52 ideas
9738 | Each line of a truth table is a model [Fitting/Mendelsohn] |
Full Idea: Each line of a truth table is, in effect, a model. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.6) | |
A reaction: I find this comment illuminating. It is being connected with the more complex models of modal logic. Each line of a truth table is a picture of how the world might be. |
9727 | Modal logic adds □ (necessarily) and ◊ (possibly) to classical logic [Fitting/Mendelsohn] |
Full Idea: For modal logic we add to the syntax of classical logic two new unary operators □ (necessarily) and ◊ (possibly). | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.3) |
9726 | We let 'R' be the accessibility relation: xRy is read 'y is accessible from x' [Fitting/Mendelsohn] |
Full Idea: We let 'R' be the accessibility relation: xRy is read 'y is accessible from x'. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.5) |
9737 | The symbol ||- is the 'forcing' relation; 'Γ ||- P' means that P is true in world Γ [Fitting/Mendelsohn] |
Full Idea: The symbol ||- is used for the 'forcing' relation, as in 'Γ ||- P', which means that P is true in world Γ. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.6) |
13136 | The prefix σ names a possible world, and σ.n names a world accessible from that one [Fitting/Mendelsohn] |
Full Idea: A 'prefix' is a finite sequence of positive integers. A 'prefixed formula' is an expression of the form σ X, where σ is a prefix and X is a formula. A prefix names a possible world, and σ.n names a world accessible from that one. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.2) |
13727 | A 'constant' domain is the same for all worlds; 'varying' domains can be entirely separate [Fitting/Mendelsohn] |
Full Idea: In 'constant domain' semantics, the domain of each possible world is the same as every other; in 'varying domain' semantics, the domains need not coincide, or even overlap. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 4.5) |
9734 | Modern modal logic introduces 'accessibility', saying xRy means 'y is accessible from x' [Fitting/Mendelsohn] |
Full Idea: Modern modal logic takes into consideration the way the modal relates the possible worlds, called the 'accessibility' relation. .. We let R be the accessibility relation, and xRy reads as 'y is accessible from x. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.5) | |
A reaction: There are various types of accessibility, and these define the various modal logics. |
9736 | A 'model' is a frame plus specification of propositions true at worlds, written < G,R,||- > [Fitting/Mendelsohn] |
Full Idea:
A 'model' is a frame plus a specification of which propositional letters are true at which worlds. It is written as |
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From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.6) |
9735 | A 'frame' is a set G of possible worlds, with an accessibility relation R, written < G,R > [Fitting/Mendelsohn] |
Full Idea:
A 'frame' consists of a non-empty set G, whose members are generally called possible worlds, and a binary relation R, on G, generally called the accessibility relation. We say the frame is the pair |
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From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.6) |
9741 | Accessibility relations can be 'reflexive' (self-referring), 'transitive' (carries over), or 'symmetric' (mutual) [Fitting/Mendelsohn] |
Full Idea: A relation R is 'reflexive' if every world is accessible from itself; 'transitive' if the first world is related to the third world (ΓRΔ and ΔRΩ → ΓRΩ); and 'symmetric' if the accessibility relation is mutual. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.7) | |
A reaction: The different systems of modal logic largely depend on how these accessibility relations are specified. There is also the 'serial' relation, which just says that any world has another world accessible to it. |
13137 | Conj: a) if σ X∧Y then σ X and σ Y b) if σ ¬(X∧Y) then σ ¬X or σ ¬Y [Fitting/Mendelsohn] |
Full Idea: General tableau rules for conjunctions: a) if σ X ∧ Y then σ X and σ Y b) if σ ¬(X ∧ Y) then σ ¬X or σ ¬Y | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.2) |
13140 | Bicon: a)if σ(X↔Y) then σ(X→Y) and σ(Y→X) b) [not biconditional, one or other fails] [Fitting/Mendelsohn] |
Full Idea: General tableau rules for biconditionals: a) if σ (X ↔ Y) then σ (X → Y) and σ (Y → X) b) if σ ¬(X ↔ Y) then σ ¬(X → Y) or σ ¬(Y → X) | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.2) |
13143 | Universal: a) if σ ¬◊X then σ.m ¬X b) if σ □X then σ.m X [m exists] [Fitting/Mendelsohn] |
Full Idea: General tableau rules for universal modality: a) if σ ¬◊ X then σ.m ¬X b) if σ □ X then σ.m X , where m refers to a world that can be seen (rather than introducing a new world). | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.2) | |
A reaction: Note that the universal rule of □, usually read as 'necessary', only refers to worlds which can already be seen, whereas possibility (◊) asserts some thing about a new as yet unseen world. |
13139 | Implic: a) if σ ¬(X→Y) then σ X and σ ¬Y b) if σ X→Y then σ ¬X or σ Y [Fitting/Mendelsohn] |
Full Idea: General tableau rules for implications: a) if σ ¬(X → Y) then σ X and σ ¬Y b) if σ X → Y then σ ¬X or σ Y | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.2) |
13148 | 4r rev-trans: a) if σ.n □X then σ □X b) if σ.n ¬◊X then σ ¬◊X [n occurs] [Fitting/Mendelsohn] |
Full Idea: System 4r reversed-transitive rules (also for S5): a) if σ.n □X then σ □X b) if σ.n ¬◊X then σ ¬◊X [where n is a world which already occurs] | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.3) |
13141 | Negation: if σ ¬¬X then σ X [Fitting/Mendelsohn] |
Full Idea: General tableau rule for negation: if σ ¬¬X then σ X | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.2) |
13138 | Disj: a) if σ ¬(X∨Y) then σ ¬X and σ ¬Y b) if σ X∨Y then σ X or σ Y [Fitting/Mendelsohn] |
Full Idea: General tableau rules for disjunctions: a) if σ ¬(X ∨ Y) then σ ¬X and σ ¬Y b) if σ X ∨ Y then σ X or σ Y | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.2) |
13142 | Existential: a) if σ ◊X then σ.n X b) if σ ¬□X then σ.n ¬X [n is new] [Fitting/Mendelsohn] |
Full Idea: General tableau rules for existential modality: a) if σ ◊ X then σ.n X b) if σ ¬□ X then σ.n ¬X , where n introduces some new world (rather than referring to a world that can be seen). | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.2) | |
A reaction: Note that the existential rule of ◊, usually read as 'possibly', asserts something about a new as yet unseen world, whereas □ only refers to worlds which can already be seen, |
13144 | T reflexive: a) if σ □X then σ X b) if σ ¬◊X then σ ¬X [Fitting/Mendelsohn] |
Full Idea: System T reflexive rules (also for B, S4, S5): a) if σ □X then σ X b) if σ ¬◊X then σ ¬X | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.3) |
13145 | D serial: a) if σ □X then σ ◊X b) if σ ¬◊X then σ ¬□X [Fitting/Mendelsohn] |
Full Idea: System D serial rules (also for T, B, S4, S5): a) if σ □X then σ ◊X b) if σ ¬◊X then σ ¬□X | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.3) |
13146 | B symmetric: a) if σ.n □X then σ X b) if σ.n ¬◊X then σ ¬X [n occurs] [Fitting/Mendelsohn] |
Full Idea: System B symmetric rules (also for S5): a) if σ.n □X then σ X b) if σ.n ¬◊X then σ ¬X [where n is a world which already occurs] | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.3) |
13147 | 4 transitive: a) if σ □X then σ.n □X b) if σ ¬◊X then σ.n ¬◊X [n occurs] [Fitting/Mendelsohn] |
Full Idea: System 4 transitive rules (also for K4, S4, S5): a) if σ □X then σ.n □X b) if σ ¬◊X then σ.n ¬◊X [where n is a world which already occurs] | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.3) |
13149 | S5: a) if n ◊X then kX b) if n ¬□X then k ¬X c) if n □X then k X d) if n ¬◊X then k ¬X [Fitting/Mendelsohn] |
Full Idea: Simplified S5 rules: a) if n ◊X then kX b) if n ¬□X then k ¬X c) if n □X then k X d) if n ¬◊X then k ¬X. 'n' picks any world; in a) and b) 'k' asserts a new world; in c) and d) 'k' refers to a known world | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 2.3) |
9739 | If a proposition is necessarily true in a world, it is true in all worlds accessible from that world [Fitting/Mendelsohn] |
Full Idea: If a proposition is necessarily true in a world, then it is also true in all worlds which are accessible from that world. That is: Γ ||- □X ↔ for every Δ ∈ G, if ΓRΔ then Δ ||- X. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.6) |
9740 | If a proposition is possibly true in a world, it is true in some world accessible from that world [Fitting/Mendelsohn] |
Full Idea: If a proposition is possibly true in a world, then it is also true in some world which is accessible from that world. That is: Γ ||- ◊X ↔ for some Δ ∈ G, ΓRΔ then Δ ||- X. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.6) |
9742 | The system K has no accessibility conditions [Fitting/Mendelsohn] |
Full Idea: The system K has no frame conditions imposed on its accessibility relation. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.8) | |
A reaction: The system is named K in honour of Saul Kripke. |
13114 | □P → P is not valid in D (Deontic Logic), since an obligatory action may be not performed [Fitting/Mendelsohn] |
Full Idea: System D is usually thought of as Deontic Logic, concerning obligations and permissions. □P → P is not valid in D, since just because an action is obligatory, it does not follow that it is performed. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.12.2 Ex) |
9743 | The system D has the 'serial' conditon imposed on its accessibility relation [Fitting/Mendelsohn] |
Full Idea: The system D has the 'serial' condition imposed on its accessibility relation - that is, every world must have some world which is accessible to it. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.8) |
9744 | The system T has the 'reflexive' conditon imposed on its accessibility relation [Fitting/Mendelsohn] |
Full Idea: The system T has the 'reflexive' condition imposed on its accessibility relation - that is, every world must be accessible to itself. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.8) |
9746 | The system K4 has the 'transitive' condition on its accessibility relation [Fitting/Mendelsohn] |
Full Idea: The system K4 has the 'transitive' condition imposed on its accessibility relation - that is, if a relation holds between worlds 1 and 2 and worlds 2 and 3, it must hold between worlds 1 and 3. The relation carries over. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.8) |
9745 | The system B has the 'reflexive' and 'symmetric' conditions on its accessibility relation [Fitting/Mendelsohn] |
Full Idea: The system B has the 'reflexive' and 'symmetric' conditions imposed on its accessibility relation - that is, every world must be accessible to itself, and any relation between worlds must be mutual. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.8) |
9747 | The system S4 has the 'reflexive' and 'transitive' conditions on its accessibility relation [Fitting/Mendelsohn] |
Full Idea: The system S4 has the 'reflexive' and 'transitive' conditions imposed on its accessibility relation - that is, every world is accessible to itself, and accessibility carries over a series of worlds. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.8) |
9748 | System S5 has the 'reflexive', 'symmetric' and 'transitive' conditions on its accessibility relation [Fitting/Mendelsohn] |
Full Idea: The system S5 has the 'reflexive', 'symmetric' and 'transitive' conditions imposed on its accessibility relation - that is, every world is self-accessible, and accessibility is mutual, and it carries over a series of worlds. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.8) | |
A reaction: S5 has total accessibility, and hence is the most powerful system (though it might be too powerful). |
9404 | Modality affects content, because P→◊P is valid, but ◊P→P isn't [Fitting/Mendelsohn] |
Full Idea: P→◊P is usually considered to be valid, but its converse, ◊P→P is not, so (by Frege's own criterion) P and possibly-P differ in conceptual content, and there is no reason why logic should not be widened to accommodate this. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.2) | |
A reaction: Frege had denied that modality affected the content of a proposition (1879:p.4). The observation here is the foundation for the need for a modal logic. |
13112 | In epistemic logic knowers are logically omniscient, so they know that they know [Fitting/Mendelsohn] |
Full Idea: In epistemic logic the knower is treated as logically omniscient. This is puzzling because one then cannot know something and yet fail to know that one knows it (the Principle of Positive Introspection). | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.11) | |
A reaction: This is nowadays known as the K-K Problem - to know, must you know that you know. Broadly, we find that externalists say you don't need to know that you know (so animals know things), but internalists say you do need to know that you know. |
13111 | Read epistemic box as 'a knows/believes P' and diamond as 'for all a knows/believes, P' [Fitting/Mendelsohn] |
Full Idea: In epistemic logic we read Υ as 'KaP: a knows that P', and ◊ as 'PaP: it is possible, for all a knows, that P' (a is an individual). For belief we read them as 'BaP: a believes that P' and 'CaP: compatible with everything a believes that P'. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.11) | |
A reaction: [scripted capitals and subscripts are involved] Hintikka 1962 is the source of this. Fitting and Mendelsohn prefer □ to read 'a is entitled to know P', rather than 'a knows that P'. |
13113 | F: will sometime, P: was sometime, G: will always, H: was always [Fitting/Mendelsohn] |
Full Idea: We introduce four future and past tense operators: FP: it will sometime be the case that P. PP: it was sometime the case that P. GP: it will always be the case that P. HP: it has always been the case that P. (P itself is untensed). | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.10) | |
A reaction: Temporal logic begins with A.N. Prior, and starts with □ as 'always', and ◊ as 'sometimes', but then adds these past and future divisions. Two different logics emerge, taking □ and ◊ as either past or as future. |
13728 | The Barcan says nothing comes into existence; the Converse says nothing ceases; the pair imply stability [Fitting/Mendelsohn] |
Full Idea: The Converse Barcan says nothing passes out of existence in alternative situations. The Barcan says that nothing comes into existence. The two together say the same things exist no matter what the situation. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 4.9) | |
A reaction: I take the big problem to be that these reflect what it is you want to say, and that does not keep stable across a conversation, so ordinary rational discussion sometimes asserts these formulas, and 30 seconds later denies them. |
13729 | The Barcan corresponds to anti-monotonicity, and the Converse to monotonicity [Fitting/Mendelsohn] |
Full Idea: The Barcan formula corresponds to anti-monotonicity, and the Converse Barcan formula corresponds to monotonicity. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 6.3) |
9725 | 'Predicate abstraction' abstracts predicates from formulae, giving scope for constants and functions [Fitting/Mendelsohn] |
Full Idea: 'Predicate abstraction' is a key idea. It is a syntactic mechanism for abstracting a predicate from a formula, providing a scoping mechanism for constants and function symbols similar to that provided for variables by quantifiers. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], Pref) |
14221 | Serious essentialism says everything has essences, they're not things, and they ground necessities [Shalkowski] |
Full Idea: Serious essentialism is the position that a) everything has an essence, b) essences are not themselves things, and c) essences are the ground for metaphysical necessity and possibility. | |
From: Scott Shalkowski (Essence and Being [2008], 'Intro') | |
A reaction: If a house is being built, it might acquire an identity first, and only get an essence later. Essences can be physical, but if you extract them you destroy thing thing of which they were the essence. Does all of this apply to abstract 'things'. |
14222 | Essences are what it is to be that (kind of) thing - in fact, they are the thing's identity [Shalkowski] |
Full Idea: The route into essentialism is, first, a recognition that the essence of a thing is "what it is to be" that (kind of) thing; the essence of a thing is just its identity. | |
From: Scott Shalkowski (Essence and Being [2008], 'Essent') | |
A reaction: The first half sounds right, and very Aristotelian. The second half is dramatically different, controversial, and far less plausible. Slipping in 'kind of' is also highly dubious. This remark shows, I think, some confusion about essences. |
14226 | We distinguish objects by their attributes, not by their essences [Shalkowski] |
Full Idea: In ordinary contexts, we distinguish objects not by their essences but by their attributes. | |
From: Scott Shalkowski (Essence and Being [2008], 'Ess and Know') | |
A reaction: Hence we have a gap between what bestows identity intrinsically, and how we bestow identity conventionally. If you could grasp the essence of something, you might predict a new attribute, as yet unobserved. |
14225 | Critics say that essences are too mysterious to be known [Shalkowski] |
Full Idea: According to critics, the thorniest problem for essentialism is the question of our knowledge of essence. It is usually at this point that terms of abuse such as 'dark', 'mysterious', and 'occult' are wheeled out. | |
From: Scott Shalkowski (Essence and Being [2008], 'Ess and Know') | |
A reaction: I'm inclined to think that the existence of essences can be fairly conclusively inferred, but that attributing a precise identity to them is the biggest challenge. |
13730 | The Indiscernibility of Identicals has been a big problem for modal logic [Fitting/Mendelsohn] |
Full Idea: Equality has caused much grief for modal logic. Many of the problems, which have struck at the heart of the coherence of modal logic, stem from the apparent violations of the Indiscernibility of Identicals. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 7.1) | |
A reaction: Thus when I say 'I might have been three inches taller', presumably I am referring to someone who is 'identical' to me, but who lacks one of my properties. A simple solution is to say that the person is 'essentially' identical. |
14223 | De dicto necessity has linguistic entities as their source, so it is a type of de re necessity [Shalkowski] |
Full Idea: De dicto necessity is a species of de re necessity. Anyone prone to countenance de dicto necessity must recognise mental and/or linguistic entities, thus counting each of them as a res to which necessity attaches. | |
From: Scott Shalkowski (Essence and Being [2008], 'Essent') | |
A reaction: This seems to rest on the Kit Fine thought that analytic necessities seem to derive from the essences of words such as 'bachelor'. I like this idea: all necessity is de re, but some of the 'things' are words. |
9220 | Lewis must specify that all possibilities are in his worlds, making the whole thing circular [Shalkowski, by Sider] |
Full Idea: If purple cows are simply absent from Lewis's multiverse, then certain correct propositions turn out to be impossible. Lewis must require a world for every possibility. But then it is circular, as the multiverse needs modal notions to characterize it. | |
From: report of Scott Shalkowski (Ontological Ground of Alethic Modality [1994], 3.9) by Theodore Sider - Reductive Theories of Modality 3.9 | |
A reaction: [Inversely, a world containing a round square would make that possible] This sounds very nice, though Sider rejects it (p.197). I've never seen how you could define possibility using the concept of 'possible' worlds. |
13725 | □ must be sensitive as to whether it picks out an object by essential or by contingent properties [Fitting/Mendelsohn] |
Full Idea: If □ is to be sensitive to the quality of the truth of a proposition in its scope, then it must be sensitive as to whether an object is picked out by an essential property or by a contingent one. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 4.3) | |
A reaction: This incredibly simple idea strikes me as being powerful and important. ...However, creating illustrative examples leaves me in a state of confusion. You try it. They cite '9' and 'number of planets'. But is it just nominal essence? '9' must be 9. |
13731 | Objects retain their possible properties across worlds, so a bundle theory of them seems best [Fitting/Mendelsohn] |
Full Idea: The property of 'possibly being a Republican' is as much a property of Bill Clinton as is 'being a democrat'. So we don't peel off his properties from world to world. Hence the bundle theory fits our treatment of objects better than bare particulars. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 7.3) | |
A reaction: This bundle theory is better described in recent parlance as the 'modal profile'. I am reluctant to talk of a modal truth about something as one of its 'properties'. An objects, then, is a bundle of truths? |
13726 | Counterpart relations are neither symmetric nor transitive, so there is no logic of equality for them [Fitting/Mendelsohn] |
Full Idea: The main technical problem with counterpart theory is that the being-a-counterpart relation is, in general, neither symmetric nor transitive, so no natural logic of equality is forthcoming. | |
From: M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 4.5) | |
A reaction: That is, nothing is equal to a counterpart, either directly or indirectly. |
14224 | Equilateral and equiangular aren't the same, as we have to prove their connection [Shalkowski] |
Full Idea: That 'all and only equilateral triangles are equiangular' required proof, and not for mere curiosity, is grounds for thinking that being an equilateral triangle is not the same property as being an equiangular triangle. | |
From: Scott Shalkowski (Essence and Being [2008], 'Serious') | |
A reaction: If you start with equiangularity, does equilateralness then require proof? This famous example is of two concepts which seem to be coextensional, but seem to have a different intension. Does a dependence relation drive a wedge between them? |
1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |
Full Idea: The Egyptians were the first to claim that the soul of a human being is immortal, and that each time the body dies the soul enters another creature just as it is being born. | |
From: Herodotus (The Histories [c.435 BCE], 2.123.2) |