Sentential Logic: Some Basic Concepts and Techniques
2.6 Conditional and Biconditional
Conditional is a binary connective, whose name is ‘→’. If we include it then, for every two sentences A and B, there is a sentence called the conditional of A and B, which we write as:
A→ B
A and B are called, respectively, the antecedent and the consequent of A→ B. ‘Conditional’, like the names of the other connectives, is used ambiguously: for the connective (i.e., the operation) as well as the resulting sentence.
Note: Do not confuse the logical antecedent–the one just defined–with the grammatical antecedent of a pronoun (the word or phrase a pronoun refers to).
The truth-table of A→ B is:
A B A→ B
T T T
T F F
F T T
F F T
Conditional corresponds to the ‘If ... then ’ formation of English. The sentence (1) If Jack is at home then Jill is at home
can be recast in sentential logic as the conditional A → B, where A represents ‘Jack is at home’, and B represents ‘Jill is at home’. Recasting (1) in this way means that we consider it false if Jack is at home and Jill is not at home, and we consider it true in all other cases–in particular, if Jack is not at home. The problems arising from this interpretation of ‘If ... then ’ are discussed in chapter 3, where the relations between natural language and sentential logic are looked into. In sentential logic → is just another connective having the above truth-table.
Obviously, the column of A→ B is the same as the column for ¬A ∨ B. Hence we have:
A → B ≡ ¬A ∨ B
Consequently,→ is expressible in terms of ¬ and ∨ (and therefore also in terms of ¬ and ∧).
We could do without → in the sentential calculus; but there are good reasons for including it, which have to do with the role of → in the context of logical implications and formal deductions.
On the other hand, ∨ (hence also ∧) is expressible in terms of ¬ and →:
A∨ B ≡ ¬A → B
Thus, ¬ and each of the binary connectives ∧, ∨, → are sufficient to express the other two binary connectives.
Note: Sometimes conditional goes under the name ‘implication’, or ‘material implication’.
The term ‘implication’ will be used for a different purpose. You should take care not to confuse the two.
Grouping Convention for →: The convention is that ‘→’ binds more weakly than any of the previous connective names. This means that in restoring parentheses we first determine the scopes of negations to be the smallest scopes consistent with the existent grouping, then the scopes of conjunctions, then those of disjunctions, and then the scopes of the conditionals.
For example, the grouping in:
¬A ∧ B ∨ B → ¬C ∨ D is:
[((¬A) ∧ B) ∨ B] → [(¬C) ∨ D]
Conditional does not have the associativity property, enjoyed by conjunction and disjunction:
(A→ B) → C and A → (B → C)
are, in general, not equivalent. The two have different values when A, B, C get all F. It also lacks commutativity:
A→ B and B → A
have different values, when A and B get different values.
Consequently, when conditionals are repeated, grouping and order are extremely important.
We cannot omit parentheses as we have done in the case of long conjunctions and disjunctions.
Note: If we toggle T and F throughout the truth-table of conditional, we get the truth-table of
¬A ∧ B
Hence, if we were to introduce a connective dual to→, then the “dual-conditional” of A and B would be logically equivalent to¬A ∧ B. This can be also seen by rewriting the conditional in the logically equivalent form¬A ∨ B and forming the dual of that.
None of the customary systems of logic has a “dual-conditional” as a primitive connective.
To form the dual of an expression involving conditionals, we should therefore replace every component of the form A→ B by ¬A ∧ B.
Homework
2.20 Consider the expression
A→ B → C → D
How many possible different sentences can we obtain from it by inserting parentheses?
Find whether any two of sentential expressions obtained in this way are logically equivalent.
2.21 In certain cases conditional can be distributed over conjunctions and disjunctions. For example,
A→ (B ∧ C) ≡ (A → B) ∧ (A → C)
But sometimes the “distributing” involves a change in the other connective (the conjunction or the disjunction over which conditional is distributed). Find the “distributive laws” for the following cases.
• A → (B ∨ C)
• (A ∨ B) → C
• (A ∧ B) → C
2.22 Using the “distributive laws” of Homework 2.21, push→ all the way in, in the following sentences:
1. (A∨ B) → (C ∨ D) 2. (A∨ B) → (C ∧ D) 3. (A∧ B) → (C ∨ D) 4. (A∧ B) → (C ∧ D)
Note: Using conditional , we can form the tautology A → A , which is perhaps simpler than our previous standard tautology A∨ ¬A.
Biconditional
We conclude our list of connectives, with biconditional. It is a binary connective whose name is ‘↔’. For every two sentences A and B, there exists a sentence that is their biconditional:
A↔ B
The interpretation of the biconditional is best expressed by expressing it as a conjunction of two conditionals:
A↔ B ≡ (A → B) ∧ (B → A) .
In terms of truth-values this means that the value of A↔ B is T if A and B have the same truth-value; it is F if they have different values.
Since conditional represents, in a way, ‘If ... then ’ , biconditional corresponds to:
If ... then , and if then ... , that is: ... iff .
For example,
(2) Jill is at home if and only if Jack is at home can be formalized as a biconditional.
More than the other connectives ↔ is dispensable as a primitive. Replacing everywhere
‘A↔ B’ by ‘(A → B) ∧ (B → A)’, may cause some, but usually minor, inconveniences.
From the truth-table of biconditional we can immediately see that it has the commutativity property:
A ↔ B ≡ B ↔ A
Also the following useful equivalences are easily verified, either by truth-table, or by algebraic manipulations (after converting the biconditional into a conjunction of two conditionals and expressing these in terms of the previous connectives).
A↔ B ≡ (A∧B) ∨ (¬A∧¬B)
¬(A ↔ B) ≡ A ↔ ¬B ≡ ¬A ↔ B
¬(A ↔ B) ≡ (A∧¬B) ∨ (¬A∧B) ≡ (A ∨ B) ∧ (¬A ∨ ¬B)
Note that the right-hand side of the first equivalence can be understood as saying: “Either both A and B are true, or both are false”. Similarly, in the third row, the second sentence can be understood as saying that A and B have different truth-values; and the third sentence as saying that one of A and B is true, and one is false.
Note that¬(A ↔ B) expresses exclusive ‘or’.
Note: If we toggle T and F in the truth-table of A↔ B we get the truth-table of ¬(A ↔ B), hence ¬(A ↔ B) can serve as the dual of A ↔ B. This can be also inferred from the fact that the rightmost sentences in the first and third rows in the above-given equivalences are duals of each other.
In addition to commutativity, biconditional has the associativity property:
(A↔ B) ↔ C ≡ A ↔ (B ↔ C)
This can be established either by truth-tables, or by algebraic manipulation using the equiv-alences given above. A short truth-value argument, which gives also some insight into the nature of repeated biconditionals, proceeds by proving first the following claim.
(A ↔ B) ↔ C gets T, under a given assignment of truth-values to the sentential variables A, B, C, iff F is assigned an even number of times (i.e., to 2 of the variables, or to none of them).
Here is the proof.
Case (i): C gets T. Then (A ↔ B) ↔ C gets T iff A and B get the same truth-value. If both get T, the number of assigned F’s is 0; and if both get F, this number is 2.
Case (ii): C gets F. Then (A ↔ B) ↔ C gets T iff A and B get different values; in this case the number of assigned F’s is 2 (one to C, one to A or to B).
It is easily seen that there are exactly four possibilities of assigning an even number of F’s. These are exactly the possibilities covered in (i) and (ii).
Hence, (A↔ B) ↔ C gets T just when the number of assigned F’s is even.
This shows also that (B ↔ C) ↔ A gets T iff F is assigned an even number of times. It follows that:
(A↔ B) ↔ C ≡ (B ↔ C) ↔ A
Since (B↔ C) ↔ A ≡ A ↔ (B ↔ C), we get the desired result.
Consequently, in expressions in which ‘↔’ is the only connective name, changes in the order and grouping yield logically equivalent expressions (as is the case with ‘∧’, or with ‘∨’). We can, therefore, ignore parentheses, e.g.,
A↔ B ↔ C ↔ D ↔ E
If in such a repeated biconditional a sentential variable, ‘A’, occurs more than once, we can, by changes of order and grouping, rewrite the biconditional as:
(A↔ A) ↔ D
where D is the rest of the repeated biconditional. Since A↔ A is a tautology, it easily follows that
(A↔ A) ↔ D ≡ D
We can therefore drop any pair of two occurrences of the same sentential variable. If every sentential variable occurs an even number of times, then the repeated biconditional is a tau-tology. If not, then by dropping all repeating pairs, we are left with a repeated biconditional in which every sentential variable occurs only once. Such a sentence is non-tautological (cf.
Homework 2.23).
Homework 2.23 Show that a sentential expression constructed using only ‘↔’ is tautological iff each sentential variable occurs in it an even number of times. (Most of the proof has already been done, you have to state it in full, supplying the last missing step).
Using the type of argument just given, one can prove the following generalization of the claim used in the proof of associativity:
If a sentential expression is built only from ‘↔’ and sentential variables, and each variable occurs only once, then the expression gets T iff the number of variables that get F is even.
Unlike our previous binary connectives, ↔ with negation is not sufficient for expressing the other connectives. It can be proved that, for any sentential expression built from two sentential variables using only ‘¬’ and ‘↔’, the number of T’s in its truth-table column is even (either 0, or 2, or 4). Therefore it cannot be equivalent say to ‘A∧ B’, which has an odd number of T’s in its column (namely, 1).
Biconditionals are handy for characterizing logical equivalence in terms of logical truth:
A ≡ B iff A↔ B is a logical truth.
Grouping: The convention is that ‘↔’ has weaker binding power than the other connective names. Hence
¬A ∨ B∧C → B ↔ A → B ∨ C should be read as:
[(((¬A) ∨ (B∧C)) → B)] ↔ [A → (B ∨ C)]
Simplifying Expressions Containing Conditionals and Biconditionals
Expressions containing conditionals and biconditionals can be transformed into equivalent ones involving only¬, ∧, and ∨, to which, in turn, we can apply our previous simplification techniques. Often, however, we can get simpler forms by retaining some conditionals, or biconditionals, in the final outcome. For example,
A → (B → C) is logically equivalent to:
¬A ∨ ¬B ∨ C but also to
A∧B → C
which has a clearer intuitive meaning. This generalizes to an arbitrary number of repeated conditionals grouped to the right:
A1 → (A2 → (. . . → (An−1 → An) . . .) ≡ (A1∧A2∧. . .∧An−1)→ An Noteworthy equivalences concerning → are:
¬A → ¬B ≡ B → A
¬(A → B) ≡ A ∧ ¬B
The above-mentioned properties concerning↔, as well as the equivalence:
¬(A ↔ B) ≡ A ↔ ¬B
can be used in simplifications of expressions containing ‘↔’ and ‘¬’ only. When other con-nectives are present, there are no straightforward simplification techniques (short of rewriting everything in terms of ¬, ∧, ∨ and applying the previous methods). But special cases may lend themselves to special treatments.
Simplify the following sentences. You may employ in the final version any of the connectives introduced here. Try to get sentences that are short or easy to grasp. You can use truth-value considerations, the algebraic methods of the previous section, or a mixture of both.
1. (A→ B) → C 2. (A→ B) → A 3. (A→ B) → B 4. (A→ B) ∨ (B → A) 5. ¬(A → B) ∨ ¬(B → A) 6. (A↔ B) ↔ A
7. A∧B ∨ ¬A∧¬B 8. ¬A∧B ∨ A∧¬B 9. (A→ B) ↔ (B → A) 10. (A → B) → (B → A) 11. (A∨ B) ↔ (A ∧ B)
12. [¬(A ↔ B)] ↔ ¬[(B ↔ C) ↔ C]
13. A∧B ↔ A∧C
14. ¬(A → B) ↔ ¬A∧¬C
15. (A → B) ∧ (B → C) ∧ (C → A) 16. (B → A) ∧ (C → A) ∧ (A → (B ∨ C)) 17. (A → B) ∧ (A → C) ∧ ((B ∧ C) → A)
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