Philosophy Lexicon of Arguments

Barcan formula: claims that from the fact that it is possible that an object has a certain property it follows that this object exists. The formula is valid only in a few systems. See also modal logic.

Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments.
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The Barcan formula (x) Na > N(x)a

Barcan formula/BF/Bigelow/Pargetter: VsBarcan: one could argue that the intended interpretations of "necessary" falsifies the barcan formula.
E.g. "N" be it logically necessary that "and suppose some of the kinds of atheism are true, according to which everything must be localized spatial-temporally. Then we have
(x) (x is spatial)
But one could add that a given spatial thing - e.g. a screwdriver - logically impossible could be non-spatial.
To put it paradoxically, if this screwdriver had been non-spatial, it would not have been that screwdriver.

N (if the screwdriver ever exists, it is spatial)
In general
(x) N (if x exists, x is spatial)
This has the form
(x) Na.
I 110
Barcan-formula/Bigelow/Pargetter: notes that
((x) NA > N (x)a).
That is, if the atheist accepts the Barcan-formula (together with the modus ponens) he is obliged to
N (x) a
That is,
N (Everything is so that if it exists, it is spatial)
Problem/VsBarcan/Bigelow/Pargetter. Many atheists would deny this. For the Barcan-formula would fix them on logical impossibility, although they proceed from a contingent fallacy.
Barcan-formula/(s): fixes the atheist to the conclusion that God is logically impossible, even if he proceeds from a contingent fact.
Barcan-formula/BF/Bigelow/Pargetter: we nevertheless plead for an acceptance of the Barcan-formula for modal realism, if it assumes the strictest interpretation of necessity.
But the reason arises only from semantics, not from logic.
N.B.: if we set up the semantics for a rejection of the Barcan-formula, we notice that we have to assume the Barcan-formula for this.
((s) question: does this not apply to any assertion of an impossibility of a thing?)
Modal Realism/Barcan Formula/BF/Bigelow/Pargetter: modal realism must therefore deny that it is contingent, what things there are. It is merely contingent on what things there are in the actual world, because it is contingent, which world is the actual world.
Possibilia: since the modal realism accepts Possibilia, it must say "there is a God or God could exist," but which is then, for him, equal with "God exists". And this already from the logical possibility! Because of its own interpretation of "there is".
"There is"/Interpretation/Bigelow/Pargetter/(s): can be interpreted differently: for modal realism it means, what is possible, exists also.
Barcan-Formula/BF/Bigelow/Pargetter: is an axiom that connects modal operators and quantifiers.
Similarly, Hughes/Cresswell's principle of predication:
Principle of Predication/Hughes/Cresswell/Bigelow/Pargetter: HC 1968, S 184-8):
(x) (Na v N~a) v (x) (Ma u M~a).
Everyday language translation/(s): all things have their properties either necessarily or possibly.
Bigelow/Pargetter: that divides all the properties (or conditions) that an object has to fulfill) in two kinds:
A) essences
B) accidents.
Principle of Predication/Hughes/Cresswell/Bigelow/Pargetter: is there to exclude properties that a thing could have essentially, but other things accidentally.
BigelowVsHughes/Cresswell/BigelowVsCresswell: you should not exclude such features! E.g. the property to be awake in the first hour of the year 1600 is accidental for Descartes, but essentially impossible for other objects.

Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.

Big I
J. Bigelow, R. Pargetter
Science and Necessity Cambridge 1990

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Ed. Martin Schulz, access date 2017-06-27