Tutorial Contents Tutorial Five: Predicate Calculus: Tableaux - The $xj rule - Bad Mistakes and Technical Mistakes - Ticking - Finished Tableaux - Sequents - Soundness and Completeness - Proof of a syntactic sequent: example
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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) ["xFx"xGx] |= "x[FxGx]
  (ii) "x[FxGx] |= ["xFx"xGx]
  (iii) ["xFx"xGx] |= "x[FxGx]
  (iv) "x[FxGx] |= ["xFx"xGx]
  (v) [$xFx$xGx] |= $x[FxGx]
  (vi) [$xFx$xGx] |= $x[FxGx]
  (vii) "xFxx |= $xFxx
  (viii) "xFxa |= $xFxa
  (ix) "x[FxGx] |= $x[FxGx]
  (x) "x[FxaGxa] |= $x[FxaGxa]
  (xi) $xFx, "x[FxGx], "x[FxGx] |=
  (xii) $xFx, "x$y[FyRxy] |= $y"x[FyRxy]
  (xiii) $xFx, "x$y[FxGy] |= $y"x[FxGy]
  (xiv) $xFx, ["xFxP] |= "x[FxP]
  (xv) $xFx, ["xFxP] |= $x[FxP]
  (xvi) [$xFxP] |= "x[FxP]
  (xvii) "x[PFx] |= [P"xFx]

2.

Show that the following sequents are incorrect by finding counterexamples.
  (i) ["xFx$xGx] |= "x[FxGx]
  (ii) "x[Fx$yGyx] |= [$xFx$x"yGxy]
  (iii) "x[[FxGx][FxHx]] |= "x[GxHx]
  (iv) "x"y$zFxyz, $x"yFxyx |= "x"y$zFyzx
  (v) $x"y[FxyFyx], "x[Gx"y[FyxGy]] |= "xGx
  (vi) $x$yRxy, "x"y"z[[RxyRyz]Rzx], "x"y"z[[RxyRxz]Ryz] |= "xRxx
  (vii)

"x[Fx"y[GyRxy]], "x[Hx$y[FyRxy]], "x"y[RxyRyx]

       |= $x[Fx"y[GyHy]]

  (viii)

"x"y[Fxy"z[GzHzy]], "x"y$z[HzxFyx]

        |= "x"y[Fxy$z[GzHzy]]

 

 
Answers

 

 

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