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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
Symbols & Translations
– Introduction and WFFs
– Negations – Conjunctions – Disjunctions – Conditionals – Biconditionals
– Less Common Sentential Connectives: XOR, NAND, NOR
Introduction and WFFs
A proposition declares something to be the case. As such, it is not a question, command, or wish. It must either be true or false. “Triangles have three sides” is a true proposition. “Squares have five sides” is a false proposition. “Shut up” is not a proposition, since it’s a command, and is neither true nor false.
A molecular proposition is formed by sentential connectives such as the “and,” “or,” “not,” or “if …, then ….” An atomic proposition has no sentential connective.
We represent an atomic proposition with a single capital letter.
“Triangles have tree sides” we’ll represent with T.
“Squares have five sides” we’ll represent with S.
Sentential connectives are symbolized.
There’s no universally agreed to list of symbols, but what follows is fairly common.
An expression can consist of capital letters, sentential connectives, and/or parentheses.
A proposition written in a grammatically correct way is a well-formed formula (wff). Pronounce “wff” as “woof.” Capital Greek letters stand for metavariables. A metavariable doesn’t represent a proposition. It only represents an expression in the language of propositional logic.
Φ is the uppercase Greek letter phi.
ψ is the uppercase Greek letter psi.
Wffs must follow these rules:
(1) Any capital letter alone is a wff.
(2) The negation of a wff is a wff.
So, ~Φ is a wff, assuming Φ is a wff.
(3) The joining together of two wffs by a sentential connective and enclosing the two in parentheses is a wff.
So, these are four good wffs,: (Φ & ψ), (Φ v ψ), (Φ → ψ), and (Φ ↔ ψ), assuming Φ is a wff and ψ is a wff.
In practice, textbooks usually adopt the convention that we can drop parentheses only if it leads to no ambiguity. There’s no reason to have (P→Q) with those parentheses, for example, when P→Q has no ambiguity to it.
What’s outside of the parentheses is the “strongest” sentential connective. (P&Q)→R is a conditional overall. In contrast, P&(Q→R) is a conjunction overall.
P&QvR is not a wff and is ambiguous. It needs parentheses to indicate the “strongest” sentential connective. P&(QvR) is a conjunction overall. (P&Q)vR is a disjunction overall.
Note that what’s grouped in parentheses is done first. With P&(QvR), we first work on QvR. Then QvR is considered in relation to P through the & sentential connective.
By default, some textbooks take the biconditional and then conditional as the “strongest” sentential connectives. This allows us to drop the parentheses in (P&Q)→R to get P&Q→R without ambiguity. Likewise, we can drop the parentheses in (P→Q) ↔R to get P→Q ↔R without ambiguity.
Negations
Negations deny. Note that ~Φ is a wff.
“Squares have five sides” is an atomic proposition.
It contains no sentential connectives within it.
We’ll translate “squares have five sides” into propositional logic.
Let’s use S to represent “squares have five sides.”
~S is a molecular proposition, since it contains the negative sentential connective.
We can translate ~S into such ordinary English sentences as:
• “It is not the case that squares have five sides.”
• “It is false that squares have five sides.”
• “Squares do not have five sides.”
S has the truth-value of being false. It thereby follows that ~S must be true.
Let’s use H to represent “man is mortal.”
We can translate ~H into:
• “Man is not mortal.”
• “Man is immortal.”
• “It is not the case that man is mortal.”
• “It is false that man is mortal.”
H has the truth-value of being true. It thereby follows that ~H must be false.
Conjunctions
Conjunctions combine two wffs with an “and.” Φ & ψ is a wff.
Let M translate “the Moon is Earth’s natural satellite.”
Let P translate “Mercury tends to be the nearest planet to Earth.”
M&P represents “the Moon is Earth’s natural satellite and Mercury tends to be the nearest planet to Earth.” The molecular proposition M&P is true because the atomic proposition M is true and the atomic proposition P is true. (Surprisingly, Mercury is closer, on average, to Earth than Venus!) M&R would be false if at least one of the atomic propositions that compose it is false.
Consider the proposition T&S, for example. T stands for “triangles have three sides” and S stands for “squares have five sides.” Since the atomic proposition S is false, the molecular proposition T&S must be false.
Here’s a mathematical example: 1 < 2 < 3.
This means 1 is less than 2 and 2 is less than 3; hence, we have a (true) conjunction.
Translating from English sentences to propositional logic (or vice versa) is more of an art form than an exact science. It takes understanding context. Furthermore, there are different expressions that indicate a conjunction, besides just the word “and.”
Disjunctions
Disjunctions combine two wffs with an “or.” Φ v ψ is a wff.
By default, the “or” is taken to be inclusive. This means that Φ v ψ is true if at least one of those options — Φ or ψ — is true. So, Φ v ψ is true in three cases: (1) Φ is true, (2) ψ is true, or (3) Φ and ψ are both true. When Φ is false and ψ is false, Φ v ψ must be false.
The inclusive “or” is akin to “and/or.”
B will stand for “George Boole is a logician.”
T will stand for “Donald Trump is a logician.”
BvT represents “Either George Boole or Donald Trump is a logician.”
Although T is false, B is true; ergo, BvT is true.
It doesn’t really matter if Φ and ψ have no relation to each other whatsoever. For example, we’ll have A stand for “Aristotle is a philosopher” and M stand for “the Moon is Earth’s natural satellite.” Since A is true and M is true, it follows that AvM is true.
Consider this mathematical example: 0 ≤ 5.
This means 0 is less than or equal to 5; hence, we have a (true) disjunction.
Conditionals
Conditionals combine two wffs through an “if. . ., then. . .” Φ → ψ is a wff.
With the proposition Φ → ψ, we call Φ the antecedent and ψ the consequent.
Terminology differs. Some math books, to my utmost dislike(!), call the antecedent the “hypothesis,” “assumption,” or “premise.” Some call the consequent the “conclusion.”
Mathematical logic has it that the conditional Φ → ψ is false only when Φ is true but ψ is false. It’s otherwise taken to be true. (To be sure, this seems puzzling! Though it preserves the consistency of mathematical logic, it leads to “paradoxes of material implication.”)
We’ll take this example: “if today is Friday, then George is at work.”
F will represent “today is Friday” and G will represent “George is at work.”
So, we have the molecular proposition F→G.
The only way F→G is false is when “today is Friday” is true, but “George is at work” is false. George not being at work on a Friday has thereby falsified the truth of the conditional.
Given how the conditional is treated in mathematical logic, how should we think of the next outrageous example (taken from Wikipedia)?
“If the Nazis had won World War Two, everybody would be happy.”
Traditional logic, I would argue, considers it false. Mathematical logic declares it true, since any conditional with a false antecedent is true. (You can see why I don’t consider mathematical logic as totally superior to traditional logic!)
Next, let’s look at another mathematical example: if 1 + 1 = 4, then 2 + 2 = 4.
It’s true, despite the false antecedent.
How about this? If 1 + 1 = 2, then 2 + 2 = 6.
It’s false, since we have a false consequent with a true antecedent.
One feature of traditional logic remains, however:
Sufficient vs. Necessary Conditions.
In general, the antecedent is said to be sufficient for the consequent. That is, if Φ really is the case, then ψ will really be the case. The consequent is said to be necessary for the antecedent. That is, if ψ is not really the case, then Φ cannot really be the case. A sufficient condition should thereby always be placed in the antecedent and a necessary condition should thereby always be placed in the consequent.
Consider R→C as representing “if it is raining, then it is cloudy.”
Rain is sufficient for it to be cloudy, though rain is not necessary for it to be cloudy (because it can be cloudy without rain). Being cloudy is necessary for it to rain (because rain cannot happen without clouds), though being cloudy is not sufficient for it to be raining.
This explains why “if it is cloudy, then it is raining” is a bad proposition.
There’s a confusion of sufficient and necessary conditions.
Note that R→C is not the same as C→R.
Also, we can falsify C→R: we can have a case where it being cloudy is true, but it raining is false; ergo, we have a false conditional.
Biconditionals
Biconditionals combine two wffs through an “if and only if.” Φ ↔ ψ is a wff.
“If and only if” is often abbreviated as iff.
The biconditional Φ ↔ ψ is equivalent to (Φ → ψ) & (ψ → Φ). Notice that a biconditional is the same as the conjunction of two conditionals, where the antecedents and consequents, so to speak, “flip.” This allows us to say, in a biconditional, that (1) Φ is both sufficient and necessary for ψ and (2) ψ is both sufficient and necessary for Φ.
Any biconditional is true when the two propositions that form it have the same exact truth value. Hence, if both Φ and ψ are true, then Φ ↔ ψ is true; likewise, if both Φ and ψ are false, then Φ ↔ ψ is true.
Put differently, the truth of the one implies the truth of the other; alternatively, the falsity of the one implies the falsity of the other.
What should we make of this example?
We’re told P→Q is true. It’s so-called converse, Q→P, is true. This implies that the biconditional P↔Q is true.
E↔B stands for “Lina ate eggs if and only if she ate bacon.”
– She ate bacon, but didn’t eat eggs. Thus, the biconditional is false.
– She ate both eggs and bacon. Thus, this time, it’s true.
– She ate eggs, but didn’t eat bacon. Thus, false.
– She didn’t eat bacon and didn’t eat eggs. Thus, true.
Mathematical definitions can be stated as biconditionals. For example, n is an even number if and only if it is of the form n = 2k, where k is an integer. 7 is not an even number, since 7/2 = k and k is not an integer.
Less Common Sentential Connectives
XOR is the exclusive or. Φ ⊕ ψ is a wff. It’s true only when (1) Φ is alone true or (2) ψ is alone true. It’s false when (1) both Φ and ψ are false or (2) both Φ and ψ are true.
In everyday conversation, we often use the “or” in an exclusive sense. With the exclusive or, if one option is true, then the other option must be false. An inclusive or, in contrast, has it that both options can be true.
XOR is not commonly used in standard logic textbooks.
More rare are the NAND and NOR. We’re more likely to see them when propositional logic is applied to engineering problems dealing with digital circuits.
We’ll also encounter them in more advanced texts that prove that any proposition in propositional logic can be represented with the NAND sentential connective alone (or the the NOR sentential connective alone).
Φ | ψ is a wff with the connective NAND.
The “|” is called the Sheffer stroke, named after Henry M. Sheffer.
Φ | ψ is true only when (1) Φ is alone true, (2) ψ is alone true, or (3) both Φ and ψ are false. It is false when Φ is true and ψ is true.
Φ↓ψ is a wff with the connective NOR.
The “↓” is called the Peirce arrow, named after C. S. Peirce.
Φ↓ψ is true only when both Φ and ψ are false.
It is false in all other cases.