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[The following is summary. Boldface (except for metavariables), underlining, and bracketed commentary are my own. I highly recommend Agler’s excellent book. It is one of the best introductions to logic I have come across.]
Summary of
David W. Agler
Symbolic Logic: Syntax, Semantics, and Proof
Ch.2: Language, Syntax, and Semantics
2.4 Disjunction, Conditional, Biconditional
Brief Summary:
A disjunction (∨) is true when at least one of the disjuncts is true, and it is false otherwise. The conditional (→) is false only when the antecedent is true but the consequent false. The biconditional (↔) is true when both sides have the same value, and it is false otherwise. The conditional is translatable using “if..., then...” formulations. However, not all “if..., then...” formulations conform to the material conditional’s truth-table. For example, when we use “if..., then...” formations to make statements of causality, both antecedent and conditional can be true and yet the whole proposition be false. Consider the example, “If John prays, then he will get an A”. Suppose that he both prayed and got an A. But suppose further that the real reason he got the A was not because he prayed (which he in fact did anyway) but rather because he cheated. Although both the antecedent and the consequent are true, the whole proposition is false; for, the proposition is stating that the cause of getting the high grade is praying, when in reality the cause is not that but instead is cheating.
Summary
2.4 Disjunction, Conditional, Biconditional
[In section 2.2, we examined negation (¬) and conjunction (∧).] Now we return to our discussion of truth-functional operators. The ones that are left are disjunction (∨), material conditional (→), and biconditional (↔).
2.4.1 Disjunction
Disjunction takes the form:
P∨Q
Each term of the pair is called a disjunct, and the ∨ symbol is called a wedge. Agler defines the truth function this way:
Disjunction = df. If the truth-value input of either of the propositions is true, then the complex proposition is true. If the truth-value input of both of the propositions is false, then the complex proposition is false.
(46)
Thus, “disjunction is true if either (or both) of the disjuncts are true and false only when both of the disjuncts are false” (46). The truth table would look like:
[Since this sort of disjunction can be true when both disjuncts are true, it is an inclusive disjunction]. The cases in English that best capture disjunction are inclusive uses of “or”, as in:
Mary is a zombie, or John is a mutant.
(46)
We can use ∨ to translate other English expressions too. For instance,
Mary is neither a zombie nor a mutant.
can be rendered
¬(Z∨M)
But, not every use of “or” in English is representable as ∨. This is because we also can use “or” in an exclusive way, as in this sentence:
Michael Jordan or Kobe Bryant is the greatest basketball player ever.
(47)
Here, for the entire proposition to be true, either disjunct must be true, but they cannot both be true. When representing such cases, we would use an extra part meaning “and not both”, so we would write for instance:
M∨K ∧ ¬(M∧K)
(Agler 48)
If we wanted, we could make a symbol for the exclusive disjunction, in this case, ⊕, and we can make the following truth table for it:
(48)
2.4.2 Material Conditional
The material conditional is:
P→Q
The → symbol is called the arrow. The left part is the antecedent and the right part is the consequent (48). The truth-function is defined:
Conditional = df. If the truth-value input of the proposition to the left of the ‘→’ is true and the one to the right is false, then the complex proposition is false. For all other truth-value inputs, the complex proposition is true.
(49)
The truth table would be:
“If..., then....” structures are the best translation for the conditional. Agler then notes these other formations, which all share the same notation:
In the case that Mary is a zombie, John is a zombie. (M→J)
Mary being a zombie means that John is a zombie. (M→J)
On the condition that Mary is a zombie, John is a zombie. (M→J)
Only if John is a zombie, Mary is a zombie (M→J)
(Agler 49)
[Note especially the “only if...” formation, which has an unexpected notation.]
Agler notes there is some debate regarding which uses of “if..., then...” corresponds to the →. [See for example chapter 7 of Graham Priest’s Logic: A Short Introduction.] We return later to see reasoning why the truth functional uses correspond to the table above.
For now we note that there are two main ways we use “if..., then...” constructions in English: a truth functional way and a non-truth-functional way (50). It is truth functional when the truth value of the whole complex proposition depends on the truth values of the component parts. “So, If John is in Toronto, then he is in Canada is true depending upon the values of John is in Toronto and John is in Canada” (50). However, we also use “if..., then...” statements in non-truth-functional ways, as for example when making causal claims. So consider this sentence:
If John jumps up, then (assuming normal conditions) he will come down.
(50)
We will assume that both the antecedent and the consequent are true. And suppose also that the causal “if..., then...” is truth functional. So this whole proposition should be true.
Now instead consider the sentence:
If John prays before his big logic exam, then he will receive an A.
(50)
Assume again that both consequent and antecedent are true. If again the formation is truth- functional, then the whole proposition should be true.
Now, still keep supposing that both John prayed and he got an A. But what if the real cause of his getting the A was not that he prayed but rather that he cheated? So the sentence again is:
If John prays before his big logic exam, then he will receive an A.
(50)
And we are assuming that it is a statement of causality [meaning that we are saying that praying is the cause whose effect is a high grade. It seems now with this new situation of cheating, the situation is that praying does not cause high grades, although cheating in fact does.] So now the whole proposition is false, even though both the consequent and the antecedent are still true (50).
Thus in some English usages of “if..., then...”, the truth values of the component parts do not always determine the truth value of the whole proposition (51).
2.4.3 Material Biconditional
The material biconditional takes the form
P↔Q
The ↔ symbol is called the double arrow (51). To name either side, we just say, “the left-hand side of the biconditional” and “the right-hand side of the biconditional”. The truth function is defined:
Biconditional = df. If the truth value input of the propositions are identical, then the complex proposition is true. If the truth value inputs differ, the complex proposition is false.
(51)
Its truth table is the following:
The best way to translate ↔ is “if and only if” (51). But it can also be rendered “just in the case that”, for instance, “Mary is a zombie just in the case that John is a vampire (52).
Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman & Littlefield, 2013.
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