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Exercise 5.4
1. |
In the case of each of the following sequents, say whether it is correct, and, if it is incorrect, give a counterexample. | |
| (i) | ["xFxÙ"xGx] |= "x[FxÙGx] | |
| (ii) | "x[FxÙGx] |= ["xFxÙ"xGx] | |
| (iii) | ["xFxÚ"xGx] |= "x[FxÚGx] | |
| (iv) | "x[FxÚGx] |= ["xFxÚ"xGx] | |
| (v) | [$xFxÙ$xGx] |= $x[FxÙGx] | |
| (vi) | [$xFxÚ$xGx] |= $x[FxÚGx] | |
| (vii) | "xFxx |= $xFxx | |
| (viii) | "xFxa |= $xFxa | |
| (ix) | "x[Fx®Gx] |= $x[FxÙGx] | |
| (x) | "x[Fxa®Gxa] |= $x[FxaÙGxa] | |
| (xi) | $xFx, "x[Fx®Gx], "x[Fx®¬Gx] |= | |
| (xii) | $xFx, "x$y[FyÙRxy] |= $y"x[FyÙRxy] | |
| (xiii) | $xFx, "x$y[FxÙGy] |= $y"x[FxÙGy] | |
| (xiv) | $xFx, ["xFx®P] |= "x[Fx®P] | |
| (xv) | $xFx, ["xFx®P] |= $x[Fx®P] | |
| (xvi) | [$xFx®P] |= "x[Fx®P] | |
| (xvii) | "x[P®Fx] |= [P®"xFx] | |
2. |
Show that the following sequents are incorrect by finding counterexamples. | |
| (i) | ["xFx®$xGx] |= "x[Fx®Gx] | |
| (ii) | "x[Fx®$yGyx] |= [$xFx®$x"yGxy] | |
| (iii) | "x[[Fx®Gx]Ú[¬Fx®Hx]] |= "x[GxÚHx] | |
| (iv) | "x"y$zFxyz, $x"yFxyx |= "x"y$zFyzx | |
| (v) | $x"y[FxyÙFyx], "x[Gx®"y[Fyx«Gy]] |= "xGx | |
| (vi) | $x$yRxy, "x"y"z[[RxyÙRyz]®Rzx], "x"y"z[[RxyÙRxz]®Ryz] |= "xRxx | |
| (vii) |
"x[Fx®"y[Gy®Rxy]], "x[HxÚ¬$y[FyÙRxy]], "x"y[Rxy®Ryx] |= $x[Fx®"y[Gy®Hy]] |
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| (viii) |
"x"y[Fxy®¬"z[Gz®Hzy]], "x"y$z[¬Hzx®Fyx] |= "x"y[Fxy«$z[GzÙ¬Hzy]] |
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