SELECCIÓN A PARTIR DE LA NORMATIVA CURRICULAR

If ∈ SEQ, then:

(i) There is no Γ and no X such that is a derivation of Γ from X

or

(ii) There is exactly one Γ and exactly one X such that is a derivation of Γ from X.

Proof: Suppose ∈ SEQ. Then there is no Γ and no X such that is a derivation of Γ

from X or there are a Γ and an X such that is a derivation of Γ from X. In the first case,

the statement holds. Now, for the second case, suppose there are a Γ and an X such that

is a derivation of Γ from X. According to Definition 3-20, we then have ∈ RCS\{∅}, Γ

= C( ) and AVAP( ) = X. We still have to show uniqueness. For this, supoose is a

derivation of Γ' from X'. Then we have Γ' = C( ) = Γ and X' = AVAP( ) = X. ■

Now, let us illutsrate this result with an example. Suppose ξ∈ VAR, Δ∈ FORM, where FV(Δ) ⊆ {ξ}, and suppose β ∈ PAR\ST(Δ). Now, let [3.1] be the following sentence sequence: Example [3.1] 0 Suppose ξ¬Δ 1 Suppose ξΔ 2 Suppose [β, ξ, Δ] 3 Suppose ξΔ 4 Therefore ξΔ∧ [β, ξ, Δ] 5 Therefore [β, ξ, Δ] 6 Therefore ¬[β, ξ, Δ] 7 Therefore ¬ ξΔ 8 Therefore ¬ ξΔ 9 Therefore ¬ ξΔ

Commentary: According to Theorem 3-11, there should either be no Γ and no X such that

[3.1]

is a derivation of Γ from X or we should be able to find unique Γ and X such that

13

For the formulation of a corresponding theorem for a regulation of the predicate '.. is a derivation of .. from ..' according to which the set of propositions named at the third place has to be a superset of the set of assumptions that actually occur in the respective sentence sequence and are not eliminated there, see footnote 4.

[3.1]

is a derivation of Γ from X. This is actually the case as [3.1] is a derivation of ¬ ξΔ from { ξ¬Δ }, where both are uniquely determined. This can be made clearer by an informal inspection of the sentence sequence. To do this, we first furnish the sen- tence sequence with comments that will then be explained.

Example [3.2] available 0 Suppose ξ¬Δ (AR) 0 1 Suppose ξΔ (AR) 0, 1 2 Suppose [β, ξ, Δ] (AR) 0, 1, 2 3 Suppose ξΔ (AR) 0, 1, 2, 3 4 Therefore ξΔ∧ [β, ξ, Δ] (CI); 2, 3 0, 1, 2, 3, 4 5 Therefore [β, ξ, Δ] (CE); 4 0, 1, 2, 3, 4, 5 6 Therefore ¬[β, ξ, Δ] (UE); 1 0, 1, 2, 3, 4, 5, 6 7 Therefore ¬ ξΔ (NI); 5, 6 0, 1, 2, 7 8 Therefore ¬ ξΔ (PE); 1, 7 0, 1, 8 9 Therefore ¬ ξΔ (NI); 1, 8 0, 9

Explanation: In the second column from the right, the rules by which one may extend an

already uttered sequence and the respective premise lines are given (cf. ch. 3.1). The ut- termost right column displays the line numbers of those lines whose propositions are available in the restriction of [3.1] on the successor of the current line number. Note that the propositions and assumptions that are available in [3.1] i (1 ≤ i ≤ 10) are always uniquely determined.

Also, we have that, for example, the inference in line 8 may only be carried out by PE and the inference in line 9 may only be carried out by NI, in both cases with uniquely determined premise lines. In line 8, NI is not an option, because, on the one hand, the proposition assumed in line 2 is still available in [3.1] 8 so that 1 cannot serve as an ini- tial assumption for NI, while, on the other hand, 3 cannot serve as an initial assumption for NI, because the proposition assumed there is not any more available in [3.1] 8 at this position. Obversly, PE may not be carried out in line 9 (and NI may be carried out), be- cause the representative instance assumption in line 2 is not any more available in [3.1] 9 at this position (and at all).

If one checks all other lines, one can easily convince oneself that [3.1]∈ RCS\{∅}. The set of the assumptions that are available in [3.1] is uniquely determined and determinable,

because, with Definition 2-26, Definition 2-28, Definition 2-29 and Definition 2-31, one can check for every proposition A that has been assumed in [3.1] whether A ∈ AVAP( [3.1]). As desired, one can easily convince oneself that AVAP( [3.1]) = { ξ¬Δ }. Obviously, we have [3.1]Dom( [3.1]_)-1 = Therefore ¬ ξΔ so that Theorem

3-11 is confirmed.

Note that the comments in the right columns do not serve to disambiguate from which set of propositions the proposition in the last line has been derived, but only serve to fa- cilitate an easier traceability and understanding. Note that the rule-commentary to [3.1] is uniquely determined by coincidence and that there are other sentence sequences for which different rule-commentaries may be produced: There are circumstances under which a transition may be carried out in accordance with different rules, e.g. UE and PE. How- ever, it is not the case that the possibility of alternative rule-commentaries has any effects on the uniqueness of the availability-commentary. Available propositions (or lines) are not determined with recourse to the rule-commentary, but according to the definition of availability and thus, eventually, according to the definition of closed segments. The separate definition of availability excludes that we arrive at different availabilities for one and the same transition, even if that transisition can be carried out in accordance with more than one rule. Thus, it is always uniquely determined and determinable if a given sentence sequence is a derivation of a certain proposition from a certain set of proposi- tions.

Closed segments emerge if and only if one may apply CdI, NI or PE (cf. Theorem 3-23 and Theorem 3-24). Thus, if a transition is covered by more than one rule, e.g. UE and PE, availabilities change as they do in a transition by PE. Thus, a user of the Speech Act Calculus is restriced in the preformance of certain inferences: For example, one is not free to carry out an assumption-discharging inference by PE as a not assumption-discharging inference by UE.

One may deem that this makes the Speech Act Calculus a bit unhandy, however, this shortcoming, if it is one, comes with the advantage that for every utterance of a sentence sequence by an author, we can uniquely determine if that author has uttered a derivation of a certain proposition from a certain set of propositions: The possibility to describe the utterance of one and the same sentence sequence in different ways so that, for example

the utterance of a sentence sequence can be described as an utterance of a derivation of

Γ from X and can also described as the utterance of a sentence sequence that is not a derivation of Γ from X, which exists for some calculi, does not exist for the Speech Act Calculus. If one utters derivations in accordance with the rules of the Speech Act Calcu- lus, one does not have to use graphical means for the marking of subderivations nor meta- theoretical rule- or dependence-commentaries: In the framework of the Speech Act Cal- culus utterances of sentence sequences are not up for interpretation.

Now, we will introduce the deductive consequence concept and some other usual meta- logical concepts. In ch. 4, we will then prove some properties of the deductive conse- quence relation, such as reflexivity, transitivity and closure under introduction and elimi- nation. Subsequently, in ch. 6, we will then provide an adequacy proof for the calculus relative to the classical model-theoretic consequence relation. This relation itself will be established in ch. 5. Now, for the definition of the consequence relation:

Definition 3-21.Deductive consequence relation

X Γ iff

X⊆ CFORM and there is an such that

(i) is a derivation of Γ from AVAP( ), and

(ii) AVAP( ) ⊆X.

With Theorem 3-9-(iii), it then follows, as usual, that for X⊆ CFORM it holds that X

Γ if and only if there is a finite Y⊆X such that Y Γ. From this and Definition 3-23, it then follows that X is consistent if and only if all finite Y⊆ X are consistent, and, with Definition 3-24, that X⊆ CFORM is inconsistent if and only if there is a finite Y⊆ X such that Y is inconsistent. Under Definition 3-20, the following theorem is equivalent to Definition 3-21:

Theorem 3-12.Γ is a deductive consequence of a set of propositions X if and only if there is a

non-empty RCS-element such that Γ is the conclusion of and AVAP( ) ⊆X

X Γ iff X⊆ CFORM and there is ∈ RCS\{∅} such that Γ = C( ) and AVAP( ) ⊆X.

Definition 3-22.Logical provability