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What is an inconsistent truth table? Zach Weber (University of - - PowerPoint PPT Presentation
What is an inconsistent truth table? Zach Weber (University of - - PowerPoint PPT Presentation
What is an inconsistent truth table? Zach Weber (University of Otago) NCM Prague - June 2015 Joint work with G Badia (Otago) and P Girard (Auckland) Introduction: Non-classical logic, top to bottom Elements of a (paraconsistent) metatheory
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Q: Can standard reasoning about logic be carried out without any appeal to classical logic?
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Q: Can standard reasoning about logic be carried out without any appeal to classical logic? A: Yes. The semantics of propositional logic can be given paraconsistently, with soundness and completeness theorems (as well as their negations).
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Q: Can standard reasoning about logic be carried out without any appeal to classical logic? A: Yes. The semantics of propositional logic can be given paraconsistently, with soundness and completeness theorems (as well as their negations). This is evidence for a more general claim: Metatheory determines object theory. When we write down the orthodox clauses for a logic, whatever logic we presuppose in the background will be the object-level logic that obtains.
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There are many non-classical logics
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There are many non-classical logics —some argued for as the correct logic.
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There are many non-classical logics —some argued for as the correct logic. But the syntax and semantics of paraconsistent and paracomplete logics—their grammar and truth tables—are always taken to be ‘classically behaved’, from Kripke 1974 to Field 2008.
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There are many non-classical logics —some argued for as the correct logic. But the syntax and semantics of paraconsistent and paracomplete logics—their grammar and truth tables—are always taken to be ‘classically behaved’, from Kripke 1974 to Field 2008. When talking about a logic, must we be working in a classical metatheory?
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How far can a logician who professes to hold that [paraconsistency] is the correct criterion of a valid argument, but who freely accepts and offers standard mathematical proofs, in particular for theorems about [paraconsistent] logic itself, be regarded as sincere or serious in objecting to classical logic? [Burgess]
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How far can a logician who professes to hold that [paraconsistency] is the correct criterion of a valid argument, but who freely accepts and offers standard mathematical proofs, in particular for theorems about [paraconsistent] logic itself, be regarded as sincere or serious in objecting to classical logic? [Burgess] Maybe “preaching to the gentiles in their own tongue” (Meyer)?
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How far can a logician who professes to hold that [paraconsistency] is the correct criterion of a valid argument, but who freely accepts and offers standard mathematical proofs, in particular for theorems about [paraconsistent] logic itself, be regarded as sincere or serious in objecting to classical logic? [Burgess] Maybe “preaching to the gentiles in their own tongue” (Meyer)? Okay ... then what is the plan for once everyone is converted to the One True (paraconsistent) Logic?
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Armchair pop-psychology claim: Classical-fallback is simply pragmatic.
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Armchair pop-psychology claim: Classical-fallback is simply pragmatic. No one really knows what e.g. a fully paraconsistently constructed truth table looks like.
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Armchair pop-psychology claim: Classical-fallback is simply pragmatic. No one really knows what e.g. a fully paraconsistently constructed truth table looks like. And so the main reason for this paper is pragmatic, too—just to show the answer.
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Logic implies logic
The work in this paper is conducted against a background inconsistent set theory.
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Logic implies logic
The work in this paper is conducted against a background inconsistent set theory. Classically, the (boolean) logic of sets generates a (boolean) semantics of logic.
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Logic implies logic
The work in this paper is conducted against a background inconsistent set theory. Classically, the (boolean) logic of sets generates a (boolean) semantics of logic. Here, a paraconsistent set theory naturally generates a paraconsistent semantics.
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Logic (Propositional Fragment)
Axioms $ ϕ Ñ ϕ $ pϕ Ñ ψq ^ pψ Ñ χq Ñ pϕ Ñ χq $ ϕ _ ϕ $ ϕ Ñ ϕ $ pϕ Ñ ψq Ñ pψ Ñ ϕq $ ϕ ^ ψ Ñ ϕ $ ϕ ^ ψ Ñ ψ ^ ϕ $ ϕ _ ψ Ø pϕ ^ ψq $ ϕ ^ pψ _ χq Ø pϕ ^ ψq _ pϕ ^ χq $ pϕ Ñ ψq ñ pϕ ñ ψq $ pϕ ñ ψq ñ pϕ Ñ ψq $ pϕ ñ ψq ^ pχ ñ ψq ñ pϕ _ χ ñ ψq $ x “ y ñ pϕpxq Ñ ϕpyqq
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Rules
ϕ, ϕ ñ ψ $ ψ ϕ, ψ $ pϕ ñ ψq Γ, ϕ $ ψ “ “ “ “ “ “ “ “ Γ $ ϕ ñ ψ Γ, ϕ, χ $ ψ “ “ “ “ “ “ “ “ Γ, χ, ϕ $ ψ Γ, ϕ, χ $ ψ Γ, ϕ ^ χ $ ψ Γ $ ψ ∆ $ ϕ Γ, ∆ $ ϕ ^ ψ Γ $ ψ Γ, ϕ $ ψ Γ $ ϕ ∆, ϕ $ ψ Γ, ∆ $ ψ
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Axiom (Ext)
@zppz P x Ø z P yq Ø x “ y
Axiom (Abs)
x P tz : ϕu Ø ϕx
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Axiom (Ext)
@zppz P x Ø z P yq Ø x “ y
Axiom (Abs)
x P tz : ϕu Ø ϕx Special case: xx, yy P tz : ϕu Ø ϕxx, yy
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Axiom (Ext)
@zppz P x Ø z P yq Ø x “ y
Axiom (Abs)
x P tz : ϕu Ø ϕx Special case: xx, yy P tz : ϕu Ø ϕxx, yy
Axiom (Choice)
A unique object can be picked out from any non-empty set.
Axiom (Induction)
Proofs by induction work for any recursively defined structure.
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Two relations are added: syntactic validity $ and semantic consequence (.
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Two relations are added: syntactic validity $ and semantic consequence (. For $, the inductive definition, supported by axiom 4, is
Definition
With Γ a set of premises, Γ $ ϕ iff ϕ follows from some subset of Γ by valid rules.
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Two relations are added: syntactic validity $ and semantic consequence (. For $, the inductive definition, supported by axiom 4, is
Definition
With Γ a set of premises, Γ $ ϕ iff ϕ follows from some subset of Γ by valid rules. The set of theorems, $ ϕ, is made up either of axioms deducible from no premises, or deducible from the axioms via the operational
- r structural rules.
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Two relations are added: syntactic validity $ and semantic consequence (. For $, the inductive definition, supported by axiom 4, is
Definition
With Γ a set of premises, Γ $ ϕ iff ϕ follows from some subset of Γ by valid rules. The set of theorems, $ ϕ, is made up either of axioms deducible from no premises, or deducible from the axioms via the operational
- r structural rules.
If this sounds (comfortingly? suspiciously?) familiar, this is prelude for what is to come.
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True vs true only: two values or three?
Tarski’s theorem: An exclusive and exhaustive partitioning of all the propositions into all-and-only the truths, versus all-and-only the non-truths, is impossible.
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True vs true only: two values or three?
Tarski’s theorem: An exclusive and exhaustive partitioning of all the propositions into all-and-only the truths, versus all-and-only the non-truths, is impossible. It would have been nice, but c’est la vie.
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True vs true only: two values or three?
Tarski’s theorem: An exclusive and exhaustive partitioning of all the propositions into all-and-only the truths, versus all-and-only the non-truths, is impossible. It would have been nice, but c’est la vie. incomplete strategy accept ‘only the truths’, leave some out
- vercomplete strategy
accept ‘all the truths’, keep some untruths in
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True vs true only: two values or three?
Tarski’s theorem: An exclusive and exhaustive partitioning of all the propositions into all-and-only the truths, versus all-and-only the non-truths, is impossible. It would have been nice, but c’est la vie. incomplete strategy accept ‘only the truths’, leave some out
- vercomplete strategy
accept ‘all the truths’, keep some untruths in Choose: untruth-avoidance or truth-seeking.
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Standard presentations of dialetheic paraconsistent logic are via a three valued functional semantics, tt, f, bu
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Standard presentations of dialetheic paraconsistent logic are via a three valued functional semantics, tt, f, bu Makes it appear that there is indeed an exclusive and exhaustive partitioning of the universe of truths,
§ all-and-only truths § all-and-only untruths § all-and-only ‘both’s
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Standard presentations of dialetheic paraconsistent logic are via a three valued functional semantics, tt, f, bu Makes it appear that there is indeed an exclusive and exhaustive partitioning of the universe of truths,
§ all-and-only truths § all-and-only untruths § all-and-only ‘both’s
If the original Tarski problem was insoluble, this new, three-tiered approach will be no less intractable.
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The three-valued approach rather encourages a common criticism —that dialetheists have lost some important expressive power, the ability to demarcate the truths (t valued) from the true contradictions (b valued).
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The three-valued approach rather encourages a common criticism —that dialetheists have lost some important expressive power, the ability to demarcate the truths (t valued) from the true contradictions (b valued). “Surely this distinction is available—there it is in your semantics!—but the object language cannot express it.”
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The three-valued approach rather encourages a common criticism —that dialetheists have lost some important expressive power, the ability to demarcate the truths (t valued) from the true contradictions (b valued). “Surely this distinction is available—there it is in your semantics!—but the object language cannot express it.” Indeed ... if not for the original problem:
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The three-valued approach rather encourages a common criticism —that dialetheists have lost some important expressive power, the ability to demarcate the truths (t valued) from the true contradictions (b valued). “Surely this distinction is available—there it is in your semantics!—but the object language cannot express it.” Indeed ... if not for the original problem: no one can in fact make this demarcation.
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The three-valued approach rather encourages a common criticism —that dialetheists have lost some important expressive power, the ability to demarcate the truths (t valued) from the true contradictions (b valued). “Surely this distinction is available—there it is in your semantics!—but the object language cannot express it.” Indeed ... if not for the original problem: no one can in fact make this demarcation. A dialetheic paraconsistentist should lead the discussion away from pre-Tarskian ideation, and use a formalism that does not invite or suggest such criticism.
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The presentation here is entirely in a two-valued relational semantics.
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The presentation here is entirely in a two-valued relational semantics. Relations can approximate the otherwise-desirable functional three valued semantics;
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The presentation here is entirely in a two-valued relational semantics. Relations can approximate the otherwise-desirable functional three valued semantics; To reiterate, this is not really a decision on our part,
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The presentation here is entirely in a two-valued relational semantics. Relations can approximate the otherwise-desirable functional three valued semantics; To reiterate, this is not really a decision on our part, but rather a requirement for any logic that can express its own metatheory.
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There are two truth values, t and f, which are duals, t “ f t “ t They are also exclusive, on pain of absurdity: t “ f ñ ϕ for any ϕ.
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Relational Truth Conditions
A truth-value assignment on PROP is any relation R0 Ď PROP ˆ tt, fu such that x P PROP ô Dypxx, yy P R0q, and xp, ty P R0 ô xp, fy R R0 xp, fy P R0 ô xp, ty R R0
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Relational Truth Conditions
A truth-value assignment on PROP is any relation R0 Ď PROP ˆ tt, fu such that x P PROP ô Dypxx, yy P R0q, and xp, ty P R0 ô xp, fy R R0 xp, fy P R0 ô xp, ty R R0 By the law of excluded middle, R0 is not empty: either xp, fy P R0, or else xp, fy R R0, in which case xp, ty P R0.
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Definition of a model
Extend R0 to R Ď FMLA ˆ tt, fu: ϕRt ô ϕRf ϕRf ô ϕRt pϕ ^ ψqRt ô ϕRt ^ ψRt pϕ ^ ψqRf ô ϕRf _ ψRf pϕ _ ψqRt ô ϕRt _ ψRt pϕ _ ψqRf ô ϕRf ^ ψRf
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Definition of a model
Extend R0 to R Ď FMLA ˆ tt, fu: ϕRt ô ϕRf ϕRf ô ϕRt pϕ ^ ψqRt ô ϕRt ^ ψRt pϕ ^ ψqRf ô ϕRf _ ψRf pϕ _ ψqRt ô ϕRt _ ψRt pϕ _ ψqRf ô ϕRf ^ ψRf Satisfies ϕRt ô pϕRfq and ϕRf ô pϕRtq
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R is called a model for extensional propositional logic.
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R is called a model for extensional propositional logic. R satisfies formula ϕ, or SatpR, ϕq, iff xϕ, ty P R.
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R is called a model for extensional propositional logic. R satisfies formula ϕ, or SatpR, ϕq, iff xϕ, ty P R.
Example
If both xϕ, ty, xϕ, fy P R, then SatpR, ϕq and SatpR, ϕq simultaneously,
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R is called a model for extensional propositional logic. R satisfies formula ϕ, or SatpR, ϕq, iff xϕ, ty P R.
Example
If both xϕ, ty, xϕ, fy P R, then SatpR, ϕq and SatpR, ϕq simultaneously, i.e. ϕ is both satisfied and not in the model R. This will be the situation with any contradiction.
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Definition
A sentence ψ is a valid consequence of ϕ0, ..., ϕn, ϕ0, ..., ϕn ( ψ iff ϕ0Rt ^ ... ^ ϕnRt ñ ψRt for all models R.
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Definition
A sentence ψ is a valid consequence of ϕ0, ..., ϕn, ϕ0, ..., ϕn ( ψ iff ϕ0Rt ^ ... ^ ϕnRt ñ ψRt for all models R. A sentence ϕ is a tautology, ( ϕ, iff ϕRt for all R.
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Definition
A sentence ψ is a valid consequence of ϕ0, ..., ϕn, ϕ0, ..., ϕn ( ψ iff ϕ0Rt ^ ... ^ ϕnRt ñ ψRt for all models R. A sentence ϕ is a tautology, ( ϕ, iff ϕRt for all R. This is as usual.
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Theorem
Any truth-value assignment R0 on PROP can be extended to a model R for propositional logic.
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Theorem
Any truth-value assignment R0 on PROP can be extended to a model R for propositional logic. Let R0 Ď PROP ˆ tt, fu be an assignment on propositional
- variables. This means that
xp, ty P R0 ô xp, fy R R0 xp, fy P R0 ô xp, ty R R0
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Theorem
Any truth-value assignment R0 on PROP can be extended to a model R for propositional logic. Let R0 Ď PROP ˆ tt, fu be an assignment on propositional
- variables. This means that
xp, ty P R0 ô xp, fy R R0 xp, fy P R0 ô xp, ty R R0 One exists: let R0 “ txp, ty, xp, fyu.
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Extend R0 with the lift R (which exists by comprehension): R “ txp, ty : pR0tu Y txp, fy : pR0fu Y txϕ, ty : xϕ, fy P Ru Y txϕ, fy : xϕ, ty P Ru Y txϕ ^ ψ, ty : xϕ, ty P R ^ xψ, ty P Ru Y txϕ ^ ψ, fy : xϕ, fy P R _ xψ, fy P Ru Y txϕ _ ψ, ty : xϕ, ty P R _ xψ, ty P Ru Y txϕ _ ψ, fy : xϕ, fy P R ^ xψ, fy P Ru
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Proof that R is a model
Show by induction that for any formula (‹) xϕ, ty P R ô xϕ, fy R R xϕ, fy P R ô xϕ, ty R R For the base case, for any proposition p and x P tt, fu, by definition pRx ô pR0x and ppRxq ô ppR0xq. Induction: assume p‹q as the inductive hypothesis. To avoid contraction, we don’t use the very same hypothesis for each inductive case. They are rather hypothesis schemata, each instance used once.
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So what does an inconsistent truth table look like?
Semantics for extensional propositional logic can be displayed as usual.
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So what does an inconsistent truth table look like?
Semantics for extensional propositional logic can be displayed as usual. The answer to our titular question is bluntly simple:
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So what does an inconsistent truth table look like?
Semantics for extensional propositional logic can be displayed as usual. The answer to our titular question is bluntly simple:
- t
f f t ^ t f t t f f f f _ t f t t t f t f
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So what does an inconsistent truth table look like?
Semantics for extensional propositional logic can be displayed as usual. The answer to our titular question is bluntly simple:
- t
f f t ^ t f t t f f f f _ t f t t t f t f
§ Such two-dimensional displays are often implicitly assumed to
be functional look-up tables.
§ No such assumption on the page. § It is simply presupposing classicality to do so. § Diagrams must be used with great care in mathematics!
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- t
f f t The tables are read as, ‘if t is among the values of ϕ, then f is among the values of ϕ’. Or more concisely, ‘if ϕ is true, then ϕ is false’.
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- t
f f t The tables are read as, ‘if t is among the values of ϕ, then f is among the values of ϕ’. Or more concisely, ‘if ϕ is true, then ϕ is false’. Such a reading is perfectly acceptable here,
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- t
f f t The tables are read as, ‘if t is among the values of ϕ, then f is among the values of ϕ’. Or more concisely, ‘if ϕ is true, then ϕ is false’. Such a reading is perfectly acceptable here, provided that additional classical presuppositions are not being made.
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- t
f f t The tables are read as, ‘if t is among the values of ϕ, then f is among the values of ϕ’. Or more concisely, ‘if ϕ is true, then ϕ is false’. Such a reading is perfectly acceptable here, provided that additional classical presuppositions are not being made. The copula—the ‘is’ of predication—is not univocal in general, and it is not here.
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Soundness
Theorem
$ ϕ ñ ( ϕ. Also, there are ϕ such that $ ϕ and * ϕ.
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Soundness
Theorem
$ ϕ ñ ( ϕ. Also, there are ϕ such that $ ϕ and * ϕ.
Corollary
For some ϕ, it is the case that $ ϕ ñ ( ϕ and p$ ϕ ñ ( ϕq.
Corollary
p( ϕ ñ Kq ñ p$ ϕ ñ Kq.
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Completeness
Notation: given R, Θϕ
R “ tpi : xpi, ty P Ru Y tpi : xpi, fy P Ru
Θ contains as many copies of pi as there are in ϕ.
Lemma
For any model R and formula ϕ, 1. xϕ, ty P R ñ Θϕ
R $ ϕ
2. xϕ, fy P R ñ Θϕ
R $ ϕ
Proof: If xψ, ty P R, then xψ, fy P R, so Θ $ ψ. If xψ, fy P R, then xψ, fy P R, so Θ $ ψ, so Θ $ ψ. Etc. l
Theorem
( ϕ ñ $ ϕ
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Non-triviality
In general, a theory is non-trivial iff there is at least one sentence that is not part of the theory.
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Non-triviality
In general, a theory is non-trivial iff there is at least one sentence that is not part of the theory. Is there an internal demonstration of non-triviality?
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Non-triviality
In general, a theory is non-trivial iff there is at least one sentence that is not part of the theory. Is there an internal demonstration of non-triviality? Why bother?
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Non-triviality
In general, a theory is non-trivial iff there is at least one sentence that is not part of the theory. Is there an internal demonstration of non-triviality? Why bother? Theorem Naive set theory is not trivial.
Proof.
Either naive set theory is trivial or not. If not, we are done. If trivial, then, since this very proof is in naive set theory, it follows that the system is not trivial—since, after all, anything follows.
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Non-triviality
In general, a theory is non-trivial iff there is at least one sentence that is not part of the theory. Is there an internal demonstration of non-triviality? Why bother? Theorem Naive set theory is not trivial.
Proof.
Either naive set theory is trivial or not. If not, we are done. If trivial, then, since this very proof is in naive set theory, it follows that the system is not trivial—since, after all, anything follows. “You can trust me.”
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In classical mathematics, consistency (and so non-triviality) is not provable.
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In classical mathematics, consistency (and so non-triviality) is not provable. Using a paraconsistent meatheory constitutes no expressive loss. Rather it makes clearer and more explicit the hard facts.
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In classical mathematics, consistency (and so non-triviality) is not provable. Using a paraconsistent meatheory constitutes no expressive loss. Rather it makes clearer and more explicit the hard facts. Indeed, in a paraconsistent system, one can prove consistency and non-triviality.
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In classical mathematics, consistency (and so non-triviality) is not provable. Using a paraconsistent meatheory constitutes no expressive loss. Rather it makes clearer and more explicit the hard facts. Indeed, in a paraconsistent system, one can prove consistency and non-triviality. This is the closest one can get to a guarantee that the proof methods themselves are reliable, by methods that are equally reliable.
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In logic, you get out what you put in
Logic does not tell us what is true. It tells us what is true, given some other truths.
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In logic, you get out what you put in
Logic does not tell us what is true. It tells us what is true, given some other truths. If you bring to the uninterpreted propositional connectives a presupposition of classical logic, then the connectives will be classical.
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In logic, you get out what you put in
Logic does not tell us what is true. It tells us what is true, given some other truths. If you bring to the uninterpreted propositional connectives a presupposition of classical logic, then the connectives will be classical. This hardly shows that metatheory ‘must’ be conducted in classical language!
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