Synthetic Tableaux with Unrestricted Cut for FirstOrder Theories
Abstract
:Cut? Don’t eliminate, introduce!
What is the method of synthetic tableaux?
1. The Method of Synthetic Tableaux for $\mathsf{CPL}$
2. System KI for $\mathsf{CPL}$
3. Completeness Proof with Respect to the Axiomatic Account of $\mathsf{CPL}$
4. Synthetic Tableaux and Other Deductive Systems for $\mathsf{CPL}$: A Note on Relative Complexity
 Gentzen system with cut ${\equiv}_{p}$ Natural Deduction ${\equiv}_{p}$ Frege systems
 Resolution ${<}_{p}$ any system from (1)
 Cutfree Gentzen system ${<}_{p}$ any system from (1)
5. The FirstOrder Case
 propositional connectives: $\neg ,\wedge ,\vee ,\to $;
 infinite set of variable symbols; we use $x,y,z,\dots $ as metasymbols for variables;
 quantifiers $(\exists x),(\forall x)$;
 function symbols of arbitrary arities; $f,g,h,\dots $ are used as metasymbols, function symbols of arity 0 are called constant symbols; and
 relation symbols of arbitrary arities; $P,Q,R,\dots $ are used as metasymbols.
 If “$A(x)$” and “$A(t)$” are written in the same context, this means that $A=A(x)$ is a formula and that $A(t)$ is $A(t/x)$; using “$A(x)$” neither presupposes that x occurs free in A nor that it occurs in A at all.
 If “$A(t)$” and “$A(s)$” are written in the same context, then this is to mean that A is a formula, x is a variable and $A(t)=A(t/x),A(s)=A(s/x)$.
 $\mathcal{M}=\langle M,{f}^{\mathcal{M}}\rangle $ for an interpretation of ${\mathcal{L}}_{\mathsf{FOL}}$, where M is the domain of $\mathcal{M}$ and ${f}^{\mathcal{M}}$ is the interpreting function; and
 $\sigma $, ${\sigma}^{*}$ for object assignments, that is, mappings from the set of variables to the domain M of $\mathcal{M}$.
5.1. Axiomatic System
 11.
 $A(t)\to (\exists x)A(x)$; and
 12.
 $(\forall x)A(x)\to A(t)$.
$$\frac{C\to A(x)}{C\to (\forall x)A(x)}GC$$

$$\frac{A(x)\to C}{(\exists x)A(x)\to C}GA$$

5.2. SyntheticTableaux System
$$\frac{A(t)}{(\exists x)A(x)}{\mathbf{r}}_{\exists}$$

$$\frac{\neg A(t)}{\neg (\forall x)A(x)}{\mathbf{r}}_{\neg \forall}$$

 UG1
 If ${\mathcal{T}}^{*}$ is a subtableau of $\mathcal{T}$ such that every open branch of ${\mathcal{T}}^{*}$ ends with formula $C\to A(x)$, where x does not occur freely in C, and no formula in ${\mathcal{T}}^{*}$ has been synthesized with the use of a premise which is not on ${\mathcal{T}}^{*}$, then $\mathcal{T}$ may be extended by adding $C\to (\forall x)A(x)$ to each open branch of ${\mathcal{T}}^{*}$.
 UG2
 If ${\mathcal{T}}^{*}$ is a subtableau of $\mathcal{T}$ such that every open branch of ${\mathcal{T}}^{*}$ ends with formula $A(x)\to C$, where x does not occur freely in C, and no formula in ${\mathcal{T}}^{*}$ has been synthesized with the use of a premise which is not on ${\mathcal{T}}^{*}$, then $\mathcal{T}$ may be extended by adding $(\exists x)A(x)\to C$ to each open branch of ${\mathcal{T}}^{*}$.
 the bold proviso
 if a formula lying on a branch of ${\mathcal{T}}^{*}$ gets here by a local rule, then the premises necessary to derive it precede it on the same branch of ${\mathcal{T}}^{*}$
 1.
 If a formula of the form $C\to A(x)$, where x is not free in C, has a proof in the STsystem for $\mathsf{FOL}$, then $C\to (\forall x)A(x)$ has it as well (rule GC).
 2.
 If a formula of the form $A(x)\to C$, where x is not free in C, has a proof in the STsystem for $\mathsf{FOL}$, then $(\exists x)A(x)\to C$ has it as well (rule GA).
 3.
 If A and $A\to B$ have proofs in the STsystem for $\mathsf{FOL}$, then there is also a proof of B (rule MP).
 UG1^{0}
 If $\mathcal{T}$ is a proof of formula $C\to A(x)$ in STsystem, where x does not occur freely in C, then each open branch of $\mathcal{T}$ may be extended with $C\to (\forall x)A(x)$. The result is a proof of $C\to (\forall x)A(x)$.
 UG2^{0}
 If $\mathcal{T}$ is a proof of formula $A(x)\to C$ in STsystem, where x does not occur freely in C, then each open branch of $\mathcal{T}$ may be extended with $(\exists x)A(x)\to C$. The result is a proof of $(\exists x)A(x)\to C$.
5.3. Derivability of Universal Generalization
 UG
 If $\mathcal{T}$ is a proof of formula $A(x)$ in the STsystem for $\mathsf{FOL}$, then each open branch of $\mathcal{T}$ may be extended by adding $(\forall x)A(x)$. The result is a proof of $(\forall x)A(x)$ in the system.
5.4. System KI for $\mathsf{FOL}$
6. Soundness of the ST System for FOL
7. Some Further Remarks on Relations between the STSystem and the Axiomatic System
1.  $(\forall x)(A\to B)\to (A\to B)$  Axiom 12 
2.  $((\forall x)(A\to B)\to (A\to B))\to ((\forall x)(A\to B)\wedge A\to B)$  Thesis of $\mathsf{FOL}$ 
3.  $(\forall x)(A\to B)\wedge A\to B$  MP:2,1 
4.  $(\forall x)(A\to B)\wedge A\to (\forall x)B$  GC:3 
5.  $((\forall x)(A\to B)\wedge A\to (\forall x)B)\to ((\forall x)(A\to B)\to (A\to (\forall x)B))$  Thesis of $\mathsf{FOL}$ 
6.  $(\forall x)(A\to B)\to (A\to (\forall x)B)$  MP:5,4 
8. STSystems for FirstOrder Theories
8.1. Universal Axioms
$$\frac{}{{B}_{1}\vee \cdots \vee {B}_{m}}({R}_{ax}^{\varnothing 1})$$

$$\frac{}{{B}_{1}\mid \cdots \mid {B}_{m}}({R}_{ax}^{\varnothing 1*})$$

8.2. First Example: Identity
8.3. Second Example: Partial Order
$$\frac{}{x\le y\vee y\le x}({R}_{li{n}_{\le}}^{\varnothing 1})$$

$$\frac{}{x\le y\mid y\le x}({R}_{li{n}_{\le}}^{\varnothing 1*})$$

9. Conclusions
Author Contributions
Funding
Conflicts of Interest
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$\frac{\mathtt{t}(C)}{\mathtt{t}(B\to C)}\mathtt{It}\to 2$  $\frac{\mathtt{t}(C)}{\mathtt{t}(B\vee C)}\mathtt{It}\vee 2$  $\frac{\mathtt{f}(C)}{\mathtt{f}(B\wedge C)}\mathtt{If}\wedge 2$  $\frac{\mathtt{f}(B)}{\mathtt{t}(\neg B)}\mathtt{It}\neg $ 
2.  $(A\to B)\to ((A\to (B\to C))\to (A\to C))$  7.  $\neg \neg A\to A$  
3.  $A\to A\vee B$  8.  $A\wedge B\to A$  
4.  $B\to A\vee B$  9.  $A\wedge B\to B$  
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LeszczyńskaJasion, D.; Chlebowski, S. Synthetic Tableaux with Unrestricted Cut for FirstOrder Theories. Axioms 2019, 8, 133. https://doi.org/10.3390/axioms8040133
LeszczyńskaJasion D, Chlebowski S. Synthetic Tableaux with Unrestricted Cut for FirstOrder Theories. Axioms. 2019; 8(4):133. https://doi.org/10.3390/axioms8040133
Chicago/Turabian StyleLeszczyńskaJasion, Dorota, and Szymon Chlebowski. 2019. "Synthetic Tableaux with Unrestricted Cut for FirstOrder Theories" Axioms 8, no. 4: 133. https://doi.org/10.3390/axioms8040133