Category Archives: Logic

A Slingshot from S4 to S5 establishing the Modal Ontological Argument?

…Or why the “strong” atheologian, i.e. the atheologian who holds that there is no omniscient, omnipotent, and omnibenevolent being, must say that ♢Θ semantically entails ☐Θ in S4.

Θ is the proposition that necessarily there is an omniscient, omnipotent, and omnibenevolent being.  

That is:

Kx ≝ x is omniscient
Px ≝ x is omnipotent
Bx ≝ x is omnibenevolent
Θ ≝ ☐(∃x)[(Kx ∧ Px) ∧ Bx]

Consider the following:

1. It is false that ♢Θ semantically entails ☐Θ in S4.

If that is true, then:

2. There is a world in which the valuation of ♢Θ at that world in S4 is true, and the valuation of ☐Θ at that world in S4 is false.

But this is just to say…

3. ♢♢Θ

That is, there is a world in which it is true that ♢Θ.  Moreover, it is an axiom of S4 that ♢♢p → ♢p, and so:

4. ♢Θ

But given our definition for “Θ”, we can say:

5. ♢☐(∃x)[(Kx ∧ Px) ∧ Bx]

Since S5 is just an extension of S4, if something is possible in S4 it is also possible in S5.  Given that ♢☐p → ☐p is an axiom in S5:

6. ☐(∃x)[(Kx ∧ Px) ∧ Bx]

And since ☐p → p in S5 (axiom M/T), we can conclude:

7. (∃x)[(Kx ∧ Px) ∧ Bx]

Hence, the committed “strong” atheologian must say that ♢Θ semantically entails ☐Θ in S4.  Moreover, since S4 is strongly complete, the atheologian is committed to: 

♢Θ ⊢S4 ☐Θ

I’d like to see that deduction. 

[Update]: One objection that I have encountered is that the move from 5 to 6 seems to switch frameworks from S4 to S5, and so the argument is invalid. The argument does not presume S4 as the framework, but rather attempts to exploit an intuition about what is semantically entailed about ♢Θ in S4. In other words, if you grant that such entailment doesn’t hold in S4, I think it follows that you are committed to ♢♢Θ in S4 and S5, which of course is just to say that you are committed to ♢Θ in S5. So from the framework of S5, and its related axioms, you would have to be committed to Θ.

In an attempt to more clearly show how I am not applying axioms of S5 in S4, here is a more formal representation of the argument. Add to our key, the following:

T ≝ true
F ≝ false
V(ω)M(P) = … the valuation at ω in model M of proposition p equals…

1. (∀p)(∀q)~[p ⊨S4 q] → (∃ω){[V(ω)S4(p) = T] ∧ [V(ω)S4(q) = F] (premise)
2. (∃ω){[V(ω)S4(p) = T] ∧ [V(ω)S4(q) = F] → ⊨S4♢p} (premise)
3. (∀p){⊨S4♢p → (∃ω){[V(ω)S5(p) = T]} (premise)
4. (∀p)(∃ω){[V(ω)S5(p) = T] → ⊨S5♢p} (premise)
5. (∀p)[⊨S5♢♢☐p → ⊢S5☐p] (premise)
6. ~[♢Θ ⊨S4 ☐Θ] (premise)
7. (∀q)~[♢Θ ⊨S4 q] → (∃ω){[V(ω)S4(♢Θ) = T] ∧ [V(ω)S4(q) = F] (1 UI)
8. ~[♢Θ ⊨S4 ☐Θ] → (∃ω){[V(ω)S4(♢Θ) = T] ∧ [V(ω)S4(☐Θ) = F] (7 UI)
9. (∃ω){[V(ω)S4(♢Θ) = T] ∧ [V(ω)S4(☐Θ) = F] (6,8 MP)
10. [V(w)S4(♢Θ) = T] ∧ [V(w)S4(☐Θ) = F (9 EI)
11. [V(w)S4(♢Θ) = T] (10 Simp)
12. (∃ω)S4(♢Θ) = T] (11 EG)
13. (∃ω){[V(ω)S4(♢Θ) = T] → ⊨S4♢♢Θ (2 UI)
14. ⊨S4♢♢Θ (12,13 MP)
15. ⊨S4♢♢Θ → (∃ω){[V(ω)S5(♢Θ) = T] (3 UI)
16.(∃ω){[V(ω)S5(♢♢Θ) = T] → ⊨S5♢♢Θ (4 UI)
17. ⊨S4♢♢Θ → ⊨S5♢♢Θ (15,16 HS)
18. ⊨S5♢♢Θ (14,17 MP)
19. ⊨S5♢♢☐(∃x)[(Kx ∧ Px) ∧ Bx] (18 Def “Θ”)
20. ⊨S5♢♢☐(∃x)[(Kx ∧ Px) ∧ Bx] → ⊢S5☐(∃x)[(Kx ∧ Px) ∧ Bx] ∧ Bx] (5 UI)
21. ⊢S5☐(∃x)[(Kx ∧ Px) ∧ Bx] ∧ Bx](19,20 MP)

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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.

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.

 

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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The Horse Head Argument

After developing three figures of the categorical syllogism, Aristotle bombastically claims, in Prior Analytics A23:

It is clear from what has been said that the deductions in these figures are made perfect by means of the universal deductions in the first figure and are reduced to them. That every deduction without qualification can be so treated, will be clear presently, when it has been proved that every deduction is formed through one or other of these figures (40b17-22, emphasis mine).

Contrary to this, Augustus De Morgan argues that there are deductions that cannot be reduced to a syllogism.

There is another process which is often necessary, in the formation of the premises of a syllogism, involving transformation which is neither done by syllogism, nor immediately reducible to it. It is the substitution, in a compound phrase, of the name of the genus for that of the species, which the use of the name is particular (FL, p. 114).

The most notorious example is, the horse head argument (HHA): ‘horse is animal, therefore the head of a horse is the head of an animal’. I should clarify that De Morgan uses ‘man’ rather than ‘horse’  in his example, but otherwise, the argument is the same. Now, our predicate logic is quite powerful and can handle compound substitutions ably. Here is an indirect proof that seems to comport to the demands of HHA:

Let

Hx – x is a horse
Ax – x is an animal
Cxy – x is the head of y

1. (∀x)(Hx ⊃ Ax) (premise)
2. ~(∀y)(∀x)[(Hx & Cyx) ⊃ (Ax & Cyx)] (IP)
3. (∃y)~(∀x)[(Hx & Cyx) ⊃ (Ax & Cyx)](2 QN)
4. (∃y)(∃x)~[(Hx & Cyx) ⊃ (Ax & Cyx)] (3 QN)
5. (∃x)~[(Hx & Cux) ⊃ (Ax & Cux)] (4 EI)
6. ~[(Hv & Cuv) ⊃ (Av & Cuv)] (5 EI)
7. ~[~(Hv & Cuv) ∨ (Av & Cuv)] (6 Impl)
8. ~~(Hv & Cuv) & ~(Av & Cuv) (7 DeM)
9. (Hv & Cuv) & ~(Av & Cuv) (8 DN)
10. Hv & Cuv (9 Simp)
11. Hv (10 Simp)
12. Hv ⊃ Av (1 UI)
13. Av (11, 12 MP)
14. ~(Av & Cuv) (9 Simp)
15. ~Av ∨ ~Cuv
16. ~~Av (13 DN)
17. ~Cuv (15, 16 DS)
18. Cuv (10 Simp)
19. Cuv & ~Cuv 17, 18 Conj)
20. (∀y)(∀x)[(Hx & Cyx) ⊃ (Ax & Cyx)] (2-19 IP)

So, the inference seems to be valid, given the rules of first order predicate calculus. Is it really the case, though, that a parallel proof cannot be rendered in a Categorical syllogism? A categorical syllogism has three terms, and two premises, yet the above argument has one premise, which leads directly to the conclusion. So we need to identify the terms that would operate in a syllogistic version of HHA. And we need to allow that the reduction will contain two premises.

We must be cautious in how we articulate this syllogism, as Aristotle warns:

It is not the same, either in fact or in speech, for A to belong to all of that to which B belongs, and for A to belong to all of that to all of which B belongs; for nothing prevents B from belonging to C, though not to every C: e.g. let B stand for beautiful, and C for white. If beauty belongs to something white, it is true to say that beauty belongs to that which is white; but not perhaps to everything that is white. If then A belongs to B, but not to everything of which B is predicated, then whether B belongs to every C or merely belongs to C, it is not necessary that A should belong, I do not say to every C, but even to C at all. But if A belongs to everything of which B is truly said, it will follow that A can be said of all of that of all of which B is said. If however A is said of that of all of which B may be said, nothing prevents B belonging to C, and yet A not belonging to every C or to any C at all. If then we take three terms it is clear that the expression ‘A is said of all of which B is said’ means this, ‘A is said of all the things of which B is said’. And if B is said of all of a third term, so also is A; but if B is not said of all of the third term, there is no necessity that A should be said of all of it (APr 49b14-31).

So we don’t want to say that because all horses are animals, everything that a horse has, like a head, is something that every animal has. Some animals, after all, could be headless! And what we really mean to say is that, since ‘animal’ is the genus of ‘horse’, and since a horse has a head, an animal has a head. Perhaps, then, we should formulate the argument as follows:

21. All horses are animals.
22. All horses are those that have heads.
∴23. Some of those that have heads are animals.

By making the conclusion particular, we do not run the risk of affirming that all animals have heads to the consternation of amoebas and sponges. The preceding argument is Darapti, and it is a valid syllogism, barring any objections on the grounds of existential import.1 However, it is not quite what HHA demands. Recall that we need to conclude that ‘the head of a horse is the head of an animal’, since ‘horse is animal’. To approximate the conclusion more closely, we might use repetition. Aristotle mentions the use of repetition in the syllogism, but stipulates how it is to be used. We find in Prior Analytics A38:

A term which is repeated in the propositions ought to be joined to the first extreme, not to the middle. I mean for example that if a deduction should be made proving that there is knowledge of justice, that it is good, the expression ‘that it is good’ (or ‘qua good’) should be joined to the first term. Let A stand for knowledge that it is good, B for good, C for justice. It is true to predicate A of B. For of the good there is knowledge that it is good. Also it is true to predicate B of C. For justice is identical with a good. In this way an analysis of the argument can be made (APr 49a11-18).

So, Aristotle sets down that A is ‘knowledge that it is good’, B is ‘good’, and C is ‘justice’. Formally, the proof would be:

24. AaB (Knowledge, that it is good, belongs to all good.)
25. BaC (Good belongs to all justice.)
∴ 26. AaC (Knowledge, that it is good, belongs to all justice.)

Let us set down that D is ‘animal, qua horse’, E is ‘horse’, and F is ‘head’. Still making use of Darapti, the argument would then be:

27. DaE (Animal, qua horse, belongs to all horse.)
28. FaE (Head belongs to all horse.)
∴29. DiF (Animal, qua horse, belongs to some head.)

Or, in a more readable English prose:

30. Every horse is an animal in virtue of being a horse.
31. Every horse is that which has a head.
∴32. Some of those which have heads are animals, in virtue of being horses.

Now one might protest that the conclusion reached here is particular, whereas in predicate calculus one reaches a universal conclusion. But what does that universal conclusion really say? It says that, for all things x and y, if x is a horse and y is the head of x, then x is an animal and y is the head of x. In effect, it is not saying that the head of a horse is an animal head, but that if something is a horse and it happens, also, to have a head, then it is an animal that happens to have a head. Is this the same as HHA? There is no real sense in which the deduction formed by predicate logic has anything to do with the relationship between genus and species, as De Morgan indicates. But the categorical syllogism that we have formed does have this information, in that a horse belongs to its genus in virtue of belonging to its species.

I grant that the conclusion of the categorical syllogism is syntactically divergent from HHA. Nonetheless, I think it captures a similar, if not the same, sense.  Perhaps this is the best we can do.

1For those who are particularly bothered by the “existential fallacy”, we could run a similar argument on Datisi.

References:

Aristotle. 1995. “Prior Analytics.” In The Complete Works of Aristotle: The Revised Oxford Translation. Trans. A.J. Jenkinson. Ed. J. Barnes. Vol. I. Princeton, New Jersey: Princeton University Press.

De Morgan, A. 1847. Formal Logic. London: Taylor and Walton Booksellers and Publishers.

A Short Reflection on Conditionals

Material implication is a very odd thing. At the very least, it reveals to me the awkward fact that what we mean when we make a conditional statement is not quite what is stated when the logician represents it with something like “p q”. It seems that something is lost in translation.

So what is the big deal? According to the logician, a conditional statement is true when both the antecedent and consequent are true or whenever the antecedent is false. This leads to there being seemingly contrary statements both being true at the same time. For instance:

A: If Mars were more massive than Earth, Mars would have a stronger gravitational pull than Earth.

This seems to be a rather uncontroversial claim, though counter-factual. So we say it is true. But what about this:

B: If Mars were more massive than Earth, Mars would not have a stronger gravitational pull than Earth.

As in A, B has a false antecedent. Accordingly, we might conclude that B is true. But how could that be? It seems that A is faithful to a certain physical understanding of the laws of the universe whereas B is just pure fantasy. B seems to presuppose not only a more massive Mars, but also an entire universe with a different set of physical laws — one where mass and gravity exhibit an inverse relationship from the one we observe every day in this universe. But that’s the paradox of material implication for you. A and B are both true since Mars is not more massive than Earth!

So there does seem to be a sense in which A is more true than B. We might even suggest that B is false, or at least extremely less likely to be true given the set of cosmological constants and physical laws we currently observe. So, how should we understand such conditionals? One way around this problem is to talk about such conditionals in terms of probabilities. A becomes something like:

C: Pr( If Mars were more massive than Earth, Mars would have a stronger gravitational pull than Earth).

or more simply,

C: Pr(M|G)

This might be helpful because we can then contrast C, with B’s probabilistic equivalent:

D: Pr(M|~G)

We can then assess the probabilities of the two statements so as to determine the likelihood that one is more probably true than the other. Of course this might give us certainty. We could imagine that in some possible world where Mars is more massive than Earth, it is also the case that greater mass diminishes the gravitational field. Such a world would be somewhat odd though. For it would not be clear how planets might naturally form. But, let us suppose that planet formation occurs by some other force, say electromagnetism. Thus we cannot conclude, with any degree of certainty that D is in some way false, just extremely implausible.

But now a more difficult matter. How can we assess conditionals where the antecedent is not logically possible. That is, there is no possible world in which the antecedent obtains. Consider, for instance:

E: If a married bachelor were to run a complete marathon, he would run at least 26.2 miles.

Is E true, false, probably true, probably false? How would we assess it? If we just depend upon material implication, then it seems the statement is true, for it is false that any married bachelors have ever run a marathon. That seems like a silly interpretation of the statement though. Here I think we must turn to modalities like possibly “” and necessarily ““. E seems to suggest some sort of relationship between running a complete marathon and the distance in miles that one would have run were one to run it. At the same time, married bachelors cannot exist. So, how do we assess this odd statement? How would we know, for instance, that in possible worlds where married bachelors exist, complete marathons are not 12.1 miles, or that such universes have dimensions where miles are intelligible? But that kind of question doesn’t really seem to help since there are no possible worlds were married bachelors exist. We cannot appeal to probabilities at all. We are left then considering whether E means:

F: ◊ (Mb⊃ R)

or

G: □ (Mb⊃ R)

Now it seems that both F and G are false, since it is neither possible, nor necessary that a married bachelor should occupy any world where running a complete marathon would imply that one has run 26.2 miles. Even though such is the case in this world, could we say that were married bachelors to exist and run a marathon, they would have run 26.2 miles? One would be forced to suppose that married bachelors possibly cohere with such a world. But how would we know that they do?

I would suggest that this has some radical philosophical implications, namely, that if a conditional is going to rise to the possibility of being true, it must contain terms that are logically coherent. Why is that radical? I will have more to say about this in posts to come. In the meantime, I would love to know if anyone disagrees with my assessment. In other words, could a probabilistic, possible, or necessary conditional be true if the antecedent and/or the consequent are logically impossible?

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