# Tag Info

14

The box and the diamond are duals (in the usual systems), so if you have the box, you can define: Definition 1. (Possibility)   ♢φ     =def    ¬▢¬ φ. If you have the diamond as primitive, you can define the box in the same way. Now, suppose we take the box as a primitive. Then the ...

13

In general □□ p and □ p are very different. Thinking in terms of Kripke frames, they only obviously coincide if the accessibility relation is transitive. This is true for Kripke frames validating S5, but not in general. Consider a frame with three worlds a,b,c where a sees b and b sees c but a doesn't see c: then given a valuation making p true at b but ...

11

As you know, the two terms mean something different, as spatio-temporal relations aren't in themselves causal relations. Being spatio-temporally isolated simply means not standing in spatio-temporal relations like “before,” “10 minutes after,” “beside,” “10 meters below.” Being causally-isolated just means not having any causal relations, such that nothing ...

11

Well, in English you would use the subjunctive tense and say, "possibly I could have had a cat in my room, but in fact I do not." That's a reasonable statement to make. "I could have been a doctor" is another statement of the same kind that is considered normal to utter. The meaning of such statements is tied to an implicit idea of ...

10

A very simple explanation: "Possibly P" means "P might be true"; "Not necessarily P" means "P might be false". These are not equivalent. For example, if P is always true, then "Possibly P" is true but "Not necessarily P" is false; if P is always false, then "Possibly P" is false but "Not necessarily P" is true. On the other hand, "Not necessarily not P" ...

10

On the prevailing extensional interpretation of modality the difference between possibility and probability is the diffference between quality and quantity, possibility is the quality quantified by probability, see Probability Distributions Over Possible Worlds by Bacchus. This interpretation can be traced back to Leibniz's determinate possible worlds, but ...

9

Logical omniscience was always only a technical problem related to formalization of epistemic logic in terms of possible worlds. Since classical possible worlds are supposed to be consistent and deductively closed they must include all the consequences along with their premises, and nothing contradicting the premises. So if we are describing acquisition of ...

8

It is a central feature of all the main formal systems that when a statement is provable, then it is provably provable. Indeed, this feature is one of the derivability conditions that is commonly used in the proof of the incompleteness theorem, and it is central to Goedel's proof of the second incompleteness theorem. But also, I might add, this principle ...

8

So the three valued logic of Łukasiewicz has three truth values {1,i,0}. Łukasiewicz was trying to solve the problem of future contigents with this logic. His view is that statements about the past and present have an unalterable truth value, so if they are true they are necessarily true, if they are false they are necessarily false. Future contingents ...

8

I understand your argument as follows: A maximally great being possibly exists. If a maximally great being possibly exists then it necessarily exists. If a maximally great being necessarily exists then it actually exists Therefore, a maximally great being actually exists. Here is a reconstruction using modal logic: ◊∃xGx ◊∃xGx → □∃xGx ...

8

If we're talking about metaphysical possibility, then normally yes. If you reject the claim that "if P then possibly P", you must also reject the claim that "if necessarily P then P". Proof: suppose we reject truth implies possibility (that is, we reject that for every formula P, if P then possibly P). Then for some formula A, we have A and not-possibly A. ...

8

You are correct about the relationship between □P → P and the reflexivity of the accessibility relation. As to whether you want to take this as an axiom, it depends entirely on your intended interpretation. If □ is to be interpreted as "it is necessarily true that" then □P → P holds, since, as you say, if a proposition is necessarily true then it ...

7

Preliminaries Priest's presentation of variable domain modal logic in An Introduction to Non-Classical Logic does utilize a free logic base. Check out ch. 15 if you can get your hands on a copy. You ask if there is a flaw in your reasoning: Then I suppose you can use Existential Instantiation at that world such that there is some constant a such that □...

7

This looks like a "lame terms" redux of Plantinga's "victorious" ontological argument from The Nature of Necessity. Here is Plantinga's explanation of why if a maximally great being exists in some possible world it exists in every possible world. He attributes the idea to Findlay (p. 214): "Those who worship God do not think of him ...

7

In a sense, Deleuze's virtual and Lewis's possible worlds compete to provide the "right" conception of the possible. The descriptions are indeed similar but this is deceptive, Deleuze and Lewis, in part reflecting their respective traditions (continental and analytic), are far apart on the possible because they are far apart on the real. Lewis's &...

7

Hughes and Cresswell's proof proceeds roughly as follows: They show that any sentence of S5 is logically equivalent to a modal conjunctive normal form (MCNF) in which a sentence takes the form of a conjunction of a series of disjunctions, with each disjunction having a specific form. They formulate a 'test', whereby a disjunction passes the test iff at ...

6

The first one (often called semantic brackets) is mostly found in formal semantics, and it's the name of the evaluation function, which maps expressions in a formal language to objects in the model of evaluation. Suppose A is the sentence "snow is white." Here's how semantic brackets are used: [["A"]] is true ≡ snow is white The second one (often ...

6

Is it an acceptable idea that each individual carries their own model of reality in their mind? Certainly! To some extent, everyone does. How many people have you come across who believe that their spouse or parent is the best in the world? Or even something as concrete as "I believe I sent you the email last week". Is there a name for the model that ...

6

The question: is there a significant geometric form of these logics? Here, "these logics" refers to Boolean algebras, Heyting algebras and modal algebra. The various representation theorems for these algebras as set-algebras related to certain topological spaces seem to provide a positive answer to this question. These representation theorems are non-...

6

An understanding of classical propositional logic and first-order logic should suffice. Some notes: Modal languages look very much like non-modal ones. For example, if you have a non-modal propositional language generated by the following grammar: (L1) φ := p | ¬φ | (φ ∧ φ), you can obtain a modal propositional language by adding ...

6

Peter Smith—who is or has been a user of this forum—has a discussion article posted called Teach Yourself Logic 2015: A Study Guide (PDF, iv + 94 pp. Last updated 1 Jan 2015). It's on his website. It lays out his informed opinions of the relative merits of the various books and resources for self-study, including the good books mentioned by Mauro Allegranza ...

6

The specific quotation you gave about Łukasiewicz refers to the fact there was an attempt to understand intuitionistic logic as a many-valued logic, but this failed because Gödel proved in the early 1930s that intuitionistic logic is not n-valent for any n. To address your last paragraph, truth and certainty are quite different things. To ask, to what ...

6

The proof does not work, I think. Your problems are premises 1 and 2. My main problem is with premise 1: it is possible for a maximally great being to exist. Why should we accept this? There is the tempting notion, mentioned in Eliran's (considerably better) answer, that anything we can imagine must be possible; but there is a great deal to be said about ...

6

Plantinga uses the concept of non-trivial properties in his transworld depravity defense of God's benevolence, see How does free will defense of God's benevolence work? Ciprotti in Theological Compatibilism and Essential Properties discusses Plantinga's trivial and non-trivial properties with PDO (power to do otherwise, a.k.a. free will) as central ...

6

To add to the existing answer, there's also epistemic logic, where □ is interpreted as knowledge (relative to a given subject). Possible worlds in such a system are worlds that are consistent with the subject's current information. Since necessity is truth in all possible worlds, it means in epistemic logic that if something is necessary then it just ...

5

Ackermann's Rule Gamma is the rule that from "├── A->B" and "├── A", one may infer "├── B," where "->" denotes the material conditional. (Equivalently, from "├── ~A V B" and "├── A", infer "├── B.") Note that this is distinct from modus ponens or disjunctive syllogism; this is the rule that if ~A V B is a theorem and A is a theorem then B, too, is a theorem....

5

"Is anything essential lost" or are the shortcomings of Meyer's system in "the exotic outer reaches of what is possible in traditional PA"? PA proves that formula if p > 2 is prime, then there is a positive integer y which is not a quadratic residue mod p; that is, ∃y ∀z: ¬(y ≡ z^2 (mod p)). That looks like a pretty unexotic bit ...

5

While I think Hunan's answer is roughly exhaustive, I'm going to give an answer that proceeds more simply. Consider ◻P. From necessarily P, it follows that ◇P [P is possible]. Or at least it does in every system of modal logic I'm aware of. Given this, it would be strange for ◇P → ~◻P. Since that would mean ◻P ↔ ~◻~P. The issue and the solution is to ...

5

We begin by recalling the basic definitions needed to settle the questions: Definitions. R is reflexive       =def   ∀w : wRw; R is symmetric    =def   ∀w, v : (wRv → vRw); R is transitive      =def   ∀w, v, u : (wRv ∧ ...

5

Some formal notes to complement Mauro's excellent answer. As one would expect in a discussion of modality, we're going to talk about modal models when defining things. Most will be familiar with logics K, S4, and so on. K and its superlogics are too sophisticated for a discussion of metaphysical modalities, so we'll begin with pre-Kripke modal models, going ...

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