An Inferentialist Account of (Implicit) Definition Dan Kaplan - - PowerPoint PPT Presentation
Definitions Inferentialism Definitions (revisited) Conclusion An Inferentialist Account of (Implicit) Definition Dan Kaplan University of Pittsburgh / Universitt Leipzig dan.kaplan@pitt.edu PhDs in Logic X May 2 nd , 2018 pitt.edu/
Definitions Inferentialism Definitions (revisited) Conclusion
Dan Kaplan
University of Pittsburgh / Universität Leipzig dan.kaplan@pitt.edu
PhDs in Logic X May 2nd, 2018
pitt.edu/∼dsk31/LogicX2018/
Definitions Inferentialism Definitions (revisited) Conclusion
Goal: Account of genuine implicit definition.
Definitions Inferentialism Definitions (revisited) Conclusion
Goal: Account of genuine implicit definition. . . . f . . .=df .___. (Explicit Definition)
Definitions Inferentialism Definitions (revisited) Conclusion
Goal: Account of genuine implicit definition.
definition
defined term
. . .
=df . ___
definiens
. (Explicit Definition)
Definitions Inferentialism Definitions (revisited) Conclusion
Goal: Account of genuine implicit definition.
definition
defined term
. . .
=df . ___
definiens
. (Explicit Definition) Stipulate that ‘. . . f . . .’ be true. (Implicit Definition)
Definitions Inferentialism Definitions (revisited) Conclusion
Goal: Account of genuine implicit definition.
definition
defined term
. . .
=df . ___
definiens
. (Explicit Definition) Stipulate that ‘. . . f . . .’ be true. (Implicit Definition) Why? Implicit definitions seem
Definitions Inferentialism Definitions (revisited) Conclusion
Goal: Account of genuine implicit definition.
definition
defined term
. . .
=df . ___
definiens
. (Explicit Definition) Stipulate that ‘. . . f . . .’ be true. (Implicit Definition) Why? Implicit definitions seem More common
Definitions Inferentialism Definitions (revisited) Conclusion
Goal: Account of genuine implicit definition.
definition
defined term
. . .
=df . ___
definiens
. (Explicit Definition) Stipulate that ‘. . . f . . .’ be true. (Implicit Definition) Why? Implicit definitions seem More common More fundamental
Definitions Inferentialism Definitions (revisited) Conclusion
Goal: Account of genuine implicit definition.
definition
defined term
. . .
=df . ___
definiens
. (Explicit Definition) Stipulate that ‘. . . f . . .’ be true. (Implicit Definition) Why? Implicit definitions seem More common More fundamental Problem: Beth’s Theorem:
Definitions Inferentialism Definitions (revisited) Conclusion
Goal: Account of genuine implicit definition.
definition
defined term
. . .
=df . ___
definiens
. (Explicit Definition) Stipulate that ‘. . . f . . .’ be true. (Implicit Definition) Why? Implicit definitions seem More common More fundamental Problem: Beth’s Theorem: ‘f ’ is implicitly definable iff ‘f ’ is explicitly definable.
Definitions Inferentialism Definitions (revisited) Conclusion
1
Definitions Explicit Definitions Implicit Definitions Beth’s Theorem
2
Inferentialism
3
Definitions (revisited) Beth’s Theorem (Revisited)
4
Conclusion
Definitions Inferentialism Definitions (revisited) Conclusion
1
Definitions Explicit Definitions Implicit Definitions Beth’s Theorem
2
Inferentialism
3
Definitions (revisited) Beth’s Theorem (Revisited)
4
Conclusion
Definitions Inferentialism Definitions (revisited) Conclusion
. . . f . . . =df . ___. (D)
Definitions Inferentialism Definitions (revisited) Conclusion
. . . f . . . =df . ___. (D) Definitions should do just and only that: define.
Definitions Inferentialism Definitions (revisited) Conclusion
Eliminability: Pragmatic constraint. All possible uses of ‘f ’ are uniquely specified by D.
Definitions Inferentialism Definitions (revisited) Conclusion
Eliminability: Pragmatic constraint. All possible uses of ‘f ’ are uniquely specified by D. A =df . A&B.
Definitions Inferentialism Definitions (revisited) Conclusion
Eliminability: Pragmatic constraint. All possible uses of ‘f ’ are uniquely specified by D. A =df . A&B. Conservativeness: Definitions should do no more. A definition should introduce no substantive truths.
Definitions Inferentialism Definitions (revisited) Conclusion
Eliminability: Pragmatic constraint. All possible uses of ‘f ’ are uniquely specified by D. A =df . A&B. Conservativeness: Definitions should do no more. A definition should introduce no substantive truths. γ =df . γ → B.
Definitions Inferentialism Definitions (revisited) Conclusion
Eliminability: Pragmatic constraint. All possible uses of ‘f ’ are uniquely specified by D. A =df . A&B. Conservativeness: Definitions should do no more. A definition should introduce no substantive truths. γ =df . γ → B. Given γ → γ: γ → γ (Taut.) γ → (γ → B) (Def. γ) γ → B (MP and Classical Taut.) γ (Def. γ) B. (MP)
Definitions Inferentialism Definitions (revisited) Conclusion Explicit Definitions
1
Definitions Explicit Definitions Implicit Definitions Beth’s Theorem
2
Inferentialism
3
Definitions (revisited) Beth’s Theorem (Revisited)
4
Conclusion
Definitions Inferentialism Definitions (revisited) Conclusion Explicit Definitions
Eliminability If L is our language and L+ is our language after f has been introduced via definition, then we say that the definition is eliminable (or f is eliminable) if for every sentence A in which f occurs, there is an equivalent sentence B (which contains no instances of ‘f ’) in L.
Definitions Inferentialism Definitions (revisited) Conclusion Explicit Definitions
Eliminability If L is our language and L+ is our language after f has been introduced via definition, then we say that the definition is eliminable (or f is eliminable) if for every sentence A in which f occurs, there is an equivalent sentence B (which contains no instances of ‘f ’) in L. Conservativeness If L is our language and L+ is our language after p has been introduced via definition, then we say that the definition is conservative if the status of all q ∈ L remains unchanged. In the primary i.e. sentential case this means that q is provable in L if and only if it is provable in L+, or a model of L satisfies q if and
Definitions Inferentialism Definitions (revisited) Conclusion Explicit Definitions
Theorem All explicit definitions with the following form will properly define a defined term: ∀(x1, . . . , xn) (#α(x1, . . . , xn) ↔ B(x1, . . . , xn)) .
Where ‘#_(. . .)’ is read as an open formula schema with α in L+ \ L and B ∈ L. In addition to the constraint that B be free of α, explicit definitions formed in this way also require that (x1, . . . , xn) be both distinct and the only free variables in B.
Definitions Inferentialism Definitions (revisited) Conclusion Explicit Definitions
Theorem All explicit definitions with the following form will properly define a defined term: ∀(x1, . . . , xn) (#α(x1, . . . , xn) ↔ B(x1, . . . , xn)) .
Where ‘#_(. . .)’ is read as an open formula schema with α in L+ \ L and B ∈ L. In addition to the constraint that B be free of α, explicit definitions formed in this way also require that (x1, . . . , xn) be both distinct and the only free variables in B.
Alternatively: α ↔ B.
Definitions Inferentialism Definitions (revisited) Conclusion Implicit Definitions
1
Definitions Explicit Definitions Implicit Definitions Beth’s Theorem
2
Inferentialism
3
Definitions (revisited) Beth’s Theorem (Revisited)
4
Conclusion
Definitions Inferentialism Definitions (revisited) Conclusion Implicit Definitions
#f . Where ‘#_’ is some sentential schema and we stipulate that ‘#f ’ be true (or similar).
Definitions Inferentialism Definitions (revisited) Conclusion Implicit Definitions
#f . Where ‘#_’ is some sentential schema and we stipulate that ‘#f ’ be true (or similar). Eliminability If L is our language and L+ is our language after f has been introduced via definition, then we say that the definition is eliminable (or f is eliminable) if, after introducing a new term f ′ (and extending L+ to L′+), f and f ′ are equivalent.
Definitions Inferentialism Definitions (revisited) Conclusion Implicit Definitions
Analogy with “reference fixing”: “Jack the Ripper is the man responsible for these murders.” (i.e. those that occurred in Whitechapel in the latter half of 1888)
Definitions Inferentialism Definitions (revisited) Conclusion Implicit Definitions
Analogy with “reference fixing”: “Jack the Ripper is the man responsible for these murders.” (i.e. those that occurred in Whitechapel in the latter half of 1888) Some (potential) problems: Existence: How do we know that anyone is referred to by Jack the Ripper?
Definitions Inferentialism Definitions (revisited) Conclusion Implicit Definitions
Analogy with “reference fixing”: “Jack the Ripper is the man responsible for these murders.” (i.e. those that occurred in Whitechapel in the latter half of 1888) Some (potential) problems: Existence: How do we know that anyone is referred to by Jack the Ripper? Uniqueness: How do we know there is a unique such person?
Definitions Inferentialism Definitions (revisited) Conclusion Implicit Definitions
Analogy with “reference fixing”: “Jack the Ripper is the man responsible for these murders.” (i.e. those that occurred in Whitechapel in the latter half of 1888) Some (potential) problems: Existence: How do we know that anyone is referred to by Jack the Ripper? Uniqueness: How do we know there is a unique such person? Possession: What does it mean that the term comes to possess this meaning.
Definitions Inferentialism Definitions (revisited) Conclusion Implicit Definitions
Analogy with “reference fixing”: “Jack the Ripper is the man responsible for these murders.” (i.e. those that occurred in Whitechapel in the latter half of 1888) Some (potential) problems: Existence: How do we know that anyone is referred to by Jack the Ripper? Uniqueness: How do we know there is a unique such person? Possession: What does it mean that the term comes to possess this meaning. Explanation: We must explain this process: how is it that asserting a sentence confers possession of a meaning to a term.
Definitions Inferentialism Definitions (revisited) Conclusion Implicit Definitions
Analogy with “reference fixing”: “Jack the Ripper is the man responsible for these murders.” (i.e. those that occurred in Whitechapel in the latter half of 1888) Some (potential) problems: Existence: How do we know that anyone is referred to by Jack the Ripper? Uniqueness: How do we know there is a unique such person? Possession: What does it mean that the term comes to possess this meaning. Explanation: We must explain this process: how is it that asserting a sentence confers possession of a meaning to a term. Suggestion: Ramsify + Carnap Conditional. Schema ‘#_’: ∃x(#x) → #f .
Definitions Inferentialism Definitions (revisited) Conclusion Beth’s Theorem
1
Definitions Explicit Definitions Implicit Definitions Beth’s Theorem
2
Inferentialism
3
Definitions (revisited) Beth’s Theorem (Revisited)
4
Conclusion
Definitions Inferentialism Definitions (revisited) Conclusion Beth’s Theorem
Theorem (Beth’s Theorem) ‘f ’ is implicitly definable iff ‘f ’ is explicitly definable (i.e. exists an explicit definition for ‘f ’ equivalent to its implicit definition).
Definitions Inferentialism Definitions (revisited) Conclusion Beth’s Theorem
Theorem (Beth’s Theorem) ‘f ’ is implicitly definable iff ‘f ’ is explicitly definable (i.e. exists an explicit definition for ‘f ’ equivalent to its implicit definition). Proof. (⇒)[Padoa’s Method] Let α ↔ B explicitly define ‘f ’. ‘α ↔ B’ may serve as ‘f ’s implicit definition.
Definitions Inferentialism Definitions (revisited) Conclusion Beth’s Theorem
(Continued) Proof. (⇐)[over-simplification from Craig’s Interpolation Lemma]
Definitions Inferentialism Definitions (revisited) Conclusion Beth’s Theorem
(Continued) Proof. (⇐)[over-simplification from Craig’s Interpolation Lemma] Suppose Craig’s Lemma holds in our logic and that α implicitly defines ‘f ’. If we introduce a second f ′ implicitly defined by α′ then they are equivalent:
Definitions Inferentialism Definitions (revisited) Conclusion Beth’s Theorem
(Continued) Proof. (⇐)[over-simplification from Craig’s Interpolation Lemma] Suppose Craig’s Lemma holds in our logic and that α implicitly defines ‘f ’. If we introduce a second f ′ implicitly defined by α′ then they are equivalent: α ↔ α′.
Definitions Inferentialism Definitions (revisited) Conclusion Beth’s Theorem
(Continued) Proof. (⇐)[over-simplification from Craig’s Interpolation Lemma] Suppose Craig’s Lemma holds in our logic and that α implicitly defines ‘f ’. If we introduce a second f ′ implicitly defined by α′ then they are equivalent: α ↔ α′. By Craig’s Lemma exists B (with proper characteristics) such that: α ⇔ B ⇔ α′
Definitions Inferentialism Definitions (revisited) Conclusion Beth’s Theorem
(Continued) Proof. (⇐)[over-simplification from Craig’s Interpolation Lemma] Suppose Craig’s Lemma holds in our logic and that α implicitly defines ‘f ’. If we introduce a second f ′ implicitly defined by α′ then they are equivalent: α ↔ α′. By Craig’s Lemma exists B (with proper characteristics) such that: α ⇔ B ⇔ α′ Explicit definition is: α ⇔ B.
Definitions Inferentialism Definitions (revisited) Conclusion Beth’s Theorem
Theorem ‘f ’ is implicitly definable iff ‘f ’ is explicitly definable (i.e. exists an explicit definition for ‘f ’ equivalent to its implicit definition). NB1: (⇒) is not trivial. NB2: As evidenced by Craig’s Lemma, we must presuppose something like classical logic to get Beth’s Theorem.
Definitions Inferentialism Definitions (revisited) Conclusion
1
Definitions Explicit Definitions Implicit Definitions Beth’s Theorem
2
Inferentialism
3
Definitions (revisited) Beth’s Theorem (Revisited)
4
Conclusion
Definitions Inferentialism Definitions (revisited) Conclusion
Recall: Goal: Account of genuine implicit definition.
Definitions Inferentialism Definitions (revisited) Conclusion
Recall: Goal: Account of genuine implicit definition. Suggestion: Move to sub-/non-classical logic. Use different theory of meaning (i.e. not truth functional).
Definitions Inferentialism Definitions (revisited) Conclusion
Suppose following lists are exhaustive: p, Γ1 ⊢ Θ1 . . . p, Γn ⊢ Θn . . . ∆1 ⊢ Λ1, p . . . ∆m ⊢ Λm, p . . . Suggestion: let us understand the meaning of ‘p’ in terms of its behavior in reasoning.
Definitions Inferentialism Definitions (revisited) Conclusion
Some useful shorthands/definitions:
Definitions Inferentialism Definitions (revisited) Conclusion
Some useful shorthands/definitions: I: Let I be the set of good implications. I.e. those sets
L is the set of all sentences).
Definitions Inferentialism Definitions (revisited) Conclusion
Some useful shorthands/definitions: I: Let I be the set of good implications. I.e. those sets
L is the set of all sentences). ⊔: Let ‘⊔’ (fuission) be a pairwise set-union operation for ordered pairs. Thus Γ, Θ ⊔ ∆, Λ =df . Γ ∪ ∆, Θ ∪ Λ.
Definitions Inferentialism Definitions (revisited) Conclusion
Some useful shorthands/definitions: I: Let I be the set of good implications. I.e. those sets
L is the set of all sentences). ⊔: Let ‘⊔’ (fuission) be a pairwise set-union operation for ordered pairs. Thus Γ, Θ ⊔ ∆, Λ =df . Γ ∪ ∆, Θ ∪ Λ. Occasionally I might use ‘⊔’ as on operation on sets
X, Y ⊆ P(L). Then we should understand X ⊔ Y as: X ⊔ Y =df . {x ⊔ y|x ∈ X, y ∈ Y }, i.e., the result of applying ‘⊔’ to x and y for each x ∈ X and y ∈ Y .
Definitions Inferentialism Definitions (revisited) Conclusion
: Let us understand ‘’ (pronounced: “vee”) as a function that maps (in the basic case) an ordered pair to a set of ordered pairs whose fuission makes a good
Γ, Θ =df . {∆, Λ|Γ, Θ ⊔ ∆, Λ ∈ I}.
Definitions Inferentialism Definitions (revisited) Conclusion
: Let us understand ‘’ (pronounced: “vee”) as a function that maps (in the basic case) an ordered pair to a set of ordered pairs whose fuission makes a good
Γ, Θ =df . {∆, Λ|Γ, Θ ⊔ ∆, Λ ∈ I}. Similarly we may define the same over sets of ordered pairs, where the result amounts to the intersection of the same of each of its members: X =df . {∆, Λ|∀Γ, Θ ∈ X(Γ, Θ ⊔ ∆, Λ ∈ I)}.
Definitions Inferentialism Definitions (revisited) Conclusion
: Let us understand ‘’ (pronounced: “vee”) as a function that maps (in the basic case) an ordered pair to a set of ordered pairs whose fuission makes a good
Γ, Θ =df . {∆, Λ|Γ, Θ ⊔ ∆, Λ ∈ I}. Similarly we may define the same over sets of ordered pairs, where the result amounts to the intersection of the same of each of its members: X =df . {∆, Λ|∀Γ, Θ ∈ X(Γ, Θ ⊔ ∆, Λ ∈ I)}. ‘⊔’ and ‘’ together give us a neat way to talk about the role that a sentence, e.g. p, plays in good implication. We can think of interpreting p thusly: p =df . {p}, ∅, ∅, {p}.
Definitions Inferentialism Definitions (revisited) Conclusion
Proper Inferential Role (PIR): Let X = Y , Z specify an inferential role (i.e. the contribution that a sentence might make to good implication). We call X a proper inferential role if X = X.
Definitions Inferentialism Definitions (revisited) Conclusion
Proper Inferential Role (PIR): Let X = Y , Z specify an inferential role (i.e. the contribution that a sentence might make to good implication). We call X a proper inferential role if X = X. Shorthand, if: A =df . X, Y . Then: AP =df .= X AC =df .= Y .
Definitions Inferentialism Definitions (revisited) Conclusion
Proper Inferential Role (PIR): Let X = Y , Z specify an inferential role (i.e. the contribution that a sentence might make to good implication). We call X a proper inferential role if X = X. Shorthand, if: A =df . X, Y . Then: AP =df .= X AC =df .= Y . Conditional (→): A → B =df . AC ∩ BP, ((AP) ⊔ (BC)). Negation (¬): ¬A =df . AC, AP.
Definitions Inferentialism Definitions (revisited) Conclusion
Proper Inferential Role (PIR): Let X = Y , Z specify an inferential role (i.e. the contribution that a sentence might make to good implication). We call X a proper inferential role if X = X. Shorthand, if: A =df . X, Y . Then: AP =df .= X AC =df .= Y . Conditional (→): A → B =df . AC ∩ BP, ((AP) ⊔ (BC)). Negation (¬): ¬A =df . AC, AP. Conjunction (&) and Disjunction (∨) defined analogously.
Definitions Inferentialism Definitions (revisited) Conclusion
Definition (Base Consequence Relation) Let | ∼0⊆ P(L0)2.
Definitions Inferentialism Definitions (revisited) Conclusion
Definition (Base Consequence Relation) Let | ∼0⊆ P(L0)2. Axiom: If Γ0 | ∼0 Θ0, then Γ0 | ∼ Θ0.
Definitions Inferentialism Definitions (revisited) Conclusion
Definition (Base Consequence Relation) Let | ∼0⊆ P(L0)2. Axiom: If Γ0 | ∼0 Θ0, then Γ0 | ∼ Θ0. Γ ⊢ Θ, A B, Γ ⊢ Θ
L→
A → B, Γ ⊢ Θ A, Γ ⊢ Θ, B
R→
Γ ⊢ A → B, Θ Γ ⊢ Θ, A
L¬
¬A, Γ ⊢ Θ A, Γ ⊢ Θ
R¬
Γ ⊢ Θ, ¬A
Definitions Inferentialism Definitions (revisited) Conclusion
Definition (Base Consequence Relation) Let | ∼0⊆ P(L0)2. Axiom: If Γ0 | ∼0 Θ0, then Γ0 | ∼ Θ0. Γ ⊢ Θ, A B, Γ ⊢ Θ
L→
A → B, Γ ⊢ Θ A, Γ ⊢ Θ, B
R→
Γ ⊢ A → B, Θ Γ ⊢ Θ, A
L¬
¬A, Γ ⊢ Θ A, Γ ⊢ Θ
R¬
Γ ⊢ Θ, ¬A Conjunction (&) and Disjunction (∨) defined analogously.
Definitions Inferentialism Definitions (revisited) Conclusion
Definition (Semantic Entailment) We say that A semantically entails B relative to a model M if closure of the fuission of A (as premise) and B (as conclusion) consists of only good implications: A M B iffdf . ((AP) ⊔ (BC)) ⊆ IM.
Definitions Inferentialism Definitions (revisited) Conclusion
Definition (Semantic Entailment) We say that A semantically entails B relative to a model M if closure of the fuission of A (as premise) and B (as conclusion) consists of only good implications: A M B iffdf . ((AP) ⊔ (BC)) ⊆ IM. We say that A semantically entails B if A M B on all models M.
Definitions Inferentialism Definitions (revisited) Conclusion
Definition (Semantic Entailment) We say that A semantically entails B relative to a model M if closure of the fuission of A (as premise) and B (as conclusion) consists of only good implications: A M B iffdf . ((AP) ⊔ (BC)) ⊆ IM. We say that A semantically entails B if A M B on all models M. NB: If A and B are sets of sentences then we read A ⊢ B as
disjunction of the elements of B.
Definitions Inferentialism Definitions (revisited) Conclusion
Definition (Base Consequence Relation) A base consequence relation is a subset of P that consists of
B ∩ P(L0)2 = B.
Definitions Inferentialism Definitions (revisited) Conclusion
Definition (Base Consequence Relation) A base consequence relation is a subset of P that consists of
B ∩ P(L0)2 = B. We say that a model M = P, I, · is fit for a base consequence relation B iff ∀∆, Λ ∈ B(∆ M Λ).
Definitions Inferentialism Definitions (revisited) Conclusion
Definition (Base Consequence Relation) A base consequence relation is a subset of P that consists of
B ∩ P(L0)2 = B. We say that a model M = P, I, · is fit for a base consequence relation B iff ∀∆, Λ ∈ B(∆ M Λ). We say that Γ semantically entails Θ relative to B iff Γ M Θ for all models M fit for B.
Definitions Inferentialism Definitions (revisited) Conclusion
Theorem (Soundness) The sequent calculus is sound: Γ ⊢B Θ ⇒ Γ B Θ. Theorem (Completeness) The sequent calculus is complete: Γ B Θ ⇒ Γ ⊢B Θ.
Definitions Inferentialism Definitions (revisited) Conclusion
1
Definitions Explicit Definitions Implicit Definitions Beth’s Theorem
2
Inferentialism
3
Definitions (revisited) Beth’s Theorem (Revisited)
4
Conclusion
Definitions Inferentialism Definitions (revisited) Conclusion
I explore five notions of “definability” (ways in which a new term may be successfully given a meaning).
Definitions Inferentialism Definitions (revisited) Conclusion
I explore five notions of “definability” (ways in which a new term may be successfully given a meaning). ∆1 Explicit Definition: A ↔ B. Introduced as: A ⊢ B and B ⊢ A.
Definitions Inferentialism Definitions (revisited) Conclusion
I explore five notions of “definability” (ways in which a new term may be successfully given a meaning). ∆1 Explicit Definition: A ↔ B. Introduced as: A ⊢ B and B ⊢ A. ∆2 Explicit (Inferential) Definition: A =df . B licenses: B, Γ ⊢ Θ ⇒ A, Γ ⊢ Θ and Γ ⊢ Θ, B ⇒ Γ ⊢ Θ, A.
Definitions Inferentialism Definitions (revisited) Conclusion
I explore five notions of “definability” (ways in which a new term may be successfully given a meaning). ∆1 Explicit Definition: A ↔ B. Introduced as: A ⊢ B and B ⊢ A. ∆2 Explicit (Inferential) Definition: A =df . B licenses: B, Γ ⊢ Θ ⇒ A, Γ ⊢ Θ and Γ ⊢ Θ, B ⇒ Γ ⊢ Θ, A. ∆3 Inferential Definition: A =df . B, C licenses: B, Γ ⊢ Θ ⇒ A, Γ ⊢ Θ and Γ ⊢ Θ, C ⇒ Γ ⊢ Θ, A.
Definitions Inferentialism Definitions (revisited) Conclusion
I explore five notions of “definability” (ways in which a new term may be successfully given a meaning). ∆1 Explicit Definition: A ↔ B. Introduced as: A ⊢ B and B ⊢ A. ∆2 Explicit (Inferential) Definition: A =df . B licenses: B, Γ ⊢ Θ ⇒ A, Γ ⊢ Θ and Γ ⊢ Θ, B ⇒ Γ ⊢ Θ, A. ∆3 Inferential Definition: A =df . B, C licenses: B, Γ ⊢ Θ ⇒ A, Γ ⊢ Θ and Γ ⊢ Θ, C ⇒ Γ ⊢ Θ, A. ∆4 Implicit Definition: A term p is defined implicitly if the stipulation that all the implications in I successfully give p a meaning (i.e. it is not possible to construct models which disagree about the meaning of p).
Definitions Inferentialism Definitions (revisited) Conclusion
I explore five notions of “definability” (ways in which a new term may be successfully given a meaning). ∆1 Explicit Definition: A ↔ B. Introduced as: A ⊢ B and B ⊢ A. ∆2 Explicit (Inferential) Definition: A =df . B licenses: B, Γ ⊢ Θ ⇒ A, Γ ⊢ Θ and Γ ⊢ Θ, B ⇒ Γ ⊢ Θ, A. ∆3 Inferential Definition: A =df . B, C licenses: B, Γ ⊢ Θ ⇒ A, Γ ⊢ Θ and Γ ⊢ Θ, C ⇒ Γ ⊢ Θ, A. ∆4 Implicit Definition: A term p is defined implicitly if the stipulation that all the implications in I successfully give p a meaning (i.e. it is not possible to construct models which disagree about the meaning of p). ∆5 Proper Inferential Role: (Compare above)
Definitions Inferentialism Definitions (revisited) Conclusion
Some interesting results:
Definitions Inferentialism Definitions (revisited) Conclusion
Some interesting results: ∆2 ⇒ ∆3 ⇒ ∆4 ⇒ ∆5.
Definitions Inferentialism Definitions (revisited) Conclusion
Some interesting results: ∆2 ⇒ ∆3 ⇒ ∆4 ⇒ ∆5. From this follows a modified version of Padoa’s result: ∆2 ⇒ ∆5.
Definitions Inferentialism Definitions (revisited) Conclusion
Some interesting results: ∆2 ⇒ ∆3 ⇒ ∆4 ⇒ ∆5. From this follows a modified version of Padoa’s result: ∆2 ⇒ ∆5. NB: None of the converses hold without further stipulation.
Definitions Inferentialism Definitions (revisited) Conclusion
Recall sentences interpreted: A =df . X, Y , where X = X and Y = Y .
Definitions Inferentialism Definitions (revisited) Conclusion
Recall sentences interpreted: A =df . X, Y , where X = X and Y = Y . Definition (Functional Completeness) We call our logic:
Definitions Inferentialism Definitions (revisited) Conclusion
Recall sentences interpreted: A =df . X, Y , where X = X and Y = Y . Definition (Functional Completeness) We call our logic: Φ1: If for any Z ⊆ P(L)2 there exists A ∈ L such that: AP = Z OR AC = Z .
Definitions Inferentialism Definitions (revisited) Conclusion
Recall sentences interpreted: A =df . X, Y , where X = X and Y = Y . Definition (Functional Completeness) We call our logic: Φ1: If for any Z ⊆ P(L)2 there exists A ∈ L such that: AP = Z OR AC = Z . Φ2: If for any Z
1
, Z
2
⊆ P(L)2 exists A ∈ L such that: A = Z
1
, Z
2
.
Definitions Inferentialism Definitions (revisited) Conclusion
The converses:
Definitions Inferentialism Definitions (revisited) Conclusion
The converses: ∆4 ⇒ ∆3: Implicit definitions collapse to inferential definitions if the underlying logic is Φ1 (see above). Stipulating contraction and weakening are sufficient to force Φ1.
Definitions Inferentialism Definitions (revisited) Conclusion
The converses: ∆4 ⇒ ∆3: Implicit definitions collapse to inferential definitions if the underlying logic is Φ1 (see above). Stipulating contraction and weakening are sufficient to force Φ1. ∆3 ⇒ ∆2: Inferential definitions collapse to explicit (inferential) definitions if the underlying logic is Φ2. It is sufficient that the logic is restricted such that it obeys reflexivity and Cut-Elimination (meaning we can find Interpolants).
Definitions Inferentialism Definitions (revisited) Conclusion
The converses: ∆4 ⇒ ∆3: Implicit definitions collapse to inferential definitions if the underlying logic is Φ1 (see above). Stipulating contraction and weakening are sufficient to force Φ1. ∆3 ⇒ ∆2: Inferential definitions collapse to explicit (inferential) definitions if the underlying logic is Φ2. It is sufficient that the logic is restricted such that it obeys reflexivity and Cut-Elimination (meaning we can find Interpolants). ∆2 ⇒ ∆1: Given transitivity.
Definitions Inferentialism Definitions (revisited) Conclusion
The converses: ∆4 ⇒ ∆3: Implicit definitions collapse to inferential definitions if the underlying logic is Φ1 (see above). Stipulating contraction and weakening are sufficient to force Φ1. ∆3 ⇒ ∆2: Inferential definitions collapse to explicit (inferential) definitions if the underlying logic is Φ2. It is sufficient that the logic is restricted such that it obeys reflexivity and Cut-Elimination (meaning we can find Interpolants). ∆2 ⇒ ∆1: Given transitivity. ∆1 ⇒ ∆2: Given reflexivity.
Definitions Inferentialism Definitions (revisited) Conclusion
The converses: ∆4 ⇒ ∆3: Implicit definitions collapse to inferential definitions if the underlying logic is Φ1 (see above). Stipulating contraction and weakening are sufficient to force Φ1. ∆3 ⇒ ∆2: Inferential definitions collapse to explicit (inferential) definitions if the underlying logic is Φ2. It is sufficient that the logic is restricted such that it obeys reflexivity and Cut-Elimination (meaning we can find Interpolants). ∆2 ⇒ ∆1: Given transitivity. ∆1 ⇒ ∆2: Given reflexivity. Note that if the consequence relation is stipulated to be supra-classical then all the above notions of definitions will collapse:
Definitions Inferentialism Definitions (revisited) Conclusion
The converses: ∆4 ⇒ ∆3: Implicit definitions collapse to inferential definitions if the underlying logic is Φ1 (see above). Stipulating contraction and weakening are sufficient to force Φ1. ∆3 ⇒ ∆2: Inferential definitions collapse to explicit (inferential) definitions if the underlying logic is Φ2. It is sufficient that the logic is restricted such that it obeys reflexivity and Cut-Elimination (meaning we can find Interpolants). ∆2 ⇒ ∆1: Given transitivity. ∆1 ⇒ ∆2: Given reflexivity. Note that if the consequence relation is stipulated to be supra-classical then all the above notions of definitions will collapse: ∆1 ⇔ ∆2 ⇔ ∆3 ⇔ ∆4.
Definitions Inferentialism Definitions (revisited) Conclusion
The converses: ∆4 ⇒ ∆3: Implicit definitions collapse to inferential definitions if the underlying logic is Φ1 (see above). Stipulating contraction and weakening are sufficient to force Φ1. ∆3 ⇒ ∆2: Inferential definitions collapse to explicit (inferential) definitions if the underlying logic is Φ2. It is sufficient that the logic is restricted such that it obeys reflexivity and Cut-Elimination (meaning we can find Interpolants). ∆2 ⇒ ∆1: Given transitivity. ∆1 ⇒ ∆2: Given reflexivity. Note that if the consequence relation is stipulated to be supra-classical then all the above notions of definitions will collapse: ∆1 ⇔ ∆2 ⇔ ∆3 ⇔ ∆4. NB: This version is independent of Craig’s Lemma.
Definitions Inferentialism Definitions (revisited) Conclusion Beth’s Theorem (Revisited)
1
Definitions Explicit Definitions Implicit Definitions Beth’s Theorem
2
Inferentialism
3
Definitions (revisited) Beth’s Theorem (Revisited)
4
Conclusion
Definitions Inferentialism Definitions (revisited) Conclusion Beth’s Theorem (Revisited)
Two analogs to Beth’s Theorem:
Definitions Inferentialism Definitions (revisited) Conclusion Beth’s Theorem (Revisited)
Two analogs to Beth’s Theorem: Theorem (Beth’s Theorem) If ⊢ is supra-classical then a definition is ∆1 iff it is ∆4. I.e. the class of explicit definitions (∆1) and the class of implicit definitions (∆4) are coextensive.
Definitions Inferentialism Definitions (revisited) Conclusion Beth’s Theorem (Revisited)
Two analogs to Beth’s Theorem: Theorem (Beth’s Theorem) If ⊢ is supra-classical then a definition is ∆1 iff it is ∆4. I.e. the class of explicit definitions (∆1) and the class of implicit definitions (∆4) are coextensive. I’ll also introduce an analog of interest (insofar as it lets us look at the relationship between notions I have introduced).
Definitions Inferentialism Definitions (revisited) Conclusion Beth’s Theorem (Revisited)
Two analogs to Beth’s Theorem: Theorem (Beth’s Theorem) If ⊢ is supra-classical then a definition is ∆1 iff it is ∆4. I.e. the class of explicit definitions (∆1) and the class of implicit definitions (∆4) are coextensive. I’ll also introduce an analog of interest (insofar as it lets us look at the relationship between notions I have introduced). Theorem (Alternative to Beth’s Theorem) If our logic is Φ2, then a definition is ∆2 iff it is ∆4. That is, the class of explicit (inferential) definitions and those of implicit definitions are coextensive.
Definitions Inferentialism Definitions (revisited) Conclusion
1
Definitions Explicit Definitions Implicit Definitions Beth’s Theorem
2
Inferentialism
3
Definitions (revisited) Beth’s Theorem (Revisited)
4
Conclusion
Definitions Inferentialism Definitions (revisited) Conclusion
Goal: Account of genuine implicit definition.
Definitions Inferentialism Definitions (revisited) Conclusion
Goal: Account of genuine implicit definition. Explored space that opens up when we move to a sub-/non-classical setting and an alternative theory of meaning.
Definitions Inferentialism Definitions (revisited) Conclusion
Goal: Account of genuine implicit definition. Explored space that opens up when we move to a sub-/non-classical setting and an alternative theory of meaning. Not explored:
Eliminability requires some notion of “equivalence”, how am I understanding this? Why do I call what I am doing “definition”. Some of what counts as definition looks quite strange.
Definitions Inferentialism Definitions (revisited) Conclusion