Title: oxford / 127 logic

Front: equinumerosity
Back: definition of equinumerosity: two sets A and B are equinumerous iff there exists a bijection (surjection and injection) between A and B.

Front: countable
Back: A set is countable has the same cardinality as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set.

Front: powerset
Back: the power set of any set S is the set of all subsets of S, including the empty set and S itself

Front: induction on formula construction
Back: induction on formula construction is used when one is trying to establish a fact of the form: every formula has a certain property p.
base case: all atomic formulae have p
inductive step: the rules of formula construction retain p
example: proof that all formulas have fnitely many sentence letters

Front: induction of the proof of a formula
Back: Induction on the proof of a formula is used when one is establishing a fact of the form: every theorem has a certain property p.
base case: axioms have p
prove that inference rules preserve p
----------
by induction on the number of proof-steps, every theorem has p
example: proof that every PL-theorem is PL-valid (soundness of PL)

Front: adequate
Back: a set of connectives is adequate iff all truth functions can be symbolized using sentences containing no connectives not in that set.

Front: scheffer stroke
Back: definition:
fact: {|} is adequate

Front: cardinality
Back: two sets have the same cardinality iff they are equinumerous

Front: PL-valuation

Front: PL-validity

Front: PL-semantic consequence

Front: shortenings in PL

Front: axiomatic system for PL

Front: LP-validities
Back: the set of LP-valid wffs is equal to the set of PL-valid wffs

Front: Łukasiewicz truth-table for ~

Front: Łukasiewicz truth-table for AND

Front: Łukasiewicz truth-table for OR

Front: Łukasiewicz truth-table for ARROW

Front: Łukasiewicz-validity

Front: Łukasiewicz semantic consequence

Front: Klenne truth-table for ARROW

Front: LP-semantic validity and consequence

Front: LP truth-tables
Back: Kleene's truth-tables

Front: MPL-model

Front: MPL-valuation

Front: types of MPL-models

Front: Validity in an MPL-model

Front: Validity and semantic consequence for MPL

Front: K-model
Back: no restriction on R

Front: D model
Back: R is serial in W

Front: T-model
Back: R is reflexive in W

Front: B-model
Back: R is reflexive in W and symmetric

Front: S4-model
Back: R is reflexive in W and transitive

Front: S5-model
Back: R is reflexive in W, symmetric, and transitive

Front: DT for PL

Front: CUT

Front: axiomatic proof from a set

Front: axiomatic proof

Front: axiomatic system K
Back: In addition, any instance of the following is a theorem of K:

Front: axiomatic system D

Front: axiomatic system T
Back: In addition, this is a theorem of T:

Front: axiomatic system B
Back: In addition, this is a theorem of B:

Front: axiomatic system S4
Back: In addition, this is a theorem of S4:

Front: axiomatic system S5
Back: In addition, any instance of the following is a theorem of S5:

Front: hierarchy of MPL-systems

Front: Graph of axioms in the MPL-systems

Front: df serial

Front: df reflexive

Front: df symmetric

Front: df transitive

Front: total relation
Back: # Sider's definition:
# Standard definition:
R is total in X iff for all a and b in X, Rab or Rba
(Terminological note: This is what Sider calls strongly connected.)

Front: precisification

Front: supervaluation

Front: validity and semantic consequence for S (the supervaluationist system)

Front: informal definition of tense operators

Front: valuation function for PTL
Back: Where M is a PTL model

Front: theorem
Back: A theorem is any formula that is the last line of an axiomatic proof (i.e. an axiomatic proof from the empty set).

Front: MPL-shortenings

Front: s-provability-from

Front: S-consistency

Front: canonical S-model

Front: truth and membership in canonical models

Front: term (PC)
Back: any variable or constant

Front: PC-model

Front: variable assignment

Front: [latex]$$``g_u^\alpha"$$[/latex]

Front: denotation

Front: PC-valuation

Front: truth in a PC-model

Front: validity and semantic consequence in PC

Front: axiomatic system for PC

Front: compactness
Back: compactnes is true of PC

Front: semantics of "="

Front: two ways of generalising PC-quantifiers
Back: [latex]
go from two cardinalities (the cardinality of $\emptyset$ and the cardinality of $D$) to any cardinality (generalised monadic quantifiers) (e.g. $\exists_{\infty}\alpha\phi$, and $\textbf{most}\alpha\phi$)
\bigskip
go from one-place quantifiers to two-place quantifiers (generalised binary quantifiers) (e.g. $\textbf{most}^2(\alpha:\phi)\psi$)
[/latex]

Front: [$]\phi^{M,g,\alpha}[/$]

Front: definition of g in second-order logic

Front: how to do validity proofs with meta-language items
Back: [latex]
Suppose for reductio that $\nvDash_{\verb|lang|}$ ...$\phi$... for some \verb|lang|-wff $\phi$ (and some ...) \\
\\ indent \\
There exists some model $M$ (interpretation $I$, variable assignemnt $g$, etc.) such that ... \\
This is a contradiction \\
de-indent \\ \\
Thus for any \verb|lang|-wff(s) $\phi$ (and any ...) $\vDash_{\verb|lang|}$ ...$\phi$...
[/latex]

Front: how to do axiomatic proofs with meta-language items

Front: the metalanguage and the object language
Back: [latex]
Object: \\
$\land$ \\
$\rightarrow$ \\
$\forall$ \\
$\alpha = \beta$ \\
Meta: \\
and \\
if...then \\
for all \\
$\in$ \\
$V_M(\phi)=1$
[/latex]

Front: configuration
Back: a mathematical representation, both of the world and of the meanings of non-logical expressions

Front: steps for an MPL-countermodel diagram

Front: components of a logical system
Back: grammar
1. primitive vocabulary
2. definition of wff
(2.1 shortenings)
semantics
3. definition of configuration (PL: interpretation, MPL, PC, QML: model)
4. definition of valuation function
5. definition of truth in an configuration [interpretation or model] (⚠️for MPL, Sider uses "validity in a model").
6. definition of validity and semantic consequence
proof theory
7. axioms
8. rules
9. theoremhood

Front: deduction theorem for PL

Front: deduction theorem for MPL
Back: Fails using old definition of provability from a set. With the new definition of S-probability from a set, it holds.

Front: SQML-model

Front: SQML-valuation

Front: axiomatic system for SQML

Front: SC-model

Front: SC-valuation

Front: The anti symmetry assumption
Back: * Stalnaker requires closeness to be anti-symmetric
* This prohibits ties in similarity
* This makes all instances of the following schema valid:
An example with ties:
"It is not the case that if Bizet and Verdi were compatriots, Bizet would be
Italian; and itisnotthe case that ifBizetand Verdiwere compatriots, Bizet
would not be Italian; nevertheless, if Bizet and Verdi were compatriots,
Bizet either would or would not be Italian." (Lewis, 1973a, p. 80)

Front: the limit assumption
Back: Lewis objects to Stalnaker’s limit assumption.
Lewis says there can, in fact, be no closest world. For example: the displayed line exactly one inch long. What is the closest world where the line is longer than an inch? There isn't one. For any world where the line is 1+k inches long, there is a closer one where the line is 1+(k/2) inches long.
Furthermore, the limit assumption is crucial to Stalnaker's system for some propositions. If we did allow a model where limit fails, but kept the Stalnaker truth-conditions, the following counterfactual:
If the line had been longer than one inch, it would have been one hundred miles long.
would come out true, for there is no closest world in which the line is more than one inch long.
Stalnaker's reply: in any practical context, there is a closest world. e.g. there is a closest world where this pen is longer than it is, because of constraints on human pen manufacturing.

Front: LC-model
Back: LC-models are defined like SC-models, except that

Front: consistency and maximality

Front: strongly connected
Back: Sider: R is strongly connected in X iff for all a and b in X, Rab or Rba.
Terminological note: most people call this property that of being total.

Front: injection

Front: surjection

Front: LC-valuation
Back: An LC-valuation is defined like an SC-valuation, except that

Front: argument for S5 as the correct logic of metaphysical necessity
Back: The basic intuition:
* If, in some fully speciﬁed state of aﬀairs, a proposition P is true, then it should be the case, not in any particular world but in general, that it’s possible that P. The possibility of P is a feature not of any particular world, but of the set of all possible worlds, taken as a whole.
* Accessibility is irrelevant; what matters is that worlds are possible at all. As Lewis says, ”Accessible or not, they're still worlds. We still believe in them. Why don’t they count?”
S5 matches this intuition:
* By making accessibility an equivalence relation, the Kripke semantics collapse into reasoning about all possible worlds simulateneously
* S5 does not introduce any apparently arbitrary restrictions on metaphysical modality
* S5 is the only logic that succeeds at capturing this notion

Front: Salmon's arguments against S4 and S5
Back: Salmon attacks the S4 schema.
Salmon argues that metaphysical necessity is not, in fact, unrestricted. In particular, it is not transitive, hence the rejection of the S4 schema.
Salmon's slogan: "Metaphysical modality should fundamentally concern what might have been, which is separate from what are the ways in which things can be".
Salmon’s argument rests on a counterexample to the (S4) schema that exploits a gradual property. (In this sense it is a Sorites-like case).
* It's not the case that I could possibly have been a fish. It's an essential property of mine that I am a non-fish. But we can imagine a long string of intermediaries between me and a fish, each only slightly different from the next. In fact, since fish and humans have a common ancestor, such a line of intermediaries must exist. But it does seem metaphysically possible for each intermediary to have been slightly different, and in fact identical to the next one in line. Hence metaphysical possibility is not transitive.

Front: Resolving the Salmon-Lewis debate
Back: For instance, it is unclear whether a counterexample to S4 can be provided without making use of intuitions about gradual diﬀerence; if not, then perhaps the failure lies not with S4 and S5 as logics for describing modality, but in our concepts as ways of describing the world in the ﬁrst place.
It seems part of the disagreement stems at least from a conﬂict of purpose: One could design a logic to capture some absolute metaphysical truth, or a logic that could be fruitfully applied to our existing language and intuitions. Plausibly, the two projects are incompatible – they seek to capture diﬀerent notions of necessity.
As such, there should be no surprise in the fact that diﬀerent logics are suited to diﬀerent projects – that S4 and S5 may be fallacious for Salmon’s purposes, but not for others.

Front: Barcan formula

Front: Barcan formula debate
Back: 1. Usual objection to BF
2. Williamson's reply
3. The rejoinder from Hayaki

Front: Barcan formula debate 1
Back: 1. Usual objection to BF

Front: Barcan formula debate 2
Back: 2. Williamson's reply
This reply has some big costs in intuitiveness:
* a radically bloated ontology. The number of possible objects is vastly larger than the number of concrete objects in any given world. This means that the overwhelming majority of objects in any given world are not ‘real’ objects, but merely possible objects. It is not the ‘ghosts’ that are extraordinary, but the concrete things!
* It is not clear, at least not without extra argumentation, that one can coherently speak of an object in this world that is not concrete, but that this very same object in a diﬀerent world is. Is Wittgenstein’s ‘possible son’, this ghost inhabiting our world, really the same object as Wittgenstein’s son in another world? That must be possible for BF to be valid.

Front: Barcan formula debate 3
Back: 3. The rejoinder from Hayaki