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 Layout Definitions 1 Explicit Definitions Implicit Definitions Beth’s Theorem Inferentialism 2 Definitions (revisited) 3 Beth’s Theorem (Revisited) Conclusion 4
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 Layout Definitions 1 Explicit Definitions Implicit Definitions Beth’s Theorem Inferentialism 2 Definitions (revisited) 3 Beth’s Theorem (Revisited) Conclusion 4
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 , Γ 1 ⊢ Θ 1 ∆ 1 ⊢ Λ 1 , p . . . . . . p , Γ n ⊢ Θ n ∆ m ⊢ Λ m , 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 of sentences related by ‘ ⊢ ’. Clearly I ⊆ P ( L ) 2 (where 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 of sentences related by ‘ ⊢ ’. Clearly I ⊆ P ( L ) 2 (where 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 of sentences related by ‘ ⊢ ’. Clearly I ⊆ P ( L ) 2 (where 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 of ordered pairs of sets of sentences, i.e. suppose 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 inference. So, for example: � 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 inference. So, for example: � 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 inference. So, for example: � 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: � A � P = df . = X � A � C = 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: � A � P = df . = X � A � C = df . = Y . Conditional ( → ): � A → B � = df . � � A � C ∩ � B � P , (( � A � P ) � ⊔ ( � B � C ) � ) � � . Negation ( ¬ ): � ¬ A � = df . � � A � C , � A � 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 . Shorthand, if: � A � = df . � X , Y � . Then: � A � P = df . = X � A � C = df . = Y . Conditional ( → ): � A → B � = df . � � A � C ∩ � B � P , (( � A � P ) � ⊔ ( � B � C ) � ) � � . Negation ( ¬ ): � ¬ A � = df . � � A � C , � A � P � . Conjunction ( & ) and Disjunction ( ∨ ) defined analogously.
Definitions Inferentialism Definitions (revisited) Conclusion Definition (Base Consequence Relation) ∼ 0 ⊆ P ( L 0 ) 2 . Let |
Definitions Inferentialism Definitions (revisited) Conclusion Definition (Base Consequence Relation) ∼ 0 ⊆ P ( L 0 ) 2 . Let | Axiom: If Γ 0 | ∼ 0 Θ 0 , then Γ 0 | ∼ Θ 0 .
Definitions Inferentialism Definitions (revisited) Conclusion Definition (Base Consequence Relation) ∼ 0 ⊆ P ( L 0 ) 2 . Let | Axiom: If Γ 0 | ∼ 0 Θ 0 , then Γ 0 | ∼ Θ 0 . Γ ⊢ Θ , A B , Γ ⊢ Θ A , Γ ⊢ Θ , B L → R → A → B , Γ ⊢ Θ Γ ⊢ A → B , Θ Γ ⊢ Θ , A A , Γ ⊢ Θ L ¬ R ¬ ¬ A , Γ ⊢ Θ Γ ⊢ Θ , ¬ A
Definitions Inferentialism Definitions (revisited) Conclusion Definition (Base Consequence Relation) ∼ 0 ⊆ P ( L 0 ) 2 . Let | Axiom: If Γ 0 | ∼ 0 Θ 0 , then Γ 0 | ∼ Θ 0 . Γ ⊢ Θ , A B , Γ ⊢ Θ A , Γ ⊢ Θ , B L → R → A → B , Γ ⊢ Θ Γ ⊢ A → B , Θ Γ ⊢ Θ , A A , Γ ⊢ Θ L ¬ R ¬ ¬ A , Γ ⊢ Θ 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 � P ) � ⊔ ( � B � C ) � ) �� ⊆ I M . iff df . A � M B
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 � P ) � ⊔ ( � B � C ) � ) �� ⊆ I M . iff df . A � M B 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 � P ) � ⊔ ( � B � C ) � ) �� ⊆ I M . iff df . A � M B 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 & A ⊢ ∨ B , i.e. the conjunction of the elements of A and the 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 only atoms. B is a base consequence relation iff B ⊆ P and B ∩ P ( L 0 ) 2 = B .
Definitions Inferentialism Definitions (revisited) Conclusion Definition (Base Consequence Relation) A base consequence relation is a subset of P that consists of only atoms. B is a base consequence relation iff B ⊆ P and B ∩ P ( L 0 ) 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 only atoms. B is a base consequence relation iff B ⊆ P and B ∩ P ( L 0 ) 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 Soundness and Completeness 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 Layout Definitions 1 Explicit Definitions Implicit Definitions Beth’s Theorem Inferentialism 2 Definitions (revisited) 3 Beth’s Theorem (Revisited) Conclusion 4
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: Φ 1 : If for any Z �� ⊆ P ( L ) 2 there exists A ∈ L such that: � A � P = Z �� � A � C = Z �� . OR
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: Φ 1 : If for any Z �� ⊆ P ( L ) 2 there exists A ∈ L such that: � A � P = Z �� � A � C = Z �� . OR ⊆ P ( L ) 2 exists A ∈ L such that: Φ 2 : If for any Z �� , Z �� 1 2 � A � = � Z �� , Z �� � . 1 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 Φ 1 (see above). Stipulating contraction and weakening are sufficient to force Φ 1 .
Definitions Inferentialism Definitions (revisited) Conclusion The converses: ∆ 4 ⇒ ∆ 3 : Implicit definitions collapse to inferential definitions if the underlying logic is Φ 1 (see above). Stipulating contraction and weakening are sufficient to force Φ 1 . ∆ 3 ⇒ ∆ 2 : Inferential definitions collapse to explicit (inferential) definitions if the underlying logic is Φ 2 . 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 Φ 1 (see above). Stipulating contraction and weakening are sufficient to force Φ 1 . ∆ 3 ⇒ ∆ 2 : Inferential definitions collapse to explicit (inferential) definitions if the underlying logic is Φ 2 . 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 Φ 1 (see above). Stipulating contraction and weakening are sufficient to force Φ 1 . ∆ 3 ⇒ ∆ 2 : Inferential definitions collapse to explicit (inferential) definitions if the underlying logic is Φ 2 . 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 Φ 1 (see above). Stipulating contraction and weakening are sufficient to force Φ 1 . ∆ 3 ⇒ ∆ 2 : Inferential definitions collapse to explicit (inferential) definitions if the underlying logic is Φ 2 . 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 Φ 1 (see above). Stipulating contraction and weakening are sufficient to force Φ 1 . ∆ 3 ⇒ ∆ 2 : Inferential definitions collapse to explicit (inferential) definitions if the underlying logic is Φ 2 . 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 Φ 1 (see above). Stipulating contraction and weakening are sufficient to force Φ 1 . ∆ 3 ⇒ ∆ 2 : Inferential definitions collapse to explicit (inferential) definitions if the underlying logic is Φ 2 . 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) Layout Definitions 1 Explicit Definitions Implicit Definitions Beth’s Theorem Inferentialism 2 Definitions (revisited) 3 Beth’s Theorem (Revisited) Conclusion 4
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 Φ 2 , 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 Layout Definitions 1 Explicit Definitions Implicit Definitions Beth’s Theorem Inferentialism 2 Definitions (revisited) 3 Beth’s Theorem (Revisited) Conclusion 4
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.
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