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- - Logic Summer School in Kobe Aug. 29, 2011 Outline 1 2 3 4 5 6 7 8 9 M ϕ M |= ϕ Example 1 R |= ∀x(x , 0 → 0 < x2), 2 C |= ∃x(x2 = −1). M ϕ M |= ϕ Example 1 R |= ∀x(x , 0 → 0 < x2), 2 C |= ∃x(x2 = −1). Example (R, 0, +, −) R (R, 0, 1, +, −, ·) R (R, 0, 1, +, −, ·, <) R Example (R, 0, +, −) R (R, 0, 1, +, −, ·) R (R, 0, 1, +, −, ·, <) R Example (R, 0, +, −) R (R, 0, 1, +, −, ·) R (R, 0, 1, +, −,