CH2 - Slides

Author

Affiliation

David W. Agler

The Pennsylvania State University

Abstract

Slides for Ch1 of my logic course.

PL — Symbols, Syntax, and Semantics

PL: The Language of Propositional Logic

PL is a formal language used to study logic. It is comprised of:

1. a set of symbols—its alphabet;
2. a syntax—its grammar, expressed as formation rules; and
3. a semantics—rules for assigning truth values such as \(T\) and \(F\) to well-formed formulas.

PL: Symbols

PL consists of the following symbols:

1. An infinite number of propositional letters: uppercase Roman letters, with or without subscripted integers, such as \(A_1, A_2, A_3, B, C, \ldots, Z\).
2. Five truth-functional operators: \(\vee, \rightarrow, \leftrightarrow, \neg,\wedge\)
3. A left and right parenthesis: ( and ) to indicate the scope of truth-functional operators.

PL - Syntax

PL Formation Rules

NoteDefinition

A well-formed formula, or wff, is any formula that can be generated by some combination of the seven formation rules below.

PL Formation Rules Listed

1. Every propositional letter of PL, such as \(A, B, C\), is a wff.
2. If \(\phi\) is a wff, then \(\neg(\phi)\) is a wff.
3. If \(\phi\) and \(\psi\) are wffs, then \((\phi\wedge\psi)\) is a wff.
4. If \(\phi\) and \(\psi\) are wffs, then \((\phi\vee\psi)\) is a wff.
5. If \(\phi\) and \(\psi\) are wffs, then \((\phi\rightarrow\psi)\) is a wff.
6. If \(\phi\) and \(\psi\) are wffs, then \((\phi\leftrightarrow\psi)\) is a wff.
7. Nothing else is a wff except what can be formed by repeated applications of rules 1–6.

Using the Formation Rules

Let’s construct \((\neg (P)\land Q)\)

Wffs in PL

1. \(P\)
2. \(\neg (P)\)
3. \((P\rightarrow Q)\)
4. \(\neg((P\vee\neg(Q)))\)
5. \(\neg(\neg(P\wedge\neg(Q)))\)

Not wffs in PL

1. \(P\neg\)
2. \(PQ\)
3. \(\vee\neg(Q)\)
4. \(\neg\neg P\wedge\neg Q\vee S\)

Exercise

See book, Ex. 2.10, pp.47

Three Types of Wffs: Atomic Wffs

NoteDefinition

An atomic wff is a wff consisting only of a single propositional letter, such as \(P, R, S, D\).

Examples:

• \(P\)
• \(Q\)
• \(A\)
• \(B_{12}\)

Three Types of Wffs: Complex Wffs

NoteDefinition

A complex wff is a wff that contains at least one propositional letter and a truth-functional operator.

Examples:

• \(\neg(P)\)
• \((P\wedge Q)\)
• \((P\rightarrow\neg(A))\)
• \((A_1\rightarrow B_8)\)

Literal Wffs

NoteDefinition

A literal wff, or literal, is either an atomic wff \(P\) or a negated atomic wff \(\neg(P)\).

Examples:

• \(P\)
• \(\neg(P)\)
• \(Q\)
• \(\neg(Q_1)\)

Exercise

See book, Ex. 2.11, pp.49

Subformulas and Proper Parts

Let \(\phi\) and \(\psi\) be any PL-wffs.

NoteDefinition: 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.

NoteDefinition: Subformula (part)

A subformula (or part) \(\phi\) of \(\psi\) is any wff occurring as a proper part of \(\psi\), including \(\psi\) itself.

Examples

• \((P\vee Q)\)
• \((\neg(P)\vee Q)\)
• \(P\)

Exercises

See book, Ex. 2.12, pp.51

Operator Occurrences

• Notice that there are two \(\neg\)’s in \(\neg(\neg(P)\rightarrow Q)\)
• Each \(\neg\) is called an occurrence of the NOT operator.
• identify an occurrence using whatever description is clearest:
  • “the leftmost NOT”
  • “the NOT next to \(P\)”; or
  • “the outer NOT.”

Exercises

See book, Ex. 2.13, pp.51

Scope

NoteDefinition

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.

Examples of Scope

• In \(\neg(P)\), the scope of \(\neg\) is \(\neg(P)\). It is the smallest subformula containing that occurrence of \(\neg\)
• In \((P\vee Q)\), the scope of \(\vee\) is \((P\vee Q)\).
• In \((\neg(P)\vee Q)\), the scope of \(\neg\) is \(\neg(P)\), while the scope of \(\vee\) is \((\neg(P)\vee Q)\).

To determine scope

• To determine scope, you can just build the wff.
• When the wff that is constructed when that operator is first introduced is the scope of that operator.
• Let’s determine the scope of the rightmost \(\neg\) in \((\neg (P)\land \neg (Q))\)

Exercises

See book, Ex. 2.14, pp.53

Main Operator

NoteDefinition

The main operator of a PL wff is the truth-functional operator whose scope is the entire wff.

Examples of Main Operator

• The main operator of \(\neg(P)\) is \(\neg\).
• The main operator of \((P\wedge Q)\) is \(\wedge\).
• The main operator of \((\neg(P)\vee Q)\) is \(\vee\).
• The main operator of \(\neg((P\rightarrow Q))\) is \(\neg\).

Exercise

See book, Ex. 2.15, p.54

Simplification

Gross: \(\neg((\neg (P)\rightarrow \neg (Q)))\)

Why parentheses?

• Parentheses exist so that the scope of the operators is precise (no ambiguity!)
• \(\neg P\land Q\) could be
  • \(\neg ((P\land Q))\). Main operator is \(\neg\), OR
  • \((\neg (P)\land Q)\). Main operator is \(\land\)

Parentheses Convention 1

NoteDefinition - 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.

1. \((P\land Q)\) to \(P\land Q\)
2. \(((P\land Q)\lor R)\) to \((P\land Q)\lor R\)
3. \(((P\land Q)\lor (S\land R))\) to \((P\land Q)\lor (S\land R)\)

Parentheses Convention 2

NoteDefinition - 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.

1. \(\neg (P)\) to \(P\)
2. \(\neg(\neg (P))\) to \(\neg\neg P\)
3. \(\neg(\neg (P\land Q))\) to \(\neg\neg (P\land Q)\)
4. NOT: \(\neg ((P\land Q))\) to \(\neg P\land Q\)

Parentheses Convention 3

NoteDefinition - 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.

1. \(\lnot ((P\land Q))\) to \(\lnot (P\land Q)\)

Exercise

See book, Ex. 2.16, pp.58

Literal Negation

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).

1. \(\phi \quad\mapsto\quad \neg(\phi)\)
2. \(\neg (\phi) \quad\mapsto\quad \phi\)

Examples of Literal Negation

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).

Literal Negations?

1. \(\neg (P\lor \neg Q)\) and \(P\lor \neg Q\)?
2. \(\neg P\land Q\) and \(P\land Q\)?

Exercise

See book, Ex. 2.17, pp.59

Types of Complex Wffs

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

Components of Complex Wffs

Some parts of complex wffs have special names:

• A conjunction \((P\wedge R)\) contains two conjuncts, \(P\) and \(R\).
• A disjunction \((P\vee R)\) contains two disjuncts, \(P\) and \(R\).
• A conditional \((P\rightarrow R)\) contains an antecedent, \(P\), and a consequent, \(R\).

Negated form of each

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

Exercise

See book, Ex. 2.18, pp.61

Symbols and Syntax - You should know

1. how to ID symbols.
2. how to ID a wff, its proper parts, subformulas.
3. how to write a wff in simplified or non-simplified form
4. how to ID scope and main operator
5. how to write a wff’s literal negation
6. the different types of wffs and types of complex wffs

Important

Syntax is fundamental so get a mastery of these skills!

PL – Semantics

PL: Semantics

• PL is a set of symbols together with formation rules for combining those symbols.
• The symbols and wffs produced by the formation rules are meaningless until PL is “interpreted”.
• Job of semantics is to assign meaning

Function

NoteDefinition

A function is a relation between two sets such that each input is related to one and only one output.

Function: Examples

Interpretation and Valuation

The semantics of PL involve two important functions:

1. Interpretation function
2. Valuation function

Interpretation Function

NoteDefinition

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.

• \(\mathscr{I}(P)=T\)
• \(\mathscr{I}(Q)=T\)
• \(\mathscr{I}(R)=F\)

Valuation Defined

NoteDefinition

For any interpretation \(\mathscr{I}\), a valuation \(v\) takes truth value(s) as input and outputs a truth value.

Interpretation and Valuation

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"]

Valuations

1. \(v(R)=\mathscr{I}(R)\)
2. \(v(\neg (\phi))=T\) iff \(v(\phi)=F\).
3. \(v(\phi\wedge \psi)=T\) iff \(v(\phi)=T\) and \(v(\psi)=T\).
4. \(v(\phi\vee \psi)=T\) iff \(v(\phi)=T\) or \(v(\psi)=T\).
5. \(v(\phi\rightarrow \psi)=T\) iff \(v(\phi)=F\) or \(v(\psi)=T\).
6. \(v(\phi\leftrightarrow \psi)=T\) iff \(v(\phi)=v(\psi)\)

Valuation: Truth Tables

Truth tables provide a graphical display of the interpretation and valuation functions.

1 Propositional letter

Interpretation	\(P\)
\(\mathscr{I}_1\)	\(T\)
\(\mathscr{I}_2\)	\(F\)

2 Propositional letters

Interpretation	\(P\)	\(Q\)
\(\mathscr{I}_1\)	T	T
\(\mathscr{I}_2\)	T	F
\(\mathscr{I}_3\)	F	T
\(\mathscr{I}_4\)	F	F

3 Propositional letters

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

4 Propositional letters

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

Valuation: Atomic wff

\(v(P)=T = \mathscr{I}(P)\)

\(\mathscr{I}(P)\)	\(v(P)\)
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.

Valuation: Negation

\(v(\neg (\phi))=T\) iff \(v(\phi)=F\)

\(\phi\)	\(\neg (\phi)\)
T	F
F	T

Intuitively:

• If \(\phi\) is T, then not-\(\phi\) is F.
• If \(\phi\) is F, then not-\(\phi\) is T.

Valuation: Conjunction

\(v(\phi\wedge \psi)=T\) 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

• If \(\phi\) and \(\psi\) are both T, then \(\phi\land\psi\) is T.
• If either \(\phi\) or \(\psi\) are F (or both), then \(\phi\land\psi\) is F.

Valuation: Disjunction

\(v(\phi\vee \psi)=T\) 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

• If at least one of the disjuncts is T, \(v(\phi\vee \psi)=T\)
• \(v(\phi\vee \psi)=F\) iff both disjuncts are F.

Valuation: Conditional

\(v(\phi\to \psi)=T\) 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

• \(v(\phi\rightarrow \psi)=F\) in one case: \(v(\phi)=T\) and \(v(\psi)=F\).

Valuation: Biconditional

\(v(\phi\leftrightarrow \psi)=T\) iff \(v(\phi)=v(\psi)\)

\(\phi\)	\(\psi\)	\(\phi\leftrightarrow \psi\)
T	T	T
T	F	F
F	T	F
F	F	T

• The biconditional is T when the truth value of both sides of the biconditional match.

Truth tables for operators

\[ \begin{array}{c|c} \phi & \neg (\phi)\\ \hline T & F\\ F & T \end{array} \]

\[ \begin{array}{c c|c c c c} \phi&\psi&\phi\wedge \psi&\phi\vee \psi&\phi \to \psi&\phi\leftrightarrow \psi\\ \hline T&T&T&T&T&T\\ T&F&F&T&F&F\\ F&T&F&T&T&F\\ F&F&F&F&T&T \end{array} \]

Exercise

See book, Ex. 2.19, p.70

Exercise

See book, Ex. 2.19, p.70

Exercise

See book, Ex. 2.20, p.72

PL – Translation

Atomic Wffs

Begin with a simple English proposition consisting of a subject and a predicate (S is P or S Ps):

• Tek is kind.
• The hat is red.
• Liz runs.

Single letter

Each simple proposition can be translated using a single propositional letter:

• Tek is kind. Trans: \(T\)
• The hat is red. Trans: \(H\)
• Liz runs. Trans: \(L\)

Choosing Letters

• “John is kind” as \(J\).
• “John is friendly” cannot also be \(J\)

Negated Wffs - Translation?

\(v(\neg\phi)=T\) iff \(v(\phi)=F\).

• What (in English) behaves just like this?
• “NOT”

Negated Wffs - Not

\(P\)	\(\neg P\)
T	F
F	T

Paul is happy	Paul is NOT happy.
T	F
F	T

Negated Wffs - Recommendation

Translate the \(\neg\) in \(\neg\phi\) using “not” or some equivalent.

• It is not the case that P
• It is false that P.
• Not-P

Conjunctions - Translation?

\(v(P\wedge Q)=T\) iff both \(P\) and \(Q\) are true.

• What (in English) behaves just like this?
• AND!

Conjunctions - And

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

• \(P\) and \(Q\) is T iff both \(P\) and \(Q\) are T.
• Just like \(P\land Q\)

Conjunctions - Recommendation

• Sentence: Paul is happy and Chris is sad.
• Form: \(P\) and \(C\)
• Translation: \(P\land C\)

Disjunctions - Translation?

\(v(P\lor Q)=T\) iff \(P\) or \(Q\) are true.

• What (in English) behaves just like this?
• Inclusive OR!

Inclusive and Exclusive “Or”

• Inclusive: One or the other, or both.
• Exclusive: One or the other, but not both.

1. Mike is at the party or Cindy is at the party (or both)
2. Mike is at the party in State College or he is visiting his parents in Rome (not both).

Disjunctions - Or

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

• \(P\) or \(Q\) is T iff \(P\) is T or \(Q\) are T, or both
• Just like \(P\lor Q\)

Disjunctions - Recommendation

• Sentence: Paul is happy or Chris is sad.
• Form: P or C
• Translation: \(P\lor C\)

Translating Exclusive Or

1. Introduce a new operator: \(v(P\oplus Q)=T\) iff exactly one of \(P\) and \(Q\) is true.
2. Use existing operators (we’ll show how later)

Conditionals - Translation?

\(v(P\rightarrow Q)=F\) iff \(v(P)=T\) and \(v(Q)=F\).

• What (in English) behaves just like this?
• IF … THEN …

Conditionals - IF THEN

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

• This sentence is false when (1) my car starts but (2) there is NO gas in it.
• If \(P\), then \(Q\) is F when P is T and Q is F.
• Just like \(P\to Q\)

Conditionals - Recommendation

• Sentence: If my client’s blood is at the scene of the crime, then the forensic team will find it.
• This sentence is false when (1) my client’s blood is at the scene but (2) the forensic team never finds it.
• Form: If B, then F.
• Translation: \(B\to F\)

Conditionals - Variations

• If P then Q
• Q, if P
• When P, Q
• Whenever P, Q

Rows 3 and 4 are weird

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

• Fine for rows 1 and 2, but rows 3 and 4 are weird.
• If my car starts, there is gas in it. True in rows 3 and 4?

Why its weird

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.

Paradoxes of Material Implication

\(\phi\)	\(\psi\)	\(\phi\rightarrow \psi\)
T	T	T
T	F	F
F	T	?
F	F	?

• There is a lot of debate about rows 3-4.
• The debate is about
  • whether (3)-(4) should be T.
  • how to teach (3)-(4) is T.

Intuition Pump - Promises

Treat conditionals like conditional promises

• false conditionals are broken promises
• true conditionals are unbroken promises

Intuition Pump - Example

• “If you are kind, then you will go to heaven”

\(K\)	\(H\)	\(K\rightarrow H\)
T	T
T	F
F	T
F	F

Biconditionals

\(v(P\leftrightarrow Q)=T\) iff \(v(P)=v(Q)\).

• Sentence: John is at the party if and only if Mary is at the party.
• Form: J iff M
• Translation: \(J\leftrightarrow M\)
• Expanded: P iff Q is short for two conjoined conditionals: “if P then Q and if Q then P”.

If P then Q vs. P iff Q

How is “If P, then Q” different than “P if and only if (iff) Q”?

• \(D\) = The number is divisible by 4.
• \(E\) = The number is even.

Consider the following:

1. If a number is divisible by 4, then it is even (\(D\to E\))
2. A number is divisible by 4 if and only if it is even (\(D\leftrightarrow E\))

If P then Q vs. P iff Q, Continued

1. If a number is divisible by 4, then it is even (\(D\to E\))
2. A number is divisible by 4 if and only if it is even (\(D\leftrightarrow E\))

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

• Sentence (1) does not say every even number is divisible by 4.
• Sentence (2) does say that every even number is divisible by 4.

If P then Q vs. P iff Q, Helpful Tips

Helpful ways to remember:

• Think of “If P then Q” as saying “Whenever P, then Q” (not vice versa)
• Think of “P iff Q” as saying “Whenever P, then Q AND Whenever Q, then P”.

Another way:

• “If P then Q” is saying “Q goes with P”, but P does not necessarily go with Q
• “P iff Q” is \(P\) and \(Q\) always have the same truth value.

Basic Translation Exercise

See book, Ex. 2.22, pp.81

PL — Complex Translations

PL: Simple Translations

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 \(Q\)	\(P\leftrightarrow Q\)

PL: Complex Translations

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 \(Q\)	\(P\)

Multiple Ands and Ors

Complex propositions chained together by “and” or “or” can be translated using multiple instances of \(\wedge\) or \(\vee\).

• John is tall and Mary is happy and Frank is sweet: \((J\wedge M)\wedge F\)
• I will go to the store, or stay home, or go to a party: \(S\vee(H\vee P)\)

Multiple Ands and Ors: Ambiguous

Mixing “ands” and “ors” in a chain create ambiguous sentences:

1. Sentence: You can have eggs and bacon or toast.
2. Translation: \(E\land B\vee T\)

Is it?

1. \((E\land B)\vee T\), or
2. \(E\land (B\vee T)\)

Neither P Nor Q

“Neither P nor Q” is “not-\(P\) and not-\(Q\)”

• (\(\neg P\wedge\neg Q\))
• John is neither happy nor angry: \(\neg H\wedge\neg A\).
• Liz is neither tired nor hungry.: \(\neg T\wedge\neg H\).
• The defendant was neither present at the scene nor involved in the planning. \(\neg P\land \lnot I\)

Not both P and Q

• “Not both P and Q” does not mean \(P\) is false and \(Q\) are false
• It means that it is not the case that both (the conjunction) are true (together).
• It says the conjunction is false.

Not both P and Q - Example

“You are not both in Chicago and Paris”

• It isn’t saying you are not in Chicago
• It isn’t saying you are not in Paris
• It isn’t saying you are in one or the other.
• It is just saying that you are not in both locations.

Not both P and Q - Table

\(C\)	\(P\)	not both \(C\) and \(P\)
T	T	F
T	F	T
F	T	T
F	F	T

Not both P and Q - Tip

• Both P and Q = \(P\land Q\)
• NOT Both P and Q = \(\neg (P\land Q)\)

\(P\)	\(Q\)	\(P\land Q\)	\(\neg (P\land Q)\)
T	T	T	F
T	F	F	T
F	T	F	T
F	F	F	T

Not Both P and Q - Another example

Tek is not both the plantiff and the defendant in this case.

• The sentence is not saying that Tek is one or the other
• Tek could have nothing to do with the case or a witness
• \(\neg (P\land D)\)

Not Both - Exclusive Or

Two senses of “or”:

• Inclusive: P or Q, or both: \(P\vee Q\).
• Exclusive: P or Q, but not both:
  • \(P\oplus Q\) or
  • \((P\vee Q)\wedge\neg(P\wedge Q)\)

1. You are in Chicago or State College, but not both \((C\lor S)\land \lnot (C\land S)\)
2. Tek is either the defendant or the plantiff, but not both. \((D\lor P)\land \lnot (D\lor P)\)

Complex Translation Exercise 1

See book, Ex. 2.26, pp.89-90,

P only if Q

“P only if Q” means

1. P requires Q
2. Whenever P is the case, Q is the case.
3. Q is a necessary condition for P
4. P is not true unless Q is true

P only if Q - Two Translations

``\(P\) only if \(Q\)’’ can be translated in two ways:

• \(P\to Q\) - Whenever \(P\) is the case, \(Q\) is the case.
• \(\lnot Q\to\lnot P\) - If \(Q\) is not the case, then \(P\) is not the case.

P only if Q - Examples

• The car will start only if there is gas in it: \(C\to G\).
• John will go to the party only if Mary goes: \(\neg M\rightarrow\neg J\).
• John will be found guilty only if he committed the crime: \(\neg C\rightarrow\neg G\).

P even if Q

“P even if Q” expresses that P is the case regardless of the truth or falsity of Q.

Translate it simply as: \(P\)

P even if Q - Examples

• John will go to the party even if Mary goes: \(J\).
• The stock market will go up even if no one buys stock: \(S\).
• You will be ruined even if you are found not guilty: \(R\).

P unless Q

“P unless Q” is ambiguous. Let’s just focus on the standard way of analyzing it.

• “P unless Q” says that “if Q is not the case, then P is the case.””
• “if not-Q, then P”
• \(\neg Q\to P\)

P unless Q - Example

• “Use the side door unless the front door is open.”
• “If the front door is not open, then use the side door”
• Doesn’t say that you should not use the side door if the front door is open.
• \(\lnot F\to S\)

\(S\)	\(F\)	\(S\) unless \(F\)	\(\neg F\to S\)
T	T	T	T
T	F	F	T
F	T	F	T
F	F	F	F

P unless Q - Real Example

“LG TV does not collect, record or transmit ambient conversations in your home, unless voice functionality has been intentionally activated by the user.”

• \(\lnot (C\lor R \lor T)\) = \(\lnot C\land \lnot R\land \lnot T\)
• \((\lnot C\land \lnot R\land \lnot T)\) unless \(A\)
• \(\lnot A\to (\lnot C\land \lnot R\land \lnot T)\)

If I didn’t intentionally activate this feature, then my TV isn’t snooping on me!

Complex Translation Exercise 1

See book, Ex. 2.28, pp.97, #3,4,5,6

Translation - You should know

1. how to translate from English to PL
2. how to translate from PL to English
3. two different senses of “or”
4. that “If P, then Q” is different than “P if and only if Q”
5. that “If P, then Q” and “P only if Q” are the same
6. that “P even if Q” is just \(P\).

Important

Further Reading

• Geoffrey Hunter, Metalogic: An Introduction to the Metatheory of Standard First Order Logic. University of California Press, 1971.
• Theodore Sider, Logic for Philosophy. Oxford University Press, 2010.
• Donald W. Loveland, Richard E. Hodel, and S. G. Sterrett, Three Views of Logic: Mathematics, Philosophy, and Computer Science. Princeton University Press, 2014.