How do we use the atomic propositions. By means of a glossary.
eg. P: Lorenz is a logician
Q : Lorenz loves Marguax
As long as it does not contain any connectives we can put basically anything into an atomic proposition.
So for any situation where we can say that P is true, Lorenz is a logician, but in any situation where P is not true, Lorenz is not a logician.
P^-Q: Lorenz is a logician and he does not love Margaux.
In any situation where Q is false, -Q is true.
In a situation where P is true and -Q is true, (P^-Q) is true.
Determinancy: given a possible situation S, every wff is either true in S or false in S.
Thus things like the liar paradox does not apply.
We assume that notions are so clear that for any situation whatsoever, the sentence is the case.
Language → World
Formal Language → Formal Model of the world
Thus, we do not have to philosophise about mathematical definitions regarding something that lacks direct connection (about connection) with the real world.
α ⊨ M
Language is in the world, but also talks about the world. In formal language however, it lacks this quality.
There are 2 truth values: T and F; or in this class, 1 and 0.
A valuation of wff’s assings to each of those wff’s exactly one truth value.
Atomic propositions are independent of each other.
If α is true and β is true then (α^β) is true as well
In all other cases (α^β) is false
(α^β)
1 0 → 0
1 1 → 1
0 0 → 0
0 1 → 0
(αvβ)
1 0 → 1
1 1 → 1
0 0 → 0
0 1 → 1
(α⊕β)
1 0 → 1
1 1 → 0
0 0 → 0
0 1 → 1
α -α
1 0 → 0
0 1 → 1
-((PvQ)^-(-QvR))
-
P :=1
-
P:= 1
-
P:= 0
-
(PvQ):=1 (from 1, 2, truth table for v)
5.-Q:=0 (from 2, truth table for -)
6.(-QvR):=0 (from 3, 5, truth table for v)
7.-(-QvR):=1 (from 6, truth table for -)
8.((PvQ)^-(-QvR)):= 1
-
-((PvQ)^-(-QvR)):= 0
Thus we can also create semantic trees, as well as syntactic trees in much the same manner.
The semantic tree shows us that the formula has an unambiguous, unique meaning.
Both trees are perfectly parallel.
Assign truth values to formulas and not the other way around.
When there does not exist a unique syntactic tree for some proposition, ie. One that is not well-formed, we cannot really do much with it.
”A string of symbols (syntactic criterion) is well-formed iff it has a unique truth value (semantic criterion).”
”An argument is syntactically valid iff it is semantically valid.”
Once a valuation has been specfiied for the atoms, it is specified fo rall wff’s
we can compute this using a derivation or a semantic tree
But usually we will do more economically, in truth tables
consider the valuation M: {P:= 1, Q:=1, R:=0}
P Q R -((PvQ)^-(-QvR)):
1 1 0 → 1 1 1 1 010 0
1
→ first negation leads to a 0
Remember to note in what order you denote the truth values
The dysjunction of 0 and ”don’t know” is 1, or ”don’t know” all the way upwards depending on the theory you use.
The connectives are truth-functional; they rely entirely on the meaning α and β, and thus are exhaustively explained by truth-tables.
Most connectives in natural language are not truth-functional
eg. α because β