The Pennsylvania State University
PL is a formal language used to study logic. It is comprised of:
PL consists of the following symbols:
( and ) to indicate the scope of truth-functional operators.Definition
A well-formed formula, or wff, is any formula that can be generated by some combination of the seven formation rules below.
Let’s construct \((\neg (P)\land Q)\)
See book, Ex. 2.10, pp.47
Definition
An atomic wff is a wff consisting only of a single propositional letter, such as \(P, R, S, D\).
Examples:
Definition
A complex wff is a wff that contains at least one propositional letter and a truth-functional operator.
Examples:
Definition
A literal wff, or literal, is either an atomic wff \(P\) or a negated atomic wff \(\neg(P)\).
Examples:
See book, Ex. 2.11, pp.49
Let \(\phi\) and \(\psi\) be any PL-wffs.
Definition: Proper part
A wff \(\phi\) is a proper part of another wff \(\psi\) if and only if \(\phi\) is a wff that is constructed by the formation rules in the process of constructing \(\psi\) but not \(\psi\) itself.
Definition: Subformula (part)
A subformula (or part) \(\phi\) of \(\psi\) is any wff occurring as a proper part of \(\psi\), including \(\psi\) itself.
See book, Ex. 2.12, pp.51
See book, Ex. 2.13, pp.51
Definition
The scope of an occurrence of an operator in a PL wff \(\phi\) is the smallest subformula of \(\phi\) that contains that occurrence of the operator.
See book, Ex. 2.14, pp.53
Definition
The main operator of a PL wff is the truth-functional operator whose scope is the entire wff.
See book, Ex. 2.15, p.54
Gross: \(\neg((\neg (P)\rightarrow \neg (Q)))\)
Definition - Convention 1
Parentheses may be omitted when applying the final formation rule for a wff \(\phi\) if that rule introduces a connective (\(\land, \lor, \to, \leftrightarrow\)). In other words, after a wff has been constructed, if the first and last symbol of a wff are parentheses, then those outer parentheses may be omitted.
Definition - Convention 2
Whenever the negation formation rule is applied, if the subformula used in its application is either a propositional letter or a negated wff, parentheses may be omitted.
Definition - Convention 3
When applying the negation formation rule to a wff that is surrounded by a pair of parenthesis, it is not necessary to add an additional pair of parentheses. In other words, if both sides of pairs of parentheses do not contain any symbols between them, a single pair of parentheses may be omitted.
See book, Ex. 2.16, pp.58
The literal negation of a proposition \(\phi\) is formed by placing parentheses around \(\phi\) and applying the negation formation rule (or performing the operation in reverse).
| Wff | Literal negation of wff |
|---|---|
| \(P\) | \(\neg(P)\) |
| \((P\rightarrow R)\) | \(\neg((P\rightarrow R))\) |
| \((\neg(P)\wedge R)\) | \(\neg((\neg(P)\wedge R))\) |
The literal negation of a wff is the negation of the entire formula, not merely a part of it (or the removal of the negation from the entire wff).
See book, Ex. 2.17, pp.59
Complex wffs can be categorized according to their main operator:
| Form | Main Operator | Type of wff |
|---|---|---|
| \(\neg(\phi)\) | \(\neg\) | Negated wff |
| \((\phi\wedge \psi)\) | \(\wedge\) | Conjunction |
| \((\phi\vee \psi)\) | \(\vee\) | Disjunction |
| \((\phi\rightarrow \psi)\) | \(\rightarrow\) | Conditional |
| \((\phi\leftrightarrow \psi)\) | \(\leftrightarrow\) | Biconditional |
Some parts of complex wffs have special names:
Complex wffs can be categorized according to their main operator:
| Form | Main Operator | Type of wff |
|---|---|---|
| \(\neg\neg (\phi)\) | \(\neg\) | Double negation |
| \(\neg(\phi\wedge \psi)\) | \(\neg\) | Negated Conjunction |
| \(\neg (\phi\vee \psi)\) | \(\neg\) | Negated Disjunction |
| \(\neg (\phi\rightarrow \psi)\) | \(\neg\) | Negated Conditional |
| \(\neg (\phi\leftrightarrow \psi)\) | \(\neg\) | Negated Biconditional |
See book, Ex. 2.18, pp.61
Important
Syntax is fundamental so get a mastery of these skills!
Definition
A function is a relation between two sets such that each input is related to one and only one output.
The semantics of PL involve two important functions:
Definition
An interpretation of PL is a function that takes propositional letters as input and assigns each one a single truth value, \(T\) or \(F\), as output.
Definition
For any interpretation \(\mathscr{I}\), a valuation \(v\) takes truth value(s) as input and outputs a truth value.
Let \(R\) be a propositional letter and \(\phi, \psi\) be any PL wff.
flowchart TD
I["Interpretation in PL"] --> I2["Assigns T or F to propositional letters"]
I2 --> V["Valuation in PL"]
V --> V2["Assigns T or F to well-formed formulas"]
Truth tables provide a graphical display of the interpretation and valuation functions.
| Interpretation | \(P\) |
|---|---|
| \(\mathscr{I}_1\) | \(T\) |
| \(\mathscr{I}_2\) | \(F\) |
| Interpretation | \(P\) | \(Q\) |
|---|---|---|
| \(\mathscr{I}_1\) | T | T |
| \(\mathscr{I}_2\) | T | F |
| \(\mathscr{I}_3\) | F | T |
| \(\mathscr{I}_4\) | F | F |
| Interpretation | \(P\) | \(Q\) | \(R\) |
|---|---|---|---|
| \(\mathscr{I}_1\) | T | T | T |
| \(\mathscr{I}_2\) | T | T | F |
| \(\mathscr{I}_3\) | T | F | T |
| \(\mathscr{I}_4\) | T | F | F |
| \(\mathscr{I}_5\) | F | T | T |
| \(\mathscr{I}_6\) | F | T | F |
| \(\mathscr{I}_7\) | F | F | T |
| \(\mathscr{I}_8\) | F | F | F |
| Interpretation | \(P\) | \(Q\) | \(R\) | \(S\) |
|---|---|---|---|---|
| \(\mathscr{I}_1\) | T | T | T | T |
| \(\mathscr{I}_2\) | T | T | F | T |
| \(\mathscr{I}_3\) | T | F | T | T |
| \(\mathscr{I}_4\) | T | F | F | T |
| \(\mathscr{I}_5\) | F | T | T | T |
| \(\mathscr{I}_6\) | F | T | F | T |
| \(\mathscr{I}_7\) | F | F | T | T |
| \(\mathscr{I}_8\) | F | F | F | T |
| \(\mathscr{I}_9\) | T | T | T | F |
| \(\mathscr{I}_{10}\) | T | T | F | F |
| \(\mathscr{I}_{11}\) | T | F | T | F |
| \(\mathscr{I}_{12}\) | T | F | F | F |
| \(\mathscr{I}_{13}\) | F | T | T | F |
| \(\mathscr{I}_{14}\) | F | T | F | F |
| \(\mathscr{I}_{15}\) | F | F | T | F |
| \(\mathscr{I}_{16}\) | F | F | F | F |
\(v(P)=T = \mathscr{I}(P)\)
| \(\mathscr{I}(P)\) | \(v(P)\) |
|---|---|
| T | T |
| F | F |
The valuation function is taking the truth value assigned to the letter \(P\) as input and then reassigning it to the wff \(P\) as output.
\(v(\neg (\phi))=T\) iff \(v(\phi)=F\)
| \(\phi\) | \(\neg (\phi)\) |
|---|---|
| T | F |
| F | T |
Intuitively:
\(v(\phi\wedge \psi)=T\) iff both \(v(\phi)=T\) and \(v(\psi)=T\).
| \(\phi\) | \(\psi\) | \(\phi\wedge \psi\) |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
\(v(\phi\vee \psi)=T\) iff \(v(\phi)=T\) or \(v(\psi)=T\) (or both)
| \(\phi\) | \(\psi\) | \(\phi\vee \psi\) |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
\(v(\phi\to \psi)=T\) iff \(v(\phi)=F\) or \(v(\psi)=T\), or both.
| \(\phi\) | \(\psi\) | \(\phi\rightarrow \psi\) |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
\(v(\phi\leftrightarrow \psi)=T\) iff \(v(\phi)=v(\psi)\)
| \(\phi\) | \(\psi\) | \(\phi\leftrightarrow \psi\) |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
\[ \begin{array}{c|c} \phi & \neg (\phi)\\ \hline T & F\\ F & T \end{array} \]
\[ \begin{array}{c c|c c c c} \phi&\psi&\phi\wedge \psi&\phi\vee \psi&\phi \to \psi&\phi\leftrightarrow \psi\\ \hline T&T&T&T&T&T\\ T&F&F&T&F&F\\ F&T&F&T&T&F\\ F&F&F&F&T&T \end{array} \]
See book, Ex. 2.19, p.70
See book, Ex. 2.19, p.70
See book, Ex. 2.20, p.72
Begin with a simple English proposition consisting of a subject and a predicate (S is P or S Ps):
Each simple proposition can be translated using a single propositional letter:
\(v(\neg\phi)=T\) iff \(v(\phi)=F\).
| \(P\) | \(\neg P\) |
|---|---|
| T | F |
| F | T |
| Paul is happy | Paul is NOT happy. |
|---|---|
| T | F |
| F | T |
Translate the \(\neg\) in \(\neg\phi\) using “not” or some equivalent.
\(v(P\wedge Q)=T\) iff both \(P\) and \(Q\) are true.
| Paul is happy | Liz is happy | Paul is happy and Liz is happy |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
\(v(P\lor Q)=T\) iff \(P\) or \(Q\) are true.
| There is cake | There is ice cream | There is cake or there is ice cream |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
\(v(P\rightarrow Q)=F\) iff \(v(P)=T\) and \(v(Q)=F\).
| My car starts | There is gas in it | If my car starts, then there is gas in it. |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Translating “if P, then Q” as \(P\to Q\) seems wrong.
| \(\phi\) | \(\psi\) | \(\phi\rightarrow \psi\) |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | |
| F | F |
When the antecedent is F, the conditional is automatically T.
- If rain is made of spaghetti sauce, then God exists.
- \(v(F\to ?)=T\)
- \(P\to Q\) is T when \(P\) is F.
When the consequent is T, the conditional is automatically T.
- If God exists, then there are dandelions on Earth.
- \(v(?\to T)=T\)
- \(P\to Q\) is T when \(Q\) is T.
| \(\phi\) | \(\psi\) | \(\phi\rightarrow \psi\) |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | ? |
| F | F | ? |
Treat conditionals like conditional promises
| \(K\) | \(H\) | \(K\rightarrow H\) |
|---|---|---|
| T | T | |
| T | F | |
| F | T | |
| F | F |
\(v(P\leftrightarrow Q)=T\) iff \(v(P)=v(Q)\).
How is “If P, then Q” different than “P if and only if (iff) Q”?
Consider the following:
Notice (1) is T and (2) is F in row (3)
| \(P\) | \(Q\) | \(P\to Q\) | \(P\leftrightarrow Q\) |
|---|---|---|---|
| T | T | T | T |
| T | F | F | F |
| F | T | T | F |
| F | F | T | T |
Helpful ways to remember:
Another way:
See book, Ex. 2.22, pp.81
| Sentence Form | PL |
|---|---|
| not-\(P\) | \(\lnot P\) |
| \(P\) and \(Q\) | \(P\land Q\) |
| \(P\) or \(Q\) | \(P\lor Q\) |
| If \(P\), then \(Q\) | \(P\to Q\) |
| \(P\) if and only if \(Q\) | \(P\leftrightarrow Q\) |
| Sentence Form | PL |
|---|---|
| \(P\) and \(Q\) and \(R\) | \((P\land Q)\land R\) |
| \(P\) or \(Q\) or \(R\) | \((P\lor Q)\lor R\) |
| Neither \(P\) nor \(Q\) | \(\lnot P\land\lnot Q\) |
| Not both \(P\) and \(Q\) | \(\lnot (P\land Q)\) |
| \(P\) or \(Q\), but not both | \((P\lor Q)\land \lnot (P\land Q)\) |
| \(P\) only if \(Q\) | \(P\to Q\) |
| \(P\) unless \(Q\) | \(\lnot Q\to P\) |
| \(P\) even if \(Q\) | \(P\) |
Complex propositions chained together by “and” or “or” can be translated using multiple instances of \(\wedge\) or \(\vee\).
Mixing “ands” and “ors” in a chain create ambiguous sentences:
Is it?
“Neither P nor Q” is “not-\(P\) and not-\(Q\)”
“You are not both in Chicago and Paris”
| \(C\) | \(P\) | not both \(C\) and \(P\) |
|---|---|---|
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | T |
| \(P\) | \(Q\) | \(P\land Q\) | \(\neg (P\land Q)\) |
|---|---|---|---|
| T | T | T | F |
| T | F | F | T |
| F | T | F | T |
| F | F | F | T |
Tek is not both the plantiff and the defendant in this case.
Two senses of “or”:
See book, Ex. 2.26, pp.89-90,
“P only if Q” means
``\(P\) only if \(Q\)’’ can be translated in two ways:
“P even if Q” expresses that P is the case regardless of the truth or falsity of Q.
Translate it simply as: \(P\)
“P unless Q” is ambiguous. Let’s just focus on the standard way of analyzing it.
| \(S\) | \(F\) | \(S\) unless \(F\) | \(\neg F\to S\) |
|---|---|---|---|
| T | T | T | T |
| T | F | F | T |
| F | T | F | T |
| F | F | F | F |
“LG TV does not collect, record or transmit ambient conversations in your home, unless voice functionality has been intentionally activated by the user.”
If I didn’t intentionally activate this feature, then my TV isn’t snooping on me!
See book, Ex. 2.28, pp.97, #3,4,5,6
Important