13. Proofs: Conjunctions, Disjunctions and Formal Proofs
We are going to learn eight validity and two invalidity rules for arguments containing conjunctions and disjunctions. Most of these rules and their violations are self-defining. But they all need to be mastered like the back of your hand. It is also best if you learn the rules by thinking through their forms using worksheets and art work in some creative fashion.
The rules of validity are argument forms. Any argument that has one of these forms is valid. We will visualize the various valid argument forms (and some invalid forms) by using p's and q's because variables (small case letters beginning with p) stand for any statement. Later when we translate a specific argument we will look for a match with one of these forms. Any argument that matches a valid form is, of course, valid (cf. "happy form" regarding the validity of categorical syllogisms).
1A.Conjunctions There are four inference rules of validity for conjunctions.
1B. Conjunctive Order (CO)
Conjunctive order means that from a conjunction, you can infer in any order with deductive validity. Thus, from p.q you can infer q.p with DV. The order can be transposed.
2B. Conjunctive Simplification (Simp)
Conjunctive simplification means that from a conjunction you can infer any part. For example, p can be inferred from p.q with DV.
3B. Adjunction (Adj)
Adjunction means that from distinct statements, you can infer their conjunction; they can be adjoined. If you have p, q as separate statements, then you can infer p.q with DV (note that the comma in "p, q" is regular English for separating a series).
4B. DeMorgan's Conjunctive Law (DeM)
The very use of the name DeMorgan should from now on be taken as signaling a negation with conjunctions or disjunctions. The sign of DeM is a negated parentheses. DeMorgan's conjunctive law means that in negating a conjunction as a unit you imply disjunction with negated parts. Our cue for DeM will be ~( ) with either a dot or a wedge inside the parentheses. Note that the tilde is a dominant operator that negates everything within the parentheses. Therefore, ~(p.q) implies ~p v ~q with DV.
We can think of the negated parentheses as the DeM base position. Implications can be made either "out from" or "into" the DeM base position although we usually want to move information out from the parentheses to make it more accessible.
We should think of things inside parentheses as "locked in" and generally unusable. Information inside a DeM parentheses is under the scope of the tilde, the tilde is in a position of dominance. DeM is a key way to get rid of the parentheses by applying the tilde to everything within its scope; you might think of the internal operator as denied as well since it is also affected by the tilde and is changed. When a tilde is distributed to each part within the parentheses, the parentheses are eliminated.
Consider the example for Joe who says, "No gal I've ever dated was both beautiful and wealthy." It translates ~(B.W) which by DeMorgan's becomes ~B v ~W and in turn implies that all those dated were either not beautiful or not wealthy (poor Joe, if she was beautiful she was not wealthy; if she was wealthy she was not beautiful).
But be alert to the fact that this is linguistically different from saying "the gals I've dated were neither beautiful nor wealthy" ( ~B . ~W ).
2A. Disjunctions
We will consider four rules of validity and two patterns of non-validity for disjunctions.
1B. Rules of validity
1C. Disjunctive order (DO)
Disjunctive order means that from a disjunction you can infer in any order. From p v q we can infer q v p. Like Simp, we have a valid transposition (movement or change in position).
2C. Disjunctive Argument (Dj Arg or DSD)
Disjunctive argument means that in denying one part of a disjunction you imply the other. We can use DSD to represent the fact that what we have here is a disjunctive singular denial (note how this differs from DeM disjunctive law below which denies the whole disjunction and not simply one part). For example, if my ad will be published in March or April and the publisher informs me that it will not be in March, then I know it will be published in April (that is, I know the proposed publication date). Thus we have: (M v A) . ~M /.. A.
Note that each side of the dot is a premise and the conclusion follows the slash. Since a simplification can be performed, this could also be written formally:
1. M v A
2. ~M / .. A
It is an equivalent substitution instance of this pattern using p's and q's:
1. p v q
2. ~p / .. q
3C. DeMorgan's Disjunctive Law
DeMorgan's Disjunctive Law means that in negating a disjunctive whole you imply conjunction with negated parts. If "it is not the case that the child is ill or spoiled then he is not ill and not spoiled" (he is neither), we then have: ~(I v S) /.. ~I . ~S
A note of caution should be given regarding "neither/nor" compounds. These compounds are conjunctive and not disjunctive. They translate to "not...and...not." The statement "neither John nor Bill will go" becomes "John will not go and Bill will not go" ( ~J . ~B ).
4C. Disjunctive addition (add)
This is a common sense rule based on the truth value of a disjunction as a whole. Namely, since the disjunctive unit is true when at least one part is true then any disjunctive unit can be formed from any true statement. So if you have p then you can create p v q, and q stands for anything. For example, if we know that Mary is a nurse then we know that it is the case that Mary is a nurse or Clinton lives in Moscow: M /.. M v C
2B. Patterns of non-validity
These are disjunctive "cannots" or fallacies.
1C. You cannot simplify a disjunction
Thus, to infer A from A v B is fallacious (or to infer B from A v B is fallacious).
2C. You cannot infer anything from affirmation
You cannot get one part by affirming the other nor can you get negation by affirming one part. Consider the Christmas tree example. You are told that a Christmas tree will be found either outside or inside the building (O v I). As you approach, you see a Christmas tree outside the building. Can you infer that there is no tree inside? No, it could be the case that if you go outside you will find a tree or if you go inside you will find a tree; it could be both. This is a fallacious attempt to get negation from affirmation with a disjunction.
We need to distinguish between weak and strong disjunction to ground this invalidity rule. Weak disjunction involves the notion that "it could be both." Strong disjunction involves the idea, "either...or, and not both." The "not both" is either implied in the context or explicitly stated as in legal documents. If there is no "not both" in the context then we must assume weak disjunction. When you have a "not both" context (as in DSD) then the affirmation of one part tells you that the other part is denied and you then know something about both disjuncts.
3A. The Formal Proof
1B. The structure of the formal proof
All formal proofs have the following structure in which a series of "mini" conclusions are reached by means of the inference rules in order to arrive at the "main" conclusion of the argument as a whole:
1. Premise
2. Premise /.. Main Conclusion (aim of the argument)
3. Mini Inference, reason (rule), reference (by lines)
4. Mini Inference, reason (rule), reference (by lines)
5. Main Conclusion, reason (rule), reference (by lines)
The last line of all formal proofs, if DV, will have the main conclusion (it will attain the aim of the argument) and how it was immediately derived from the preceding information. There may be more, or less, "mini" inferences between the premises and the final conclusion than two exemplified here by lines 3, 4 (i.e., we could have another intermediate inference on line 5 and then the main conclusion would be line 6).
2B. The procedure of the formal proof
Beginning with a set of premises and a conclusion, we seek to draw out solid pieces of information by applying the inference rules. Each piece of information that is drawn out is listed as another premise and assigned a number, defended by citing the rules and the lines above where the rules were put to work to draw it out. A formal proof is like an extended chain argument in which each conclusion becomes a premise. As each mini conclusion unfolds, it is compared with all the previous lines of evidence to see if any further inferences can now be made. This goes on until the main conclusion is reached. If the main conclusion is reached through valid steps, then it too will be valid. For example, let's see what we can determine about an argument that begins with the following:
1. ~(T v R)
2. G v R /.. G Recall that line 2, which is premise 2, ends at the slash.
We want to see if enough information can be drawn out of these premises to reach the conclusion, G. Our first mini conclusion will be the result of DeMorgan applied to line 1:
inference reason reference
3. ~T . ~R by DeM, from line 1
Next we can simplify line three which gives us each conjunct from line 3 on the separate lines, 4 and 5:
4. ~T by simp, from line 3
5. ~R by simp, from line 3
We now look back over all the lines 1-5 to see if there is any combination that will get us to the conclusion or at least give us another step in the process of reasoning toward the conclusion. Here is the argument so far:
1. ~(T v R)
2. G v R /.. G
3. ~T . ~R by DeM from line 1
4. ~T by simp from line 3
5. ~R by simp from line 3
The only G in the premises is in line 2. So we enquire as to how we can break up the information of a disjunction; we must think of a disjunctive rule that applies; let's try DSD from line five to line two. What do we get? We deny one part so we get the other part, which is G. Note that the last line of a valid formal proof will always be the main conclusion, how it was immediately reached, and the lines used for this final inference.
6. G by DSD, from line 5 and 2
Proofs ws1
Applying Conjunction and Disjunction validity rules
(The answers are given below)
Directions:
1. Identify the rule by repetition and perhaps by morphological art work (doodling, visualizing).
2. Determine DV or ~DV and state why.
3. Check your answers.
4. Repeat for rule mastery by thinking through the rules.
1. ~( b v w ) 2. ( d v s ) . ~d
.. ~b . ~w .. s
3. ( T v P ) . D 4. ~( J . F )
~P /.. T .. ~J v ~F
5. w . s 6. O . M
.. s . w .. M
7. ~ ( s . c ) 8. ~( w v s )
.. ~s . ~c .. ~w . ~s
9. C v G 10. ~B . ~U
.. G .. ~B
11. B v U 12. ( I v C ) . ~C
U / .. ~B .. I
13. ~F . ( F v S ) 14. F . ( F v S)
.. S .. ~S
15. ( R v S ) . (~R . ~V ) 16. ( ~A v S ) . ~A
.. ( R v S ) . ~R .. S
.. S
17. ( ~d v s ) . d 18. ~ ( x v y )
.. s .. ~x v ~y
19. Q v Z /.. Z v Q 20. ~A v ~B
.. ~ ( A . B )
21. ~w . ~s 22. w . ~s
.. ~ ( w v s ) .. ~~w . ~s
23. 1. (A . B) v ~T 24. 1. ~(D v W)
2. T / .. B 2. T v W
3. 3. ~T v Z / .. Z
4. 4.
5.
6.
7.
25. 1. K 26. 1. P . E /.. ~ ( )
2. ~( K . C ) / .. ~C 2.
3.
4.
27. 1. T v ~N /.. ~ ( )
2.