Extended Game-Theoretical Semantics

Abstract

A new version of Game-Theoretical Semantics (GTS) is put forward where game rules are extended to the non-logical constants of sentences. The resulting theory, together with a refinement of our criteria of identity for functions, provide the technical basis for a game-based conception of linguistic meaning and interpretation.

Appendices

Appendix 1: eGTS Rules

In eGTS, every FO sentence φ evaluated relative to a structure $\mathbf{M}=\langle D,I \rangle$ is associated to a game $eG(\varphi,\mathbf{M})$. This new game is identical to the game $G(\varphi,\mathbf{M})$ played according to the standard GTS rules when it is a molecular game, i.e. when φ is a complex formula. If φ is an atomic formula or an identity, one reaches an atomic game. The original game rule for atomic sentences in GTS, (R.At), is replaced by the following specific four rules:

• (R.At ^*) In the atomic game $eG(\alpha,\mathbf{M})$, if either ∃loise is the current verifier and α is false in M, or she is the current falsifier and α is true, then ∀belard wins and ∃loise loses; else there are two cases:

  • (i) α is of the form $Rt_1\ldots t_n$, R being a n-ary relation symbol and the t _i s being terms: ∀belard picks out an index $i \in \{0, 1, \ldots,n\}$; if i = 0 then the rest of the game is as in $eG(R,\mathbf{M})$, else it is as in $eG(t_i,\mathbf{M})$.

  • (ii) α is of the form $(t_1 = t_2)$, the t _i s being terms: ∀belard picks out an index $i \in \{1,2\}$; the rest of the game is as in $eG(t_i,\mathbf{M})$.

• (R.Rel) In the relation game $eG(R,\mathbf{M})$, where R is a n-ary relation symbol, ∀belard chooses a n-tuple $\langle d_1, \ldots, d_n\rangle \in D^n$, and ∃loise answers $\mathtt{yes}$ or $\ mathtt{no}$. If she answers $\ mathtt{yes}$ and $\langle d_1, \ldots, d_n \rangle \in I(R)$, or she answers $\mathtt{no}$ and $\langle d_1, \ldots, d_n \rangle \notin I(R)$ then ∃loise wins and ∀belard loses; else ∀belard wins and ∃loise loses.

• (R.Term) In the term game $eG(t,\mathbf{M})$, where t is a term, one of the following three cases occurs:

  • (i) t is a new constant previously introduced through the play i.e. not occurring in the original formula; then ∃loise wins and ∀belard loses;

  • (ii) t is an individual constant; then ∃loise chooses an object $d\in D$; if $d=I(t)$ then ∃loise wins and ∀belard loses; if $d\neq I(t)$ vice versa;

  • (iii) t is a complex term involving a n-ary function symbol: $t=f(t_1, \ldots, t_n)$; then ∀belard picks out an index $i\in\{0,1,\ldots,n\}$. If $i > 0$ then rest of the game is as in $eG(t_i,\mathbf{M})$, and if i = 0 it is as in $eG(f,\mathbf{M})$.

• (R.Fun) In the function game $eG(f,\mathbf{M})$, f being a n-ary function symbol, ∀belard chooses a n-tuple $\langle d_1, \ldots, d_n\rangle \in D^n$, then ∃loise chooses an object $d\in D$. If $d=I(f)(d_1, \ldots, d_n)$ then ∃loise wins and ∀belard loses; else vice versa.

Appendix 2: Second-Order Skolem Forms

There is a simple device to yield a result equivalent to extended GTS with no resort to (sub)atomic games, but only to skolemization. In what follows we will assume that the models under consideration contain at least two distinct elements, and that second-order formulas get a standard (full) semantic interpretation. Let:

$$\Phi(a_1, \ldots, a_k, R_1, \ldots, R_l, x_1, \ldots, x_m)$$

((9.11))

be a first-order sentence, where $a_1, \ldots, a_k$ are the k individual constant symbols, $R_1, \ldots, R_l$ the l relation symbols, and $x_1, \ldots, x_m$ the m variables occurring inside Φ. Being first-order Φ can be put into prenex normal form:

$$Q_1 x_1\ldots Q_m x_m\Phi^\otimes(a_1, \ldots, a_k, R_1, \ldots, R_l, x_1, \ldots, x_m)$$

((9.12))

with $Q_i \in \{\exists, \forall\}$. Φ can also be skolemized, and is equivalent to:

$$\begin{array}{l} \mathbf{2Sk}[\Phi]=\exists f_1\ldots\exists f_n \forall t_1\ldots\forall t_p\\ \,\hspace{1.5cm} \Phi^\circ(a_1, \ldots, a_k, R_1, \ldots, R_l,f_1(\vec{t_1}), \ldots, f_n(\vec{t_n}), t_1,\ldots, t_p) \end{array}$$

((9.13))

where $\{t_1,\ldots, t_p\}\subseteq\{x_1, \ldots, x_m\}$ is the subset of the universally quantified variables of $\Phi^\otimes$, $\vec{t_i}\subseteq\{t_1,\ldots, t_p\}$ is the set of the universally quantified variables of Φ^⊗ on which the existentially quantified variable replaced by f _i depends, and Φ^° results from Φ^⊗ by a mere permutation of its arguments so that the Skolem functions appear first.

Now we can replace the relation and individual constant symbols in Φ by second-order quantified variables. So Φ is equivalent to the following formula:

$$\begin{array}{l} \exists g_1\ldots\exists g_k \exists X_1\ldots\exists X_l [\Phi(g_1, \ldots, g_k, X_1, \ldots, X_l, x_1, \ldots, x_m)\\ \wedge [g_1=a_1]\wedge\ldots\wedge[g_k=a_k] \wedge [X_1=R_1]\wedge\ldots\wedge[X_l=R_l]] \end{array}$$

((9.14))

The same existential generalization can be done within 2SkΦ, reaching the following formula:

$$\begin{array}{l} \exists g_1\ldots\exists g_k \exists X_1\ldots\exists X_l\exists f_1\ldots\exists f_n \forall t_1\ldots\forall t_p\\ \,\hspace{1.5cm} [\Phi^\circ(g_1, \ldots, g_k, X_1, \ldots, X_l,f_1(\vec{t_1}), \ldots, f_n(\vec{t_n}), t_1,\ldots, t_p)\\ \,\hspace{1.5cm} \wedge [g_1=a_1]\wedge\ldots\wedge[g_k=a_k] \wedge [X_1=R_1]\wedge\ldots\wedge[X_l=R_l]] \end{array}$$

((9.15))

Furthermore, let us change each relation variable X _i into a corresponding (indicator) function variable h _i and correlatively modify Φ^° into Φ^⋆—so that each token of $X_i t_1 \ldots t_{n_i}$ be replaced by one of $h_i (t_1\ldots t_{n_i})=1$. Hence Φ is equivalent to its extended second-order Skolem form:

$$\begin{array}{l} \mathbf{2eSk}[\Phi]_1\;=\; \exists g_1\ldots\exists g_k\exists h_1\ldots\exists h_l \exists f_1\ldots\exists f_n \forall t_1\ldots\forall t_p\,\hspace{2cm} \\ \,\hspace{2cm} [\Phi^\star(g_1, \ldots, g_k, h_1, \ldots, h_l,f_1(\vec{t_1}), \ldots, f_n(\vec{t_n}), t_1,\ldots, t_p)\\ \,\hspace{2cm} \wedge [g_1=a_1]\wedge\ldots\wedge[g_k=a_k] \wedge [h_1=\mathbf 1_{R_1}]\wedge\ldots\wedge[h_l=\mathbf 1_{R_l}]] \end{array}$$

((9.16))

A similar first-order skolemization is then given by:

$$\begin{array}{l} \mathbf{eSk}[\Phi]\;=\;\forall t_1\ldots\forall t_p\,\\ \hspace{0,2cm} [\Phi^\star(\mathbf{g}_1, \ldots, \mathbf{g}_k, \mathbf{h}_1, \ldots, \mathbf{h}_l,\mathbf{f}_1(\vec{t_1}), \ldots, \mathbf{f}_n(\vec{t_n}), t_1,\ldots, t_p)\\ \,\hspace{2cm} \wedge [\mathbf{g}_1=a_1]\wedge\ldots\wedge[\mathbf{g}_k=a_k] \wedge [\mathbf{h}_1=\mathbf 1_{R_1}]\wedge\ldots\wedge[\mathbf{h}_l=\mathbf 1_{R_l}]] \end{array}$$

((9.17))

Another second-order Skolem form, equivalent to (9.16), obtains if we restore the original constants in Φ^⋆:

$$\begin{array}{l} \mathbf{2eSk}[\Phi]_2\;=\;\exists g_1\ldots\exists g_k\exists h_1\ldots\exists h_l \exists f_1\ldots\exists f_n \forall t_1\ldots\forall t_p\,\hspace{2cm} \\ \,\hspace{1.5cm} [\Phi^\circ(a_1, \ldots, a_k, R_1, \ldots, R_l,f_1(\vec{t_1}), \ldots, f_n(\vec{t_n}), t_1,\ldots, t_p)\\ \,\hspace{1.5cm} \wedge [g_1=a_1]\wedge\ldots\wedge[g_k=a_k]\\ \,\hspace{1.5cm} \wedge [h_1=\mathbf 1_{R_1}]\wedge\ldots\wedge[h_l=\mathbf 1_{R_l}]] \end{array}$$

((9.18))

which can be written in a condensed form as:

$$\begin{array}{ll} \mathbf{2eSk}[\Phi]_2=&\exists g_1\ldots\exists g_k\exists h_1\ldots\exists h_l\\ &\,[\mathbf{2Sk}[\Phi] \wedge [g_1=a_1]\wedge\ldots\wedge[g_k=a_k] \wedge\\ &\,[h_1=\mathbf 1_{R_1}]\wedge\ldots\wedge[h_l=\mathbf 1_{R_l}]] \end{array}$$

((9.19))