In addition to being able to represent facts, or real-
world statements, as formulas, we want to be able to
manipulate facts, e.g., derive new facts from a set of
statements. For example, if we know that
If it is raining, my dog is wet.
If my dog is wet, there will be water on the floor.
There is no water on the floor.
We should be able to conclude that it is not raining.
In propositional form, this argument would look like:
P -> Q
Q -> R
Is there a formal (a systematic, unambiguous way) of
deriving new facts? We could have a set of rules that
are apply to formulas that can be used to create
(derive) new formulas which follow the original
ones. One such rule is Modus Ponens:
P -> Q
Another such rule is called Modus Tollens:
P -> Q
We could use two applications of Modus Tollens to
show that it is not raining.
In fact, there are other rules, some very obvious, so
not so obvious, that we use to derive new facts from
old ones. Three obvious ones deal with conjunction:
P P∧Q P∧Q
Q P Q
In the previous slide, the blue box introduces an
additional assumption which is at the top of the box.
The scope of the assumption is within the box. After
the box, the formula is no longer assumed to be true
and may not be used. There may be additional
formulas inside the box between the first and last
lines. Also, boxes may be nested to introduce several
Two rules involve implication. The second is MP:
Q P -> Q
P -> Q Q
false ⌐P false ⌐⌐P
⌐P false P P
In general, for each operator there are two (sets of)
rules: an introduction rule which introduces the
operator, that is, the operator appears in the
conclusion, and an elimination rule which removes
an operator, that is, the operator appears in one of the
hypotheses. The is one conjunction introduction rule,
and two conjunction elimination rules, one
disjunction elimination rule (proof by cases) and two
disjunction introduction rules.
Rules Names (cont'd)
Implication elimination is the Modus Ponens rule and
there is a corresponding implication introduction
(assume that P is true, prove Q, and you have the
implication P->Q. The other four rules are negation
introduction, negation elimination, contradiction
elimination, and double negation elimination.
In all, there are twelve rules (Modens Tollens can be
derived from the other rules). These rules codify one
approach that people take toward deduction (hence
the name Natural Deduction). A natural deduction
proof is a sequence of steps where each step is either
a hypothesis (formula assumed to be true) or is
derived from previous formulas by one of the rules.
Here is a proof that if we have hypothesis P->Q, and
Q->R, we can derive the formula P->R:
1. P ->Q Hyp.
2. Q-> R Hyp.
3. P Assume
4. Q MP
5. R MP
6. P -> R Implication intro
Modus Tollens is a derived rule, that is, any proof
which uses Modus Tollens could be done by using
the other rules instead. To see this, consider:
1. P -> Q
3. P Assume
4. Q -> Elim: 3,1
5. false ⌐ Elim: 4, 2
6. ⌐P ⌐ Intro: 3-5
Since we have a system of natural deduction, we
have a way to derive new facts from a set of given
facts (hypotheses). We construct a proof which each
line is either a hypothesis or derived from previous
lines by a proof rule. The last line of the proof is the
conclusion. If such a proof exists, we say that the
conclusion C can be derived from the hypotheses H,
H ⊢ C (read “H derives C.”)
We also have a way of relating a formula to a set of
formulas which we assume to be true (the
hypotheses). If every interpretation which makes the
hypotheses true (satisfies the hypotheses), also makes
the conclusion true, i.e., whenever the hypotheses are
true, the conclusion is also true, we say the the
hypotheses semantically entails the conclusion, H ⊨
Tautologies and Contradictions
A formula that is true under every interpretation is
said to be valid and is also called a tautology.
A formula that is false under every interpretation is
call a contradiction and said to be unsatisfiable.
A formula that is true under some interpretation is
said to be satisfiable.
Proof vs. Truth
In short, we have two notions: Proof and Truth. We
have facts that we can prove from the set of
hypotheses (using the rules of natural deduction) and
we have facts that we know are true given that the
hypotheses are true (which can check by constructing
the truth table). It turns out that for the natural
deduction method for proposition calculus, these two
Soundness and Completeness
If we can prove formula C from the set of hypotheses
H, then H must also semantically entail C. In other
words, if we can prove it, it must be true. This is
known as soundness.
If C is semantically entailed from H, then there is a
proof of C from hypotheses H. In other words, if it is
true, we can prove it. This is known as completeness.
Are All Systems Sound and/or
No. Consider a system with one rule:
that is, you can prove anything. Such a system is
complete (if it's true you can prove it because you can
prove anything), but it is not sound (you can prove
things that aren't true).
To construct a system that is not sound, delete one of
the twelve rules of natural deduction.