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The Amateur Logician Tutorial:
Symbols & Translations | Basic Truth Tables | Complicated Translations | Introducing Rules of Inference | Equivalences (Replacement Rules) | Conditional & Indirect Proofs
Introduction
Translating is an art, not an exact science. Context is everything. Strictly speaking, we could translate “A or B, if C and D” simply as P. There’s not much we can do with P in propositional logic, though! It’s better as C&D→AvB, since it captures more of the formal structure of the sentence. There are sententional connectives within that sentence, and that sentence expresses a proposition with multiple sub-propositions in it.
Review
Remember, a proposition written in a grammatically correct way is a well-formed formula. The main sentential connective will be outside of parentheses, although we’ll follow the convention to have the biconditional and then conditional as the “strongest” sentential connectives, allowing us to drop parentheses in those cases without ambiguity.
There are five main sentential connectives we have to look out for: negation, conjunction, disjunction, conditional, and biconditional. There are “key” words that tend to be associated with these connectives. What follows can’t be exhaustive, but it’s helpful.
Examples
Example 1. “A or B, if C and D.”
Let’s work with the example from the introduction.
It’s a conditional overall. “C and D” is immediately after the “if,” which indicates it is part of the antecedent, not the consequent. The consequent is thereby the “A or B.” The antecedent is a conjunction with the “and” and the consequent is a disjunction with the “or.”
So, we get (C&D)→(AvB). By our convention, we can drop parentheses in this case. This gives us C&D→AvB.
Example 2. “Either Steve works in produce, or works in meat and the deli.”
P = Steve works in produce
M = Steve works in meat
D = Steve works in the deli
Notice key leading words and the comma’s placement. The first word is “either,” indicating that the main sentential connective is a disjunction, which means we must place this connective outside of any parentheses. The word “or” is immediately after a comma, which also indicates that the main sentential connective is a disjunction.
There’s a conjunction within this, too, as indicated with the word “and.”
So, we get: Pv(M&D)
Example 3. “Nate is laughing but Karina is mad.”
N = Nate is laughing
K = Karina is mad
The key word there is “but.” Hence, we have a conjunction.
This gives us: N&K
Example 4. “If neither Cassandra nor Shane are working, then Paul is not happy.”
C = Cassandra is working
S = Shane is working
P = Paul is happy
Clearly, it’s a conditional overall. The antecedent, while negated with the “neither,” has a disjunction option with the “nor.” The consequent has a “not” in it, which means we have a negation.
Thus, we get: ~(CvS) →~P
Example 5. “George will work if David or Cornellius but not both work.”
G = George will work
D = David works
C = Cornellius works
This is another conditional with the “if.” What follows the “if” is the antecedent, which happens to be a disjunction with the “or.” Yet this is an exclusive disjunction; so, we have to deny both options happening together in a conjunction.
Hence, (DvC)&~(D&C)→G.
Example 6. “Justified belief is a necessary condition for possessing knowledge.”
J = justified belief
K = possessing knowledge
Conditionals tend to be trickiest. “Necessary condition” is a dead giveaway. What’s necessary is in the consequent, not antecedent.
We should get K→J.
Example 7. “The business will grow and profits will increase only if services improve, and that will happen only if more training takes place.”
B = business will grow
P = profits will increase
S = services improve
T = training takes place
Observe that there’s an “and” immediately following a comma. This suggests we have a conjunction overall. Within that conjunction are two conditionals, as the “only if” suggests. There’s also a conjunction in one of those antecedents with the “and.”
Parentheses are important here so as to indicate that the main connective is a conjunction. Putting things together, we get (B&P→S)&(S→T).