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Truth-tables
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The best of all possible worlds
I
In the Discourse on Metaphysics (1686) Leibniz argues that
this world is “the best of all possible worlds”.
I
A response to the problem of evil.
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The best of all possible worlds
I
(1) God is perfect in every respect.
I
(2) If this is not the best of all possible worlds, then God is
imperfect in some respect.
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(3) So, this is the best of all possible worlds.
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“God has chosen the most perfect world, that is, the one
which is at the same time the simplest in hypotheses and
the richest in phenomena, as might be a line in geometry whose construction is easy and whose properties and
effects are extremely remarkable and widespread.”
[Leibniz (1686), Discourse,§6]
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Leibniz and possible worlds
“I call ‘World’ the whole succession and the whole agglomeration
of all existent things” (Theodicy, 1710)
“there is an infinity of possible ways in which to create the world,
according to the different designs which God could form” (Letter
to Arnauld, 1686)
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Tractatus Logico-Philosophicus
“Die Welt ist alles, was der Fall ist.” (Wittgenstein, 1922)
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Tractatus Logico-Philosophicus (1922)
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1. The world is all that is the case.
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1.1 The world is the totality of facts, not things.
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1.13 The facts in logical space are the world.
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2. What is the case, the fact, is the existence of atomic facts.
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5. Propositions are truth-functions of elementary propositions.
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5.123 If a god creates a world in which certain propositions are true,
then by that very act he also creates a world in which all the
propositions that follow from them come true. And similarly he
could not create a world in which the proposition ‘p’ was true
without creating all its objects.
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5.6 The limits of my language mean the limits of my world.
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7. Whereof one cannot speak, thereof one must be silent.
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Imagine that I’m about to throw two dice, thus displaying two numbers
face up. As a competent speaker of English you know in which of the
following situations this sentence would be true:
The dice add up to eleven.
a.
b.
c.
d.
Out of the thirty-six possible states of the pair of dice—it is only true in
situations b and d. The set of possible situations provides the truth
conditions: {b, d}
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“We all learned in school how to compute the probabilities
of various events. . . Now in doing these school exercises in
probability, we were in fact introduced at a tender age to a
set of (miniature) ‘possible worlds’. The thirty-six possible
states of the dice are literally thirty-six ‘possible worlds’. . . ”
(Kripke 1980, Naming and Necessity ).
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Worlds
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Let S be the set of all sentence letters = {P, Q, R, . . . }
I
Let W be the set of all worlds = S × {T , F }
W =
w1
w2
w3
...
P
T
T
T
...
Q
T
T
F
...
R
T
F
F
...
...
...
...
...
...
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A plurality of worlds
possible worlds:
A possible world w is a complete and possible scenario, i.e.
it is a way that the world (viz. the entire universe) could
be. There are infinitely many possible worlds, since there
are infinitely many ways that the world could be. Yet, not all
scenarios are possible, e.g. there are no possible worlds where
both P and ˜P are true.
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There are an infinite number of logically possible worlds, but
they fall into two classes.
I
One class consists of situations in which P is true, and the
other consists of situations in which P isn’t true.
I
In any situation in the first class, P ∨ ˜P is true because it is
a disjunction whose first disjunct is true.
I
In any situation in the second class, P is not true in that
situation, and so its negation, ˜P is true, and again P ∨ ˜P is
true because it is a disjunction which has a true disjunct.
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So in either class of situations P ∨ ˜P is true, thus P ∨ ˜P no
matter which world is actual.
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P (P ∨ ∼P)
T T T F
F F T T
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Tautology: A sentence is a tautology if no matter what
truth values are assigned to its simple parts, the definitions of the connectives used in the sentence determine
that the sentence is true.
Notice: every theorem of L2 is a tautology and every tautology can
be derived by the rules from no premises.
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Truth tables
[Ludwig Wittgenstein (1921) Tractatus Logico-Philosophicus]
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P
T
T
F
F
Q (¬P ∧ Q)
T F F T
F F F F
T T T T
F T F F
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P
T
T
F
F
Q ((P ∨ Q) → P)
T
F
T
F
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