least been a part of the process of minting real money.  Still another theist might say something like, since God is the creator of all matter and energy, God is the remote cause of U.S. currency—though not the proximate cause.  Really the legal currency example really boils down to a logical incoherency.  An omnipotent being cannot create something that is defined as not having been created by an omnipotent being.  So if legal U.S. currency is, by definition, currency not created by omnipotent beings, there is little God could do, short of becoming the building, the printing machine, the employees, and the U.S. treasurer, so that God could be said to have fully created the dollar bill all by himself.  And even if he did all those things, there would still be some question as to whether the minted bill was really created by a genuine U.S. mint of a counterfeited one.

My interest in this puzzle is with those who might find this first solution dissatisfying.  They might insist that omnipotence is the ability to do anything, including the logically impossible.  So, if God cannot make a stone so big that even God cannot lift it, then omnipotence is impossible and so is God.  So rather than insist that the previous definition of omnipotence is the only one on the table, I would like to offer the following counter-dilemma to those who might think that these paradoxes are real defeaters for theism.  The dilemma is as follows:

Either omnipotence is the ability to do anything but the logically impossible or omnipotence is the ability to do anything including the logically impossible.  If omnipotence is limited to doing the logically possible, then God cannot make a stone so heavy that he cannot lift it simply for the reasons stated above.  If omnipotence is not limited by the logically impossible, then God CAN make a stone so heavy that he cannot lift it.  But this is not very problematic either.  For if one insists that omnipotence ought to include the ability to break the laws of logic, then it must be reassessed as to whether the inability to lift a stone qualifies as something which precludes an entity from being omnipotent.  I argue that we have no logical footing from which to make such an assessment.  If there is a logically impossible world where omnipotent beings can create objects so heavy that they cannot be lifted, then we are just as likely to infer that an omnipotent being is unlimited in action as we are to say that such a being is limited in action.  The laws of logic no longer apply to such a being, right?  So, if an omnipotent being could make a stone so heavy that a being that can lift anything cannot lift it, then the same omnipotent being certainly would have the power to stipulate the definition of omnipotence around any objections.  If one were to object to such a move as illogical it’s just too bad, for the objector has already insisted that an omnipotent being can do that which is illogical.

Is the ability to do the logically impossible logically impossible?  Squared-circles cannot exist because squareness and circularity are contrary attributes that cannot cohere in the same object at the same time.  But, is the ability to make squared-circles itself logically impossible?  So it might not be the case that the ability to do the logically impossible it itself logically impossible.

These questions aside, the theist is perfectly within her right to insist that omnipotence means only that God can do the logically possible.  If the atheologian insists that omnipotence requires the ability to do the logically impossible, then it is the atheologian who has walked through the dialethistic door of admitting the possible impossible.  And if the only reason for insisting that omnipotence means the ability to do the logically impossible is to conclude that omnipotence is itself logically incoherent and cannot exist, then one has merely begged the question through stipulating omnipotence in this manner.  That is, one has stipulated omnipotence to be defined as an impossible attribute which cannot exist.  That is not a very compelling reason to think omnipotence doesn’t exist, especially since there are competing definitions out there.  So I think a theist is perfectly within his or her rational right to think omnipotence can and does exist.

Computer Validates Anselm’s Proof for God’s Existence

Paul Oppenheimer and Edward Zalta (2011) have used Prover9 to check the validity of Anselm’s ontological argument.  The program not only proved the argument’s validity, but went on to derive a simplified version of the argument containing only one non-logical premise and avoiding many of the metaphysical pitfalls of Anselm’s original formulation.  Their article, A Computationally-Discovered Simplification of the Ontological Argument, appears in the June issue of the Australian Journal of Philosophy.

In plain English, the one non-logical second premise of the simplified argument reads:

If the conceivable thing than which nothing greater is conceivable fails to exist, then something greater than it is conceivable (348).1

[Spoiler Alert] Oppenheimer and Zalta do not claim that this simplified version of the argument is sound.  They think the premise has some prima facie plausibility (ibid.).  Further analysis reveals that the defender of the ontological argument must provide an independent argument to think the premise is true.  Still, they seem to suggest that such an independent argument could be constructed out of their previous work.  They write:

Our 1991 analysis of the argument is still relevant, since it shows how the ontological arguer could justify Anselm’s use of the definite description. The present analysis
shows why the use of the definite description needs independent justification.  Consequently, though the simplified ontological argument is valid, Premise 2 is questionable and to the extent that it lacks independent justification, the simplified argument fails to demonstrate that God exists.  The use of computational techniques in systematic metaphysics has illuminated the relationship between Premise 2 of the ontological argument and the conclusion that God exists (349).

So the argument in plain English would run something like this:

1. Nothing greater is conceivable than the conceivable thing than which nothing greater is conceivable.

2. If the conceivable thing than which nothing greater is conceivable fails to exist, then something greater than it is conceivable.

3.  Therefore, the conceivable thing than which nothing greater is conceivable does not fail to exist.

Premise One does seem to be a priori true.  So, in this formulation, the question of the soundness of the argument really does come down to Premise 2.  All else being equal, is a conceivable thing that exists greater than a conceivable thing that fails to exist?  Kant would say no.  Existence is not a real predicate!  But I don’t remember Kant really giving an argument for this.  All he gives is a weak analogy about thalers, the Prussian currency of his day.  Kant argues that 100 real thalers does not contain a coin more than 100 imagined thalers.  Thus, by analogy, “that God exists” adds nothing to the concept of God.  Now it might be true that 100 imagined thalers have as many coins as 100 real thalers.  But the for the analogy to hold, the question is not with regard to the equality of coins between the two, but which is greater.  So I offer the following prize: if you can prove that 100 imagined dollars are just as great as 100 real dollars, you win 100 imagined dollars.  Once you have submitted the proof just close your eyes and imagine that green-hued Benjamin Franklin with his perturbed expression.  Are you not motivated to win the prize?  Would you prefer that I offer you a real 100 dollar bill for the proof?  Why?

[An Aside] The fact that a computer program was able to refine the ontological argument is quite intriguing to me.  Suppose the singularity occurs, as predicted by some futurists.  What if the resulting super-intelligent machines were able to demonstrate the soundness of the ontological argument?  Would these super-intelligent machines develop religion?

1P. Oppenheimer & E. Zalta. (2011). “A Computationally-Discovered Simplification of the Ontological Argument”. Australasian Journal of Philosophy 89 (2): 333-349.

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