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18

The common axiom systems for intuitionistic logic are both sound and complete. It is interpretable as an S4 modal logic or as a weakening of classical logic (essentially you just drop the law of excluded middle and double negation elimination and then tweak the quantifier rules). Since it is both sound and complete it is not incomplete. The fact that they ...


10

Kripke hesitates a little bit when it comes to fictional entities. The issue partially boils down to the following question: "Could fictional entities like Sherlock Holmes and Mickey Mouse exist?" On the one hand, you might think "Yeah, of course! Sherlock Holmes isn't contradictory or anything; surely there's a possible world where he could have existed." ...


8

They are in opposition, as Quine and Kripke generally are on interpreting modal logic, and much of what is related to it. Rigid designators are defined as those picking out the same object in all possible worlds, so unsurprisingly Quine and Kripke do not see eye to eye on this issue in particular. To pick out the same object we must agree on how it is done, ...


7

The two notions (completeness and incompleteness) are not opposites but very much connected (not only by Godel's name in the name of the two theorems). Do take into account that Godel's Completeness Th of First-Order Logic is : if a sentence is true in all the models of the axioms (i.e. it is a logical consequence of the axioms) then it is also formally ...


4

Technically, actually, your sentence doesn't even have to be true. Consider: "There's a man in Memphis who paints pictures of dogs in the park for a living. I don't know what his name is, but let's just call him Jerry." If I were to say next "Jerry's name is 'Jerry'," I would (probably) be speaking falsely, because 'Jerry' is just the name that we, as ...


4

Like perception, introspection, memory, and testimony, a priori justification is fallible. One might be justified in believing something a priori, e.g., that every event has a cause, that is actually false. Many physicists think that some subatomic events occur at random and so have no cause. Besides being fallible, it seems that a priori justification is ...


4

Another insightful reference for the semantics of dual-intuitionistic logics is Yaroslav Shramko's paper on the "logic of scientific research" (Studia Logica 2005). Just to add a brief comment on @TMF's answer, for the benefit of the author of the original question, it is worth noting that Priest's "Da Costa Logic" was indeed proposed as a fragment of ...


4

Holism is an epistemological position, and externalism is a semantic one. Of course, some degree of interaction is to be expected, but not only is it possible to hold them together, it is not particularly challenging. The appearance of incompatibility comes from the misleading use of the word "meaning". In the holism, especially Quine's and Davidson's, "...


4

The TL;DR version is that Kripke has misunderstood Lewis’ counterpart theory, and so his criticism is off-base. The longer version follows. To understand what is going on here, it helps to have a little background. Modal logic, with the box/diamond notation, was originally conceived by C I Lewis to express modal properties of propositions, e.g. to say of ...


3

You surely cannot do this in System K. Instead of focusing on symbol manipulation, it's important to understand the semantics of these sentences. System K is a normal modal logic, so we may dispense with axioms and focus on the frame conditions instead: ~◇◻P is saying that there is no accessible world W, such that every world that W can access satisfies P....


3

It does make sense in a way. Kripke's thesis is that proper names have essences, properties that belong to them of necessity, not accidentally. Putnam extended this thesis to "natural kinds" of objects, whose essential properties "carve nature at the joints", in Plato's metaphor, rather than according to our cultural biases, practical needs, etc., those ...


3

Shane's answer is perfectly correct as to the question as worded, but I figure we could use some added background. I think the question doesn't seem to grasp what Kripke means by rigid designator. A rigid designator by definition is a term that picks out only one thing and continues to pick out the same thing regardless of everything else. That is its ...


3

(◊ ∃x Fx) ↔ (∃x ◊ Fx) can be seen as a conjunction of (◊ ∃x Fx) → (∃x ◊ Fx) (the Barcan formula in the narrower sense) and (∃x ◊ Fx) → (◊ ∃x Fx) (the converse Barcan formula). The forward direction, (◊ ∃x Fx) → (∃x ◊ Fx), says that no new objects come into existence when going from one possible world to another: If there is an ...


2

Most of the "dualized" intuitionistic logics in the literature, e.g., Priest-da Costa and anti-intuitionistic, are fragments of Cecylia Rauszer's Heyting-Brouwer logic, in which all connectives---not merely negation---are given duals. It's probably worth your time to review Rauszer's 1974 "Semi-Boolean Algebras and Their Application to Intuitionistic Logic ...


2

Descriptivism requires that proper names have their reference fixed in virtue of a description attached to them, which singles out a unique object in the world, and which is analytic: the meaning of a proper name can be expressed by a definite description. Analytic means: in virtue of the meaning of words. It is implicit here that the meaning of words is ...


2

The paper H.P. Sankappanavar, "Heyting algebras with dual pseudocomplementation", published in Pacific journal of Mathematics 117 (1985), 405–415, I believe, provides an algebraic semantics for what Priest calls as "Da Costa Logic".


2

What you are looking for is called dynamic epistemic logic, it was developed starting in late 1980s to represent changes in knowledge. Internet Encyclopedia of Philosophy gives a nice overview with many references: "The modal knowledge operators in epistemic logic are formally interpreted by employing binary accessibility relations in multi-agent Kripke ...


2

The main problem with your suggestion is not philosophical but mathematical. Let's denote quus by # and plus by +. Even without any skeptical thesis, you simply cannot move from 57 # b = 5 to 57 = 5 - b. With plus, such a move is made by subtracting b from both sides. For instance, going from a+b=c to (a+b)-b=c-b to a=c-b. But such a move is valid ...


2

Kripke's argument for a casual-historical view of names is first and foremost about proper names. It is a theory to explain how proper names are used in natural languages. Kripke argues that rigid designators are central to our use of language and how we, in this world as opposed to a possible world, use language. To illustrate, from Marc Cohen: The claim ...


2

~◇◻P → ◇◇~P Following is the proof. - ~◇◻P - ∴ ◻~◻P - ∴ ◻◇~P From ◻◇~P we can apply an axiom from Modal System D, stating that everything that is necessary is also possible, that is if P is necessarily the case then P cannot be impossible. D: ◻A → ◇A Then from ◻◇~P using Axiom D, it follows that : ◇◇~P You can find an application of System D to ...


2

Yes. Reading Kripke's works is very different than reading a math book on set theory, principally because his interests are in "meta" issues, and his works (books), are comprised of his lecture series, in many cases. But if, you do have an understanding of Philosophy of Language and Logic that's derived from your understanding of Frege, Russell, and ...


2

Long comment I'm puzzled also, but for a different reason... From (∃y) ((x) ◊(x ≠ y)), using a fresh term a, we have, by (∃-elim): (x) ◊(x ≠ a). Thus, using (∀-elim) with a (legitimate) we have: ◊(a ≠ a) and finally with (∃-intro): (∃y) ◊(y ≠ y). But how we can say that the premise is satisfiable, if we can derive from it: ◊(a ≠ a) ?


1

If a meter is the same thing in all possible worlds, we're referring to an abstract measurement, divorced from any definition that can vary. The meter has been defined as a tenth of a millionth of the distance from North Pole to equator going through Paris, or by a platinum-iridium bar with scratches on it, or as a certain number of wavelengths of a ...


1

There's really no translation to be done. Here's an outline of a proof. Assume P ⊨c Q. To show ⊨k 򪪪(P → Q), assume for contradiction that 򪪪(P → Q) is not valid. This means that there's a world which falsifies P → Q, that is, in which P is true but Q is false. But that contradicts the assumption that Q follows classically from P.


1

Kripke models can be used to prove that a formula is not valid. Reagrading your example, this means, to show that the antecedent: (p & q → (p → q)) is true in w (the "actual" world) and the consequnr □(p → q) is not, i.e. (p → q) is false in some world w' accessible from w (i.e. such that wRw'). If q is false in w (written: w ⊮ q) we have that p & ...


1

Conifold's answer seems to contain all the relevant materials, but I'll try to arrange them a bit differently. So yes, there seems to be a certain collision between semantic externalism and holism. The reason is not because holism implies internalism (as your first point suggests) but because holism disputes the very distiction between internal and external ...


1

Kripke has more recently come to hold that socalled fictional entities are real entities, and more specifically Kripke holds that e.g. Scherlock Holmes is an abstract object created by the author A.C. Doyle. This strategy may solve many riddles concerning socalled empty names, and perhaps create new riddles. Kripke, Saul A. 2011. “Vacuous Names and ...


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