Hand Handou outt No. No. 1 Phil. 015 Zoltan Domotor Department of Philosophy, University of Pennsylvania January 29, 2016 DERIVATIONS IN SENTENTIAL LOGIC derivations (proofs) of logical This handout includes a large variety of derivations theorems and argument-forms in the language of sentential logic LSL, using the traditional ten basic rules of derivation or inference (recalled below) and some previously proven proven results. results. We proceed from simple theorems and arguments to increasingly more complex ones that often depend on other theorems theorems and argumen arguments. ts. Feel free to use any of these these ‘solv ‘solved ed problem problems’ s’ in your work by quoting their labels, e.g., Theorem T1,T2, and so on.
Handout No.1 is intended to summarize and simplify the treatment of natural deduction in G. Forbes’ MODERN LOGIC, without altering the scope of the classical sentential logic framework. The basic rules of derivation derivation – an introduction rule rule and a corresponding elim rule for each of the five basic logical connectives ( & , ∨, ↔, →, ∼) ination rule are symbolized in a simple argument-form (or inference-form) as follows: 1. Introduct Introduction ion and Eliminat Elimination ion of Conjuncti Conjunction: on: Introduction of & , symbolized by I & :
(Note (Note that that MODE MODERN RN LOGIC LOGIC uses uses a revers reversed ed notatio notation n & I in which which the connective comes first and the introduction/el introduction/eliminati imination on symbol comes last. This is of course a matter of convention. Feel free to use either notation.) From any formulas P and Q in LSL1 that we may infer their conjunction P & Q: 1
Since, P,Q,R, are sentential letters in the object language LSL, in all rules of inference we should use Greek capital letters Φ, Ψ, , , serving as sentence variables belonging to the metalanguage, so that Φ stands for P or P → Q or P & Q, etc., and likewise for Ψ and . However, for the sake of simplicity, in Hando Handout ut No. 1, the metalanguage notation notation for sentant sentantial ial variables variables will not be used. Of course, you should remember remember that in all rules of inference the symbols P and Q stand any sentence in LSL. · · ·
· · ·
1
P Q P & Q
∴
Elimination of & , symbolized by E & : From any conjunction P & Q in LSL we may infer either conjuncts:
(i)
(ii) P & Q ∴
P & Q
P
∴
Q
In view of the truth-table definition of conjunction, the foregoing rules are intuitively obvious, since by definition of & from a conjunction we can easily infer either conjunct, and from a pair of given premises we can always infer their conjunction. However, we want these rules to be used rigorously in a proof protocol (derivation). A proof of the validity of a formal argument (argument-form) is a finite sequence of consecutively numbered lines, e.g., (1) (2) (3) (7) (illustrated below), each consisting of a formal sentence (formula) together with a list of premise numbers (on the far left) and a list of used rules (on the far right) such that each sentence is introduced according to one of 10 rules of inference. The conclusion of the argument appears on the last line of the sequence in which the premise numbers (on the far left) are only the line numbers of assumptions in the argument. · · ·
To illustrate the application of rules of inference we first rewrite the conjunction and elimination rules into a proof protocol form: a1 , . . . , a a1 , . . . , a
n
n
(i) P & Q ( j ) P
i & E
On the left-hand size of the first row of the rule we have the premisenumbers a1 , . . . , a that indicate how sentence P & Q on line (i) is arrived at. The rule & E or rather E & (elimination of conjunction) together with the line number (i) fully justifies line (j) with sentence P. n
Since this form of presentation of rules may appear to be too sophisticated to some, we shall considerably simplify the natural deduction system and its proof protocols. But first we complete the list of rules for all logical (sentential) connectives.
2
2. Introduction and Elimation of Disjunction: Introduction of ∨, symbolized by I ∨: From any sentence P we may infer the disjunction of P with any sentence Q:
(i)
(ii) P ∴
P
P∨Q
∴
Q∨P
Elimation of ∨, symbolized by E ∨: P∨Q P→R Q→R ∴
R
3. Introduction and Elimination of Biconditional: Introduction of ↔, symbolized by I ↔:
P→Q Q→P ∴
P
↔
Q
Elimination of ↔, symbolized by E ↔:
(i)
(ii) P↔Q ∴
P↔Q
P→Q
∴
Q
→
P
4. Introduction and Elimination of Conditional: Introduction of →:
This introduction rule is called a Conditional Proof and is symbolized by C or I→. Given a derivation of a sentence R from
3
hypothesis2 P and perhaps other premises Q, we may discharge the hypothesis P from the list of supplementary assumptions and infer from premise Q only the conditional conclusion P → R, with the hypothesis P now placed as the antecendent (first term) of the conditional. (i) If under hypothesis P P Q ∴
(ii) then P can be discharged as
H
Q ∴
P
→
R
H
R
We indicate the removal of hypothesis P from the list of supplementary premises by a slashed H : H . Elimination of →:
This elimination rule is called Modus Ponens and is symbolized by MP or E →. P P→Q ∴
Q
5. Introduction and Elimation of Negation: Introduction of ∼:
This rule is called Reductio ad Absurdum and is symbolized by RAA. Given a derivation of a contradiction R & ∼ R from a hypothesis P and possibly other assumptions Q, we may discharge the hypothesis P and infer ∼ P from Q alone: (i) If under hypothesis P P Q ∴
R &
(ii) then we have
H
Q ∴
∼
∼
P
H
R
2
In MODERN LOGIC, Forbes uses the term assumption instead of ‘hypothesis’, and what we call ‘assumptions’ he calls premises . The term ‘hypothesis’ seems to fit the patterns of scientific reasoning better than the term ‘assumption’. Be that as it may, in some cases logicians do not have an agreed-upon universal terminology.
4
Elimation of ∼, symbolized by E ∼: ∼∼ ∴
P
P
A proof or derivation in sentential logic is a finite sequence of consecutively numbered lines, each consisting of a formula in the language textbfLSL of sentential logic together with its line number (on the left) and rules used (on the right), such that on each line the sentence is a given assumption or a temporary hypothesis, or it follows from some previous lines in accordance with any of the ten rules itemized above. If based on assumptions P1 , P2 , . . ., sentence Q appears on the last line in which all premises belong to P1 , P2 , . . ., then the sequence of sentences under consideration is called the derivation (proof) of Q from P1 , P2 , . . .. SOLVED PROBLEMS IN SENTENTIAL LOGIC
T0
⊢
P → P
Proof.
1 P H 2 P → P 1,1 C, H This is about the simplest proof we can have in sentential logic! Here Line 1 serves as both the hypothesis and conclusion (antecendent and consequent of the conditional). The hypothesis is conditionalized out in Line 2. Of course, a sentence validly follows from itself, so that P → P is logically true! Please notice that we use a backslashed H to indicate that we have discharged the hypothesis. This is a useful reminder when there are several hypotheses involved in the proof. Eventually we will simplify even this sort of pedantic proof protocol. Also, observe that (unlike Forbes) we do not list the premise numbers on the far left of each line, and we do not put the line numbering in parantheses. Finally, the single turnstile ⊢ in front of a formal sentence indicates that it is a logically true (i.e., follows from the 10 basic rules without any assumptions). The argument-form P ⊢ P
has the simplest proof: 5
Proof.
1 P A (assumption) Here Line 1 is both the assumption and conclusion. Another triviality is: T1
∼
P → P ⊢ P
Proof.
1 ∼ P → P 2 ∼ P 3 P 4 P & ∼ P 5 ∼∼ P 6 P
A H (for RAA) 1,2 MP,H 2,3 I & ,H 2-4 RAA,H 5, E ¬
Because a hypothesis in sentential logic can be discharged either by conditioning or by RAA, to help the reader, we may indicate the purpose of our hypothesis in parantheses on the line where it is introduced. Notice also that we keep the hypothesis (whether directly relevant or not) on all subsequent lines, until it is discharged. Finally, as a small exercise, please prove the following variant of the foregoing theorem: ∼ P → P ⊢ P . You will need only one hypothesis. Later on, we shall indicate only the introduction of a hypothesis by entering H on the right-hand side of the line of the hypothesis, and then enter symbol H on the line, where the hypothesis is discharged. Sometimes we need two (or more) hypotheses, as in the next logical theorem. T2
⊢
Q
→
(P → Q)
Proof.
1 2 3 4 5 6
Q P Q→Q Q P → Q Q → (P → Q)
H (for C) H’(for C) T0,H, H’ 1,3 MP,H,H’ 2,5 C, H, H 1,5 C, H ′
On Line 3 we have introduced a substitution instance of a previously proven theorem, namely T0 . This is legitimate because after Line 3 we could simply insert the proof of T0 and then continue with the proof as before. This is of course not necessary because we could have copied Line 1 and then conditionalize. The objective here is to show some extra degrees of freedom in proving logical theorems. Finally, observe 6
that the theorem is quite intuitive: Given Q and then given anything else, say P , as antecendents, then the consequent is trivially Q. If there are no assumptions, one may need several hypotheses, as in the theorem below, that must be conditioned out before the proof is complete. T3
⊢
(P → Q) → [(Q → R) → (P → R)]
Proof.
1 2 3 4 5 6 7 8 T4
P → Q Q→R P Q R P → R (Q → R) → (P → R) (P → Q) → [(Q → R)
H (for C) H’(for C) H”(for C) 1,3 MP,H, H’, H” 2,4 MP, H, H’, H” 3,5 C,H, H’, H 2,6 C,H, H 1,7C,H ′
→
(P → R)]
(P & Q) → R ⊢ P → (Q → R) Proof.
1 2 3 4 5 6 7
(P & Q) → R A P Q P & Q R Q→R P → (Q → R)
H(for C) H’(for C) 2,I & ,H, H’ 1,4 MP, H, H’ 3,5 C,H, H 2,6 C, H ′
The foregoing is called the Law of Exportation . Its point is that if assumptions are made conjunctively, then they can be made sequentially. The converse is also true, but we need more ‘assistant’ theorems (or lemmas) before we can prove it. The next theorem illustrates Modus Tollens in the form of a theorem: If you deny the consequent, then you must deny the the antecendent as well, given the conditional P → Q. T5
P → Q, ∼ Q ⊢∼ P
Proof.
1 2 3 4 5 6
P → Q ∼Q P Q Q &∼Q ∼ P
A A H(for RAA) 1,3 MP, H 2,4I & ,H 3-5, RAA, H
7
This theorem gives the popular so-called Contraposition Law , proven basically in the same way. T6
⊢
(P → Q) → (∼ Q →∼ P )
Proof.
1 2 3 4 5 6 7 8
H (for C) H’ (for another C) H”(for RAA) 1,3 MP, H,H’, H” 2,4I & ,H,H’, H” 3-5, RAA, H,H’, H 2,6 C,H, H 1,7 C, H
P → Q ∼Q P Q Q &∼Q ∼ P ∼ Q →∼ P (P → Q) → (∼ Q →∼ P )
′′
′
Upon inspecting the proof you will note that for reasons of proof nesting the hypotheses are discharged in the reversed order: the one introduced last is discharged first. We can think of proofs and their subproofs as boxes within boxes. Let us return to the Law of Contradiction , stating that from contradictory assumptions anything follows. T7
P, ∼ P ⊢ Q
Proof.
1 P 2 ∼ P 3 ∼Q 4 P & ∼ P 5 ∼∼ Q 6 Q
A A H (for RAA) 1,2 I & , H 3-4 RAA, H 5, E∼
The same theorem can be formulated as follows: T8
⊢
P → (∼ P → Q)
Proof.
1 P 2 ∼ P 3 Q 4 ∼ P → Q 5 P → (∼ P → Q)
H (for C H’ (for C) 1,2 T7,H, H’ 2,3 C, H, H 1,4 C, H ′
Clearly, the sister theorem ⊢∼
P → (P → Q)
8
can be shown to be valid in the same way as above. Soon we will need the reverse condition for double negation: T9
⊢
P →∼∼ P
Proof.
1 P 2 ∼ P 3 P & ∼ P 4 ∼∼ P 5 P →∼∼ P
H (for C H’ (for RAA) 1,2 I & ,H, H’ 2,3 RAA, H, H 1,4 C, H
′
Next, we prove the dual Contraposition Law : T10
⊢ (∼
P →∼ Q) → (Q → P )
Proof.
1 ∼ P →∼ Q 2 Q 3 ∼∼ Q 4 ∼∼ P 5 P 6 Q → P 7 (∼ P →∼ Q) → (Q → P )
H (for C) H’ (for another C) 2, T9 MP, 1,3 T5,H, H’ 4E ,H,H’ 2,5 C, H, H 1,6 C,H ¬
′
Here is a ‘mixed’ variant of the law above: T11
⊢ (∼
P → Q)
→ (∼
Q → P )
Proof.
1 ∼ P → Q 2 ∼Q 3 ∼∼ P 4 P 5 Q → P 6 (∼ P → Q) → (∼ Q → P )
H (for C) H’ (for another C) 1,2 T5,H, H’ 3E∼,H,H’ 2,5 C, H, H 1,5 C,H ′
As an easy exercise, you should prove ⊢
(P →∼ Q) → (Q →∼ P )
The next two theorems are very useful within the context of negated conditionals: T12
⊢∼
(P → Q) → P
Proof.
9
1 2 3
(P → Q) T8 (a variant of T8) [−P → (P → Q)] → [∼ (P → Q) → P ] T11 1,2 MP ∼ (P → Q) → P −P →
Observe that in this proof all we do is quote some substitution instances of previously proven theorems and apply Modus Ponens to them. We prove the dual of the foregoing in the same way: T13
(P → Q) →∼ Q
⊢∼
Proof.
1 Q → (P → Q) T2 2 [Q → (P → Q)] → [∼ (P → Q) →∼ Q] T6 3 ∼ (P → Q) →∼ Q 1,2 MP The next theorem has many applications. T14
P ∨ Q, ∼ P ⊢ Q
Proof.
1 2 3 4 5 6 7
P ∨ Q ∼ P P Q P → Q Q→Q Q
A A H (for C) 2,3 T7, H 3,4 C, H T0 1,5,6E∨
Often it is useful to convert a conditional into a disjunction. Here is the theorem for it. T15
P → Q
⊢∼
P ∨ Q
Proof.
1 2 3 4 5 6 7 8 9 10 11
P → Q ∼ (∼ P ∨ Q) P Q ∼ P ∨ Q (∼ P ∨ Q) & ∼ (∼ P ∨ Q) ∼ P ∼ P ∨ Q (∼ P ∨ Q) & ∼ (∼ P ∨ Q) ∼∼ (∼ P ∨ Q) ∼ P ∨ Q
A H (for RAA) H’ (for another RAA) 1,3 MP H, H’ 4 I∨,H,H’ 2,5 I & , H, H’ 3-6 RAA, H,H 7, I ∨, H 2,8 I & , H 2-9 RAA, H 10 E∼
Here is the first half of De Morgan’s Law : 10
′
T16
∼
(P & Q) ⊢∼ P ∨ ∼ Q
Proof. The proof requires far more lines than usual. One way to proceed
is to prove a couple of lemmas and then put together the entire proof using all relevant lemmas. We will have such examples in the second handout. The proof of this theorem is on the next page. 1 ∼ (P & Q) A 2 ∼ (∼ P ∨ ∼ Q) H (for RAA) 3 ∼ P H’ (for another RAA) 4 ∼ P ∨ ∼ Q 3 I∨,H, H’ 5 (∼ P ∨ ∼ Q) & ∼ (∼ P ∨ ∼ Q) 2,4 I & ,H,H’ 6 ∼∼ P 2-5 RAAH, H 7 P 6 E∼, H 8 ∼Q H” (for a new RAA) 9 ∼ P ∨ ∼ Q) 8 I∨,H, H” 10 (∼ P ∨ ∼ Q) & ∼ (∼ P ∨ ∼ Q) 2,9 I & , H, H” 11 ∼∼ Q 8-10 RAA,H, H 12 Q 11E∼ , H 13 P & Q 7,12 I & H 14 (P & Q) & ∼ (P & Q) 13,1 I & , H 15 ∼∼ (∼ P ∨ ∼ Q) 2-14 RAA, H 16 ∼ P ∨ ∼ Q 15 E ∼ Next, we prove the Law of Importation : ′
′′
T17
P → (Q → R)
⊢
(P & Q) → R
Proof.
1 P → (Q → R) A 2 ∼ [(P & Q) → R] H (for RAA) 3 P & Q 2,T12 MP, H 4 ∼R 2 T13 MP, H 5 P 3 E & , H 6 Q 3 E & , H 7 R 1, 5, 6 MP ×2, H 8 R &∼R 4,7 I & 9 ∼∼ [(P & Q) → R] 2-8 RAA, H 10 (P & Q) → R 9 E∼ In this example the proof protocol is considerably simplified. For example, in Line 3 we quote a theorem without presenting it on the previous line and simultaneosuly apply Modus Ponens to it. On Line 7, we apply Modus Ponens twice without writing out the obvious steps. These are some of the simplifications in proofs that we shall use in predicate logic. 11
