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Pruss and Rasmussen on Modal Logic

Chapter two of Pruss and Rasmussen's Necessary Existence can be seen as preliminary to the main project of the book. The core aim of the chapter is the explanation and defense of a picture of metaphysical modality that is already (so it seems to me) standard among analytic metaphysicians. The chapter concludes with a brief demonstration of a proposition that will be a crucial lemma in many of the arguments throughout the book: if a necessary being is possible, then a necessary being is actual. Those who are already immersed in analytic metaphysics and accept the modal system S5 could...
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Finkish Backtracking Abilities

A disposition or ability is said to be 'finkish' iff, were the conditions for its exercise actual, the disposition/ability would be lost. (See Martin and Lewis.) For instance, imagine a sorcerer casts a spell on a fragile glass that will make it cease to be fragile if it is ever struck or dropped. (This example is due to Vihvelin. Realistic, non-magical examples are possible but more complex.) A fragile object is one that is disposed to break if struck, dropped, etc. The intuition is supposed to be that, given that the glass is intrinsically qualitatively identical to any other fragile...
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Chillingworth on Strange Extractions in Chymistry and Logick

In your sixt parag. I let all passe saving only this, That a perswasion that men of different Religions ... may be saved, is a most pernitious heresy, and even a ground of Atheisme. What strange extractions Chymistry can make I know not, but sure I am, he that by reason would inferre this conclusion, That there is no God, from this ground, That God will save men in different Religions, must have a higher strain in Logick, then you or I have hitherto made shew of. In my apprehension, the other part of the contradiction, That there is a...
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Locke's Populist Logic

Again, your lordship [Stillingfleet] charges me, that I do not place certainty in syllogism; I crave leave to ask again, and does your lordship? ... And if you do, I know nothing so requisite, as that you should advise all people, women and all, to betake themselves immediately to the universities, and to the learning of logic, to put themselves out of the dangerous state of scepticism: for there young lads, by being taught syllogism, arrive at certainty; whereas, without mode and figure, the world is in perfect ignorance and uncertainty, and is sure of nothing. The merchant cannot be...
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Sets and Possible Worlds

This semester I'm directing an independent study on modal logic with a couple of students with strong math background. Yesterday some questions about sets and possible worlds came up, so I wrote up some notes for my students on the subject. This blog post is adapted from those notes. Introduction The development of axiomatic set theory was launched by consideration of Russell's Paradox: let A be the set of all sets that do not contain themselves. Does A contain itself or not? (This was on Existential Comics just yesterday!) The collection of axioms mathematicians developed to avoid paradox has the...
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Counterpossible Reasoning in Philosophy of Religion (and Elsewhere)

The latest (July 2011) Faith and Philosophy contains an excellent article by Jeff Speaks on some difficulties related to establishing the consistency of certain claims (he uses as examples the existence of human freedom and the existence of evil) with the existence of an Anselmian God. The basic idea is this: since an Anselmian God is, by definition, a necessary being, establishing the possibility of an Anselmian God is tantamount to establishing the necessary, and therefore actual, existence of an Anselmian God. But these compatibility arguments typically, in one way or another, assume the possibility, and so the actuality, of...
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Validly Affirming the Consequent

I'm grading some logic exercises from an intro class today. The students were supposed to give examples of valid and invalid arguments, with true and false premises and conclusions, and so forth. One student turned in the following fantastic example (I have edited it to remove some ambiguities):

(P1) If P1, then C
(P2) C
:. (C) P1

The student, understandably, thought the argument was invalid, since it has the form of affirming the consequent. However, due to the self-reference, the argument is valid. The student just wrote 'the premise' and 'the conclusion', so I'm not sure if this is the intended interpretation, but still pretty clever.
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Kant, Strawson, and Conditionals

P. F. Strawson is not one of Kant's more sympathetic interpreters: Kant's faculty psychology, he thinks, is no more than a historical curiosity. The account of logic is likewise a mess. Above all, transcendental idealism is sheer nonsense. Also, of course, Kant's arguments notoriously rely on the claim that Euclidean geometry is known a priori to be the geometry of the sensible world, whereas we now know that this claim is not only not known a priori, but is actually false. (James Van Cleve has argued, however, that Kant needs only the existence of some a priori geometrical knowledge, and...
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Explanatory Principles and Infinite Propositions

In the course of his discussion of cosmological arguments, Sobel argues against the Principle of Sufficient Reason and similar strong explanatory principles. In particular, he argues that even a weak principle like "there is a deductive explanation that has only true premises for every contingent truth" will result in modal collapse (p. 218). In Sobel's terminology, an argument 'deductively explains' its conclusion iff (1) the argument is sound, and (2) the conclusion does not entail the premises (p. 219; condition (2) applies to contingent conclusions only). Sobel now introduces the following two premises: (3) If there is any true contingent...
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A Leibnizian Cosmological Argument

Sobel's sixth chapter is devoted to, as he says, "proofs a contingentia mundi" (from the contingency of the world). The chief exponent here is Leibniz, though Sobel also considers Hume's Demea and his probable source, Samuel Clarke. Sobel argues that Leibniz's argument is valid ... by contradiction explosion. That is, he argues that Leibniz's premises are inconsistent. In this post, I show how to fix the argument using Leibnizian resources. In the next post, I will give another version of the argument which uses premises which I consider to be anti-Leibnizian, but which I think are more widely held than...
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The Dialectical Appropriateness of Ontological Arguments

After, for some reason or other, spending some 30 dense pages of Logic and Theism on the laughable ontological arguments of Descartes and Spinoza, Sobel moves on to the more interesting argument advanced by Anselm. (The next chapter deals with versions of the argument set in modern modal logic, such as those of Hartshorne and Plantinga.) In my view, the Descartes and Spinoza arguments don't even look good; the Anselm version at least produces puzzlement, insofar as the reasoning looks valid, yet it seems, intuitively, that no such strong conclusion could ever be derived from such weak premises. Sobel (fairly uncontroversially...
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Quote of the Day: Aristotle on the Law of Contradiction

The most certain principle of all [is one] it is impossible to be mistaken about ... A [principle] one must have in order to understand any being whatsoever - this is not a [mere] hypothesis! ... Next we will state what this principle is: it is impossible for the same thing at the same time to exist and not to exist in the same [subject] and in the same respect (and however many other [qualifications] we [previously] defined, let them be defined [here] on account of the logical difficulties). ... But we now have accepted that it is impossible for a being to be and not be at the same time, and we showed that this was the most certain of all principles. In fact, some people, because [they are] uneducated, think that even this ought to be proven. [Someone who] doesn't know that it is necessary to prove some things and not others is uneducated...
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Calvinism and Arminianism: On Making the Right Objection

I want to make an important point about something that is either a reasoning mistake (if done accidentally) or an underhanded rhetorical trick (if done intentionally). I've seen it a lot (and done it myself, accidentally) in debates between Calvinists and Arminians (mostly on a popular level, but sometimes even in the writings of philosophers and theologians), so I'm going to use this debate to provide examples ... The issue is this: all of us believe implicit contradictions, because we are unable to determine all the consequences of our beliefs. This means that there is a big difference between rejecting a belief p and accepting a belief q which, unbeknownst to you, logically entails not-p. So, if you believe ...
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Truth-Makers, Truth-Conditions, and Middle Knowledge

Middle knowledge is a problem that has been bothering me for quite some time now. It goes like this: middle knowledge is knowledge of the truth or falsity of counterfactuals of freedom, where a counterfactual of freedom (sometimes called a counterfactual of creaturely freedom) is a statement about what some agent having libertarian free will would do in a purely hypothetical situation, i.e. one that never has and never will occur. Libertarian free will means that one is free because one could do otherwise than one actually does. So, for instance, if human beings (including me) have libertarian free will...
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Blogging Parmenides

I feel the need to point to this post about Parmenides over at Mathetes simply because ... well, because I approve of blogging about Parmenides! The post gives a good overview of Parmenides' argument for the establishment of monism. To which let me add three things: This is the oldest deductively valid argument in surviving literature. It is contained in a hexameter poem (written, presumably, in imitation of Homer and Hesiod) which begins with an appeal to divine revelation (a narrative about being carried in a chariot to meet a strange goddess who promises to reveal "the way of truth"...
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