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In this paper Bruce Aune states the following

Taylor would probably want to maintain that (p ≺ q) ⊃ — (p → q), where "→" represents physical, or natural, implication

So I'm not clear what "physical, or natural, implication" means. Nor do I know what "—" means though I suspect it is the logical negation and I also do not know what ≺ means though I suspect it has something to do with causation.

The original paper is here: https://www.jstor.org/stable/2183464?seq=1#page_scan_tab_contents

  • Impossible to understand without further context: we have three "implicational" symbols in place: ≺, ⊃, →: too many. One is the "material conditional" (the truth functional one): I think . The "natural" one : must be some sort of non-truth-functional rendering of the natural language use of "if..., then...". – Mauro ALLEGRANZA Mar 16 '18 at 8:44
  • Maybe is a sort of causal impliocation or strict conditional, often symbolized with a sort of harpoon. – Mauro ALLEGRANZA Mar 16 '18 at 8:54
  • The long dash is indeed negation (it's just a result of whatever typesetting was used because they chose to use "-" which reads as a dash). The paper outlines (or at least attempts to outline) a difference between logical implication and physical implication and it's relation to physical necessity. Mauro is correct that ⊃ is the truth functional material conditional. The ram's head (it's curlier in the paper than in html) implication is logical implication and the arrow is physical implication, that much is explained in the sentence that comes after the one you've quoted. – Not_Here Mar 16 '18 at 11:32
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    Correct: A ⥽ B was introduced by C.I.Lewis as an abbreviation for ☐(A⊃B). – Mauro ALLEGRANZA Mar 16 '18 at 12:11
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    @Not_Here - see Graham Priest, An Introduction to Non-Classical Logic, Cambridge UP (2nd ed 2008), page 72. – Mauro ALLEGRANZA Mar 16 '18 at 12:30
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The hyphen does indeed mean negation (for this answer I'll use ~). The ending of the paragraph that you've quoted gives the precise meanings of the symbols you are asking about. Granted, I do not understand why Aune didn't introduce the symbols first (they are not found in Taylor's paper, but maybe they were common in literature at that time).

The ram's head implication, ≺, stands for logical implication. The superset implication, ⊃, stands for the regular truth functional material implication. The arrow implication, →, as is stated, represents physical implication.

Here are the relevant parts of the paragraph that explain this:

Taylor would probably want to maintain that (7) (p ≺ q) ⊃ ~(p → q), where "→" represents physical, or natural, implication.

Yet when one considers that no one would want to maintain that a logically impossible state of affairs could still be physically possible, the falsity of (7) becomes apparent at once.

Here Aune is saying that (7) cannot be true. Parsing what he said, he is saying that no one would want it to be the case where ~(p ≺ q) ^ (p → q), (p does not logically entail q AND p physically entails q), under this symbolism, is true. Since (False ⊃ True) results in True, it would be possible for ~(p ≺ q) ^ (p → q) to be the case, if (p ≺ q) ⊃ ~(p → q) is granted.

For beginning with the premise that physical possibility entails logical possibility, that is, beginning with (8) "PM(p) ⊃ LM(p)" (where "M" represents "possibility"), we may infer (9) " ~ LM(p) ⊃ ~ PM(p)."

This is important because Aune has just shown that he is now using "PM" to represent physically possible, and "LM" to represent "logically possible. The lines that show that his weird use of symbols in the beginning are exactly the same as these lettered abbreviations are:

we may obtain from (9) the equivalent (10) "LN(~p) ⊃ PN(~P)."

Since (10) holds for all values of "p," we may substitute " (p. ~ q) " for "p" and then infer (11) "LN(p ⊃ q) ⊃ PN(p ⊃ q).

But because (11) may also be written as (12) "p ≺ q :⊃: p → q," it is clear that we are committed to deny that logical necessity never entails physical necessity.

Notice that he states (11) can be rewritten as (12), and I honestly have absolutely no idea why he randomly jumps to using Principia colon punctuation instead of parenthetical, but (12) is identical to (7) (given that the delineating punctuation doesn't have some secret meaning that eludes me) he is saying that (11) is also identical to (7), and reading (11) out loud you get the statement that ≺ is logical implication LM, and → is physical implication PM.

Thus, given that "A ≺ A" is true, we must also accept "A → A" as true; and this contradicts the result we get from the assumption, held by many philosophers today, that logical necessity never implies physical necessity.

This answer was very hard to do without LaTex and I'm sure it is hard to parse, (I am also very tired as I write this and might have made a mistake, so I am thankful to any edits that can fix them if they exist), but hopefully I can now give a summary that explains what is going on.

As stated, I don't know why he did not introduce the symbols before giving a derivation of what they mean, but he uses ≺ to mean a logical implication, ⊃ in a regular material implication way (notice how (11) is written with ⊃), and → as physical implication. ≺ and → are both material implications, but they are about different things (this is why (11) could be written with both). The relevant subject matter of the paper, and the wider literature about fatalism, is engrossed in understanding alethic modality so it makes sense to try to distinguish what things are logically modal and what things are physically modal. I don't know why Aune introduced the symbols the way he did, but the explanation of what they mean is what the rest of that paragraph discusses.

One last caveat, again alluding to the fact that I am very tired, I might have misunderstood some of the arguments that were being presented (the more that I think about the switch to Principia punctuation, the more I second guess myself). This being said, I am absolutely confident that the last move from (11) to (12) confirms what I have explained the symbols to mean.

  • Thank you for your answer. However, I'm still not entirely sure what "physical implication" means.... though I think it might mean something roughly like "Although it's not a contradiction of logic that my hand doesn't suddenly turn into a ball of iron, it is a contradiction of laws of nature.... namely, conservation laws and so on. Likewise it's not a logical contradiction that I suddenly flap my arms and fly, but it is a physical contradiction." I assume if this capture "physical contradiction" then "physical implication" follows in an analogous manner.... but it's not entirely clear to me – Squirtle Mar 16 '18 at 18:56

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