Setting the bounds Dave Ripley University of Connecticut - - PowerPoint PPT Presentation

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Apr 23, 2023 469 likes •943 views

1/ 44 Setting the bounds Dave Ripley University of Connecticut http://davewripley.rocks Victoria University of Wellington March 2015 davewripley@gmail.com Setting the bounds 2/ 44 Historical prelude From bounds to meaning Rumfitts

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Setting the bounds

Dave Ripley

University of Connecticut http://davewripley.rocks

Victoria University of Wellington March 2015

davewripley@gmail.com Setting the bounds

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Historical prelude From bounds to meaning Rumfitt’s objection

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Historical prelude Gentzen’s sequents 3/ 44

Historical prelude

Gentzen’s sequents

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Historical prelude Gentzen’s sequents 4/ 44

Gentzen’s dissertation was a landmark for proof theory Key notions introduced: natural deduction and sequent calculus Sequent calculus for classical logic worked on things of the form Γ ⇒ ∆, where Γ and ∆ are finite lists of formulas.

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Historical prelude Gentzen’s sequents 5/ 44

Gentzen: “The sequent A1, . . . , An ⇒ B1, . . . , Bm has the same meaning as the formula (A1 ∧ . . . ∧ An) ⊃ (B1 ∨ . . . ∨ Bn).’ By organizing his calculus in this way, Gentzen was able to do lots of nice things.

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Historical prelude Gentzen’s sequents 6/ 44

Gentzen seemed to think this was all a technical trick. But what if it’s more than that?

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From bounds to meaning Multiple conclusions 7/ 44

From bounds to meaning

Multiple conclusions

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From bounds to meaning Multiple conclusions 8/ 44

A sequent Γ ⇒ ∆ can be seen as representing an argument with premises Γ and conclusions ∆. This can take a bit of practice; recall that the conclusions are disjunctively combined.

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From bounds to meaning Multiple conclusions 9/ 44

When is such an argument valid? The key idea here is from Restall: Restall (2005, 2008, 2009, 2013): An argument is valid iff: asserting all its premises and denying all its conclusions clashes. Other phrasing: ‘out of bounds’, ‘incoherent’, ‘self-defeating’.

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From bounds to meaning Multiple conclusions 9/ 44

When is such an argument valid? The key idea here is from Restall: Restall (2005, 2008, 2009, 2013): An argument is valid iff: asserting all its premises and denying all its conclusions clashes. Other phrasing: ‘out of bounds’, ‘incoherent’, ‘self-defeating’.

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From bounds to meaning Multiple conclusions 10/ 44

Example: Asserting and denying the same thing is out of bounds. So A ⊢ A. Example: Asserting ‘Auckland is bigger than Wellington’ and ‘Wellington is bigger than Palmy’ while denying ‘Auckland is bigger than Palmy’ is out of bounds. So A >> W, W >> P ⊢ A >> P.

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From bounds to meaning Multiple conclusions 11/ 44

A position is a collection of assertions and denials. It is positions that are in or out of bounds. Γ ⇒ ∆ is valid iff the position that asserts the Γs and denies the ∆s is out of bounds.

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From bounds to meaning Multiple conclusions 12/ 44

This gives a way to understand Gentzen’s (and others’) sequent rules: Some example rules Γ ⇒ ∆

KL:

A, Γ ⇒ ∆ Γ ⇒ ∆

KR:

Γ ⇒ ∆, A A/B, Γ ⇒ ∆

∧L:

A ∧ B, Γ ⇒ ∆ Γ ⇒ ∆, A Γ ⇒ ∆, B

∧R:

Γ ⇒ ∆, A ∧ B A, Γ ⇒ ∆

TL:

TA, Γ ⇒ ∆ Γ ⇒ ∆, A

TR:

Γ ⇒ ∆, TA Rules 3–6 give the meanings of ∧, T.

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From bounds to meaning Multiple conclusions 13/ 44

All that’s well and good (let’s suppose). But what’s a clash in the first place?

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From bounds to meaning Vocabulary-independent 14/ 44

From bounds to meaning

Vocabulary-independent

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From bounds to meaning Vocabulary-independent 15/ 44

Recall Gentzen: “The sequent A1, . . . , An ⇒ B1, . . . , Bm has the same meaning as the formula (A1 ∧ . . . ∧ An) ⊃ (B1 ∨ . . . ∨ Bn).’ We don’t want this kind of approach to clashes. Adding vocabulary to the sequent —∧, ∨, ⊃, whatever— is a bad idea, for three reasons.

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From bounds to meaning Vocabulary-independent 16/ 44

First: There are perfectly sensible applications of this approach to languages that lack ∧, ∨, ⊃, etc—eg English. Possible response: see such languages as fragments of fuller languages that do contain the needed vocabulary? But that’s not always possible.

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From bounds to meaning Vocabulary-independent 17/ 44

Second: Infinite collections of premises or conclusions don’t require infinitary connectives. Again, seeing these as a fragment of fuller languages is not always possible.

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From bounds to meaning Vocabulary-independent 18/ 44

Third: The sequent rules, interpreted via clashes, explain the meanings of ∧, ∨, ⊃, etc. Dragging their meanings into the interpretation would give an explanatory circle.

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From bounds to meaning Possible truth? 19/ 44

From bounds to meaning

Possible truth?

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From bounds to meaning Possible truth? 20/ 44

Another nonstarter: A sequent Γ ⇒ ∆ clashes when it’s impossible for all the Γs to be true while all the ∆s are false.

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From bounds to meaning Possible truth? 21/ 44

Trouble: Whether it’s possible for all the Γs to be true while the ∆s are false depends on what the Γs and ∆s mean. So this would again result in an explanatory circle.

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From bounds to meaning What sets the bounds? 22/ 44

From bounds to meaning

What sets the bounds?

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From bounds to meaning What sets the bounds? 23/ 44

Where do the bounds come from, then? Not from implication, not from possible truth-and-falsity, so . . .

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From bounds to meaning What sets the bounds? 24/ 44

The bounds are a social kind: they are created and sustained by the place they occupy in our social practices.

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From bounds to meaning What sets the bounds? 25/ 44

Some norms on assertion and denial: — Assert only what’s true — Deny only what’s false — Assert only what you have warrant for — Deny only what you have warrant against — Assert or deny only what’s relevant . . .

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From bounds to meaning What sets the bounds? 26/ 44

Norms involving the bounds: — Don’t adopt a position that’s out of bounds — What’s out of bounds is discountable Discountable: it’s ok to ‘leave one’s flank open’ to risks from this angle.

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From bounds to meaning What sets the bounds? 27/ 44

Three characteristic responses to clashes:

Reinterpretation
Clarification
Dismissal

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Example clash: Someone asserts both: ‘Napoleon died in 1821’ ‘Napoleon organized a coup in 1851’ In a context where we’re not taking zombie Napoleon seriously, this is probably a clash.

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From bounds to meaning What sets the bounds? 29/ 44

Three responses: Reinterpret: ‘They must be talking about two different Napoleons’ Clarify: ‘But I thought you said he had died in 1821?’ Dismiss: ‘This asshole is just talking nonsense’ These responses are not exclusive, and they shade into each other. They all mark a standing back from what’s been claimed.

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Discountability: Given that Napoleon died in 1821, there’s no risk that he led a coup in 1851. Nor is there any risk he’ll lead a coup tomorrow.

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Discountability works in ‘what-ifs’ too: What if Palmy were bigger than Auckland? Then there’d be no risk of it being smaller than Wellington.

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From bounds to meaning What sets the bounds? 32/ 44

Words have the meanings we give them; we give meaning by treating things as clashing or not. (Reinterpreting, clarifying, dismissing, discounting)

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From bounds to meaning What sets the bounds? 33/ 44

This is a stance approach to the bounds: the crucial notion is treating something as a clash. A natural way to think about social kinds.

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Rumfitt’s objection Moore’s paradox 34/ 44

Rumfitt’s objection

Moore’s paradox

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Rumfitt’s objection Moore’s paradox 35/ 44

Rumfitt (2008) offers two related objections. One is patterned after Moore’s paradox: Rumfitt: “A thinker who accepts that it is raining but denies that he accepts that it is raining will be making a mistake as to the facts. But the statement ‘He accepts that it is raining’ is not a consequence of the statement ‘It is raining’.. . . This sort of case is a problem for those who seek to explicate consequence in normative terms.”

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Rumfitt’s objection Moore’s paradox 36/ 44

Suppose someone asserts ‘It’s raining’ and denies ‘I’ve asserted that it’s raining’. They’ve surely done something wrong.

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Rumfitt’s objection Lack of grounds 37/ 44

Rumfitt’s objection

Lack of grounds

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Rumfitt’s objection Lack of grounds 38/ 44

The second is similar: Rumfitt: “The mental state that consists of accepting that there will never be grounds for accepting or rejecting ‘There is a god’, while rejecting that very statement, is self-defeating. But ‘There is a god’ is in no sense a consequence of ‘There will never be sufficient grounds for accepting or rejecting “There is a god”’.”

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Rumfitt’s objection Lack of grounds 39/ 44

Suppose someone asserts ‘There will never be sufficient grounds for asserting or denying that there is a god’ and denies ‘There is a god’. Again, they’ve surely done something wrong.

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Rumfitt’s objection Answering Rumfitt 40/ 44

Rumfitt’s objection

Answering Rumfitt

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Rumfitt’s objection Answering Rumfitt 41/ 44

Both cases violate norms. But neither violates the bounds. Moore paradox: violates ‘assert only what’s true’ Groundlessness: violates ‘assert only what’s warranted’

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Rumfitt’s objection Answering Rumfitt 42/ 44

Discountability shows the difference: Moore: Can’t discount that it’s raining but that I didn’t assert it. (This happens all the time!) Grounds: Can’t discount that there’s no god and no grounds for this. (This may well be the case!)

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Rumfitt’s objection Answering Rumfitt 43/ 44

Clashes are a particular social kind. There is a norm: ‘don’t adopt a clashing position’ This doesn’t mean anything that violates a norm is a clash.

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Rumfitt’s objection Answering Rumfitt 44/ 44

Multiple-conclusion consequence can be understood in terms of

clashes.

If clashes are understood the right way, this gives foundations for

a theory of both consequence and meaning.

Clashes are a social kind.
Two key norms: don’t clash, and discount clashes.

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