In addition to negation, conjunction, and alternation, another connective role is played in rapid, precise dialogue: conditionality. Conditional statements come in the form
That a false antecedent implies every statement sounds tricky and perhaps wrong at first. What you need to realize is that we are handling statements only on a truth-functional level - only looking at their truth-values and not the details of any actual semantic contents. Thus it is not so that "Louis is a woman" proves the truth of "God does not exist". Rather, the truth-value of the whole conditional is handled as if it were true, since the antecedent is known to be false (and there may be controversy over the consequent). This might feel weird to you, but it is no less weird, according to Mr. W. V. Quine, than treating the truth-value of a conditional with a false antecedent as false. In everyday language we typically consider ourselves as simply not being committed to any claims when our antecedents turn out false (except in cases where we're intentionally making a contrafactual conditional). But in formal logic we just treat conditionals with false antecedents as true compounds.
These same ends could be accomplished by the means presented in §2 ("Truth Functions").
In elaboration, Quine offers the example
Thus, the above example should be seen as affirming a series of material conditionals:
The arguably ineffable relationship between the antecedent and consequent of a contrafactual conditional is similar to that which may be argued to exist between the components of any meaningful alternation. This phenomenon is extremely fertile ground for exploration and controversy.
If p then q.The "p" component is referred to as the "antecedent" and the "q" component the "consequent".
Material Conditionals
Conditionals of this form are called "material conditionals", and may be written like the following:p ⊃ qConditionals, insofar as they are treated as proper truth functions, admit as wholes of truth when the antecedent and consequent are both true, and when the antecedent is false (regardless of the truth of the consequent). Conditionals, when treated as truth functions, admit as wholes of falsity only when the antecedent is true and the conditional is false.
That a false antecedent implies every statement sounds tricky and perhaps wrong at first. What you need to realize is that we are handling statements only on a truth-functional level - only looking at their truth-values and not the details of any actual semantic contents. Thus it is not so that "Louis is a woman" proves the truth of "God does not exist". Rather, the truth-value of the whole conditional is handled as if it were true, since the antecedent is known to be false (and there may be controversy over the consequent). This might feel weird to you, but it is no less weird, according to Mr. W. V. Quine, than treating the truth-value of a conditional with a false antecedent as false. In everyday language we typically consider ourselves as simply not being committed to any claims when our antecedents turn out false (except in cases where we're intentionally making a contrafactual conditional). But in formal logic we just treat conditionals with false antecedents as true compounds.
These same ends could be accomplished by the means presented in §2 ("Truth Functions").
In elaboration, Quine offers the example
If anything is a vertebrate, it has a heart.The reason that the above statement is not like the other truth functions is that its component parts are not independent statements, whose truth or falsity sufficiently determines the truth or falsity of the whole. Quine explains that "it has a heart" is not a statement unto itself.
Thus, the above example should be seen as affirming a series of material conditionals:
If a is a vertebrate, a has a heart;Or, as Quine puts it,
If b is a vertebrate, b has a heart,
etc.
No matter what x may be, it is not the case that x both is a vertebrate and does not have a heart.As a method of logic then, the precise meaning of
p ⊃ qis
~(p~q)However situations arise wherein it doesn't seem right to treat conditionals as traditional truth functions in this manner..
Contrafactual Conditionals
It is not uncommon to use "if-then" in the subjunctive mood, instead of the indicative mood. Such cases tacitly carry the admittance of the falsity of the antecedent. For exampleIf I had not walked to Java this morning, then I would not be drinking coffee as I write this example.Such contrafactual conditionals are intended to transmit information despite the falsity of their antecedents. Thus, some contrafactual conditionals with false antecedents may be argued to be true, while others may be argued to be false. They therefore do not qualify as being truth functional. Rather, there are kindred causal relationships to explore during examinations of contrafactual conditionals. Quine briefly follows this rabbit trail in §3.
The arguably ineffable relationship between the antecedent and consequent of a contrafactual conditional is similar to that which may be argued to exist between the components of any meaningful alternation. This phenomenon is extremely fertile ground for exploration and controversy.
Biconditionals
Simply put, a material biconditional is of the formp if and only if qor
p ≡ qand it is true only when the truth values of the components match (either both p and q are true, or both p and q are false). This truth function is identical to affirming
(p ⊃ q)(q ⊃ p)Like "v" and "⊃", the particular symbol and use of "≡" is superfluous, yet helpful.
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By at least the 4th Edition of this text, Quine includes a historical note to §3. The Conditional, which briefly chronicles the use of ⊃ and ≡. He explains why he ultimately abandons them for → and ↔, respectively.
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