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Consider the following interpretation:

Domain = {1, 2}

Assignment of constants: a = 1 and b = 2

Assignment of functions: f(1) = 2 and f(2) = 1

Assignment for predicate P: P(1, 1) = T; P(1, 2) = T; P(2, 1) = F; P(2, 2) = F

Evaluate the truth value of following formulas in the above interpretation:

a. P(a, f(a)) ∧ P(b, f(b))

b. (∀x)(∃y)P(y, x)

c. (∀x)(∀y)(P(x, y) → P(f(x), f(y))

My attempt at solution:

a)

P(1,2) | P(2,1) | P(1,2) and P(2,1)
____________________________________
  T    |   F    |          F

b)

 x |  y  |  P(y,x)
____________________
 1 |  1  |  T
 1 |  2  |  F
 2 |  1  |  T
 2 |  2  |  F

But I'm not sure if this is correct. I haven't seen anyone make a truth table yet using variables in the table. Is that doable? What else do I have to say to evaluate the truth of the entire statement?

c)

 x | y | P(x,y) | P(f(x),f(y) | P(x,y) implies P(f(x), f(y)
____________________________________________________________
 1 | 1 |   T    |     F       |        F
 1 | 2 |   T    |     F       |        F
 2 | 1 |   F    |     T       |        T
 2 | 2 |   F    |     T       |        T

Again, same with b), I don't know if I'm taking the right approach.

Thanks!

dibdub
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  • The "truth table" for $P$ is part of the diagram of the model (interpretation) — the set of all atomic formulas and negations of atomic formulas that are true in the model, the basic facts which determine the truth or falsity of all other more complex sentences. Your approach is correct. For b), the only other thing you have to do is check that for all $x\in {1,2}$ there is a $y\in {1,2}$ such that the row for $x,y$ has T in the last column. – BrianO Jan 27 '16 at 23:29
  • Ok, is there a certain way I need to demonstrate that I checked that? Can I just write below the table that for all x∈{1,2} there is a y∈{1,2} that creates a true statement?

    Do I need to do that same for c) as well?

     – dibdub Jan 27 '16 at 23:35
  • For b), just list the cases — there are only two. For c), it's a universal statement so I'd say no it's not necessary to elaborate further. – BrianO Jan 27 '16 at 23:39
  • Great, thanks for the help! – dibdub Jan 27 '16 at 23:42
  • Actually I retract that: yes it is necessary to say one more thing. The last sentence is just a single universally quantified statement, $\forall x,y[...]$, and it's either true or false in the model. The last column of your table for $[...]$ shows that the universally quantified sentence is false. – BrianO Jan 27 '16 at 23:45

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