Ideas from 'Beginning Logic' by E.J. Lemmon [1965], by Theme Structure

[found in 'Beginning Logic' by Lemmon,E.J. [Nelson 1979,017-712040-1]].

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4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic

9535	'Contradictory' propositions always differ in truth-value

			Full Idea: Two propositions are 'contradictory' if they are never both true and never both false either, which means that ¬(A↔B) is a tautology.
			From: E.J. Lemmon (Beginning Logic [1965], 2.3)

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / a. Symbols of PL

9508	The sign |- may be read as 'therefore'

			Full Idea: I introduce the sign |- to mean 'we may validly conclude'. To call it the 'assertion sign' is misleading. It may conveniently be read as 'therefore'.
			From: E.J. Lemmon (Beginning Logic [1965], 1.2)
			A reaction: [Actually no gap between the vertical and horizontal strokes of the sign] As well as meaning 'assertion', it may also mean 'it is a theorem that' (with no proof shown).

9509	That proposition that both P and Q is their 'conjunction', written P∧Q

			Full Idea: If P and Q are any two propositions, the proposition that both P and Q is called the 'conjunction' of P and Q, and is written P∧Q.
			From: E.J. Lemmon (Beginning Logic [1965], 1.3)
			A reaction: [I use the more fashionable inverted-v '∧', rather than Lemmon's '&', which no longer seems to be used] P∧Q can also be defined as ¬(¬P∨¬Q)

9514	If A and B are 'interderivable' from one another we may write A -||- B

			Full Idea: If we say that A and B are 'interderivable' from one another (that is, A |- B and B |- A), then we may write A -||- B.
			From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9510	That proposition that either P or Q is their 'disjunction', written P∨Q

			Full Idea: If P and Q are any two propositions, the proposition that either P or Q is called the 'disjunction' of P and Q, and is written P∨Q.
			From: E.J. Lemmon (Beginning Logic [1965], 1.3)
			A reaction: This is inclusive-or (meaning 'P, or Q, or both'), and not exlusive-or (Boolean XOR), which means 'P, or Q, but not both'. The ∨ sign is sometimes called 'vel' (Latin).

9511	We write the conditional 'if P (antecedent) then Q (consequent)' as P→Q

			Full Idea: We write 'if P then Q' as P→Q. This is called a 'conditional', with P as its 'antecedent', and Q as its 'consequent'.
			From: E.J. Lemmon (Beginning Logic [1965], 1.2)
			A reaction: P→Q can also be written as ¬P∨Q.

9512	We write the 'negation' of P (not-P) as ¬

			Full Idea: We write 'not-P' as ¬P. This is called the 'negation' of P. The 'double negation' of P (not not-P) would be written as ¬¬P.
			From: E.J. Lemmon (Beginning Logic [1965], 1.2)
			A reaction: Lemmons use of -P is no longer in use for 'not'. A tilde sign (squiggle) is also used for 'not', but some interpreters give that a subtly different meaning (involving vagueness). The sign ¬ is sometimes called 'hook' or 'corner'.

9513	We write 'P if and only if Q' as P↔Q; it is also P iff Q, or (P→Q)∧(Q→P)

			Full Idea: We write 'P if and only if Q' as P↔Q. It is called the 'biconditional', often abbreviate in writing as 'iff'. It also says that P is both sufficient and necessary for Q, and may be written out in full as (P→Q)∧(Q→P).
			From: E.J. Lemmon (Beginning Logic [1965], 1.4)
			A reaction: If this symbol is found in a sequence, the first move in a proof is to expand it to the full version.

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL

9529	A wff is 'inconsistent' if all assignments to variables result in the value F

			Full Idea: If a well-formed formula of propositional calculus takes the value F for all possible assignments of truth-values to its variables, it is said to be 'inconsistent'.
			From: E.J. Lemmon (Beginning Logic [1965], 2.3)

9531	'Contrary' propositions are never both true, so that ¬(A∧B) is a tautology

			Full Idea: If A and B are expressible in propositional calculus notation, they are 'contrary' if they are never both true, which may be tested by the truth-table for ¬(A∧B), which is a tautology if they are contrary.
			From: E.J. Lemmon (Beginning Logic [1965], 2.3)

9534	Two propositions are 'equivalent' if they mirror one another's truth-value

			Full Idea: Two propositions are 'equivalent' if whenever A is true B is true, and whenever B is true A is true, in which case A↔B is a tautology.
			From: E.J. Lemmon (Beginning Logic [1965], 2.3)

9516	A 'well-formed formula' follows the rules for variables, ¬, →, ∧, ∨, and ↔

			Full Idea: A 'well-formed formula' of the propositional calculus is a sequence of symbols which follows the rules for variables, ¬, →, ∧, ∨, and ↔.
			From: E.J. Lemmon (Beginning Logic [1965], 2.1)

9518	A 'theorem' is the conclusion of a provable sequent with zero assumptions

			Full Idea: A 'theorem' of logic is the conclusion of a provable sequent in which the number of assumptions is zero.
			From: E.J. Lemmon (Beginning Logic [1965], 2.2)
			A reaction: This is what Quine and others call a 'logical truth'.

9528	A wff is a 'tautology' if all assignments to variables result in the value T

			Full Idea: If a well-formed formula of propositional calculus takes the value T for all possible assignments of truth-values to its variables, it is said to be a 'tautology'.
			From: E.J. Lemmon (Beginning Logic [1965], 2.3)

9530	A wff is 'contingent' if produces at least one T and at least one F

			Full Idea: If a well-formed formula of propositional calculus takes at least one T and at least one F for all the assignments of truth-values to its variables, it is said to be 'contingent'.
			From: E.J. Lemmon (Beginning Logic [1965], 2.3)

9532	'Subcontrary' propositions are never both false, so that A∨B is a tautology

			Full Idea: If A and B are expressible in propositional calculus notation, they are 'subcontrary' if they are never both false, which may be tested by the truth-table for A∨B, which is a tautology if they are subcontrary.
			From: E.J. Lemmon (Beginning Logic [1965], 2.3)

9533	A 'implies' B if B is true whenever A is true (so that A→B is tautologous)

			Full Idea: One proposition A 'implies' a proposition B if whenever A is true B is true (but not necessarily conversely), which is only the case if A→B is tautologous. Hence B 'is implied' by A.
			From: E.J. Lemmon (Beginning Logic [1965], 2.3)

9519	A 'substitution-instance' is a wff formed by consistent replacing variables with wffs

			Full Idea: A 'substitution-instance' is a wff which results by replacing one or more variables throughout with the same wffs (the same wff replacing each variable).
			From: E.J. Lemmon (Beginning Logic [1965], 2.2)

9517	The 'scope' of a connective is the connective, the linked formulae, and the brackets

			Full Idea: The 'scope' of a connective in a certain formula is the formulae linked by the connective, together with the connective itself and the (theoretically) encircling brackets
			From: E.J. Lemmon (Beginning Logic [1965], 2.1)

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL

9399	∧E: Given A∧B, we may derive either A or B separately

			Full Idea: And-Elimination (∧E): Given A∧B, we may derive either A or B separately. The conclusions will depend on the assumptions of the premiss.
			From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9398	∧I: Given A and B, we may derive A∧B

			Full Idea: And-Introduction (&I): Given A and B, we may derive A∧B as conclusion. This depends on their previous assumptions.
			From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9401	∨E: Derive C from A∨B, if C can be derived both from A and from B

			Full Idea: Or-Elimination (∨E): Given A∨B, we may derive C if it is proved from A as assumption and from B as assumption. This will also depend on prior assumptions.
			From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9396	DN: Given A, we may derive ¬¬A

			Full Idea: Double Negation (DN): Given A, we may derive ¬¬A as a conclusion, and vice versa. The conclusion depends on the assumptions of the premiss.
			From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9395	MTT: Given ¬B and A→B, we derive ¬A

			Full Idea: Modus Tollendo Tollens (MTT): Given ¬B and A→B, we derive ¬A as a conclusion. ¬A depends on any assumptions that have been made
			From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9393	A: we may assume any proposition at any stage

			Full Idea: Assumptions (A): any proposition may be introduced at any stage of a proof.
			From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9397	CP: Given a proof of B from A as assumption, we may derive A→B

			Full Idea: Conditional Proof (CP): Given a proof of B from A as assumption, we may derive A→B as conclusion, on the remaining assumptions (if any).
			From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9402	RAA: If assuming A will prove B∧¬B, then derive ¬A

			Full Idea: Reduction ad Absurdum (RAA): Given a proof of B∧¬B from A as assumption, we may derive ¬A as conclusion, depending on the remaining assumptions (if any).
			From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9400	∨I: Given either A or B separately, we may derive A∨B

			Full Idea: Or-Introduction (∨I): Given either A or B separately, we may derive A∨B as conclusion. This depends on the assumption of the premisses.
			From: E.J. Lemmon (Beginning Logic [1965], 1.5)

9394	MPP: Given A and A→B, we may derive B

			Full Idea: Modus Ponendo Ponens (MPP): Given A and A→B, we may derive B as a conclusion. B will rest on any assumptions that have been made.
			From: E.J. Lemmon (Beginning Logic [1965], 1.5)

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL

9522	'Modus ponendo tollens' (MPT) says P, ¬(P ∧ Q) |- ¬Q

			Full Idea: 'Modus ponendo tollens' (MPT) says that if the negation of a conjunction holds and also one of its conjuncts, then the negation of the other conjunct holds. Thus P, ¬(P ∧ Q) |- ¬Q may be introduced as a theorem.
			From: E.J. Lemmon (Beginning Logic [1965], 2.2)
			A reaction: Unlike Modus Ponens and Modus Tollens, this is a derived rule.

9525	We can change conditionals into negated conjunctions with P→Q -||- ¬(P ∧ ¬Q)

			Full Idea: The proof that P→Q -||- ¬(P ∧ ¬Q) is useful for enabling us to change conditionals into negated conjunctions
			From: E.J. Lemmon (Beginning Logic [1965], 2.2)

9523	De Morgan's Laws make negated conjunctions/disjunctions into non-negated disjunctions/conjunctions

			Full Idea: The forms of De Morgan's Laws [P∨Q -||- ¬(¬P ∧ ¬Q); ¬(P∨Q) -||- ¬P ∧ ¬Q; ¬(P∧Q) -||- ¬P ∨ ¬Q); P∧Q -||- ¬(¬P∨¬Q)] transform negated conjunctions and disjunctions into non-negated disjunctions and conjunctions respectively.
			From: E.J. Lemmon (Beginning Logic [1965], 2.2)

9521	'Modus tollendo ponens' (MTP) says ¬P, P ∨ Q |- Q

			Full Idea: 'Modus tollendo ponens' (MTP) says that if a disjunction holds and also the negation of one of its disjuncts, then the other disjunct holds. Thus ¬P, P ∨ Q |- Q may be introduced as a theorem.
			From: E.J. Lemmon (Beginning Logic [1965], 2.2)
			A reaction: Unlike Modus Ponens and Modus Tollens, this is a derived rule.

9524	We can change conditionals into disjunctions with P→Q -||- ¬P ∨ Q

			Full Idea: The proof that P→Q -||- ¬P ∨ Q is useful for enabling us to change conditionals into disjunctions.
			From: E.J. Lemmon (Beginning Logic [1965], 2.2)

9527	The Distributive Laws can rearrange a pair of conjunctions or disjunctions

			Full Idea: The Distributive Laws say that P ∧ (Q∨R) -||- (P∧Q) ∨ (P∧R), and that P ∨ (Q∨R) -||- (P∨Q) ∧ (P∨R)
			From: E.J. Lemmon (Beginning Logic [1965], 2.2)

9526	We can change conjunctions into negated conditionals with P→Q -||- ¬(P → ¬Q)

			Full Idea: The proof that P∧Q -||- ¬(P → ¬Q) is useful for enabling us to change conjunctions into negated conditionals.
			From: E.J. Lemmon (Beginning Logic [1965], 2.2)

4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables

9538	A truth-table test is entirely mechanical, but this won't work for more complex logic

			Full Idea: A truth-table test is entirely mechanical, ..and in propositional logic we can even generate proofs mechanically for tautological sequences, ..but this mechanical approach breaks down with predicate calculus, and proof-discovery is an imaginative process.
			From: E.J. Lemmon (Beginning Logic [1965], 2.5)

9537	Truth-tables are good for showing invalidity

			Full Idea: The truth-table approach enables us to show the invalidity of argument-patterns, as well as their validity.
			From: E.J. Lemmon (Beginning Logic [1965], 2.4)

4. Formal Logic / B. Propositional Logic PL / 4. Soundness of PL

9536	If any of the nine rules of propositional logic are applied to tautologies, the result is a tautology

			Full Idea: If any application of the nine derivation rules of propositional logic is made on tautologous sequents, we have demonstrated that the result is always a tautologous sequent. Thus the system is consistent.
			From: E.J. Lemmon (Beginning Logic [1965], 2.4)
			A reaction: The term 'sound' tends to be used now, rather than 'consistent'. See Lemmon for the proofs of each of the nine rules.

4. Formal Logic / B. Propositional Logic PL / 5. Completeness of PL

9539	Propositional logic is complete, since all of its tautologous sequents are derivable

			Full Idea: A logical system is complete if all expressions of a specified kind are derivable in it. If we specify tautologous sequent-expressions, then propositional logic is complete, because we can show that all tautologous sequents are derivable.
			From: E.J. Lemmon (Beginning Logic [1965], 2.5)
			A reaction: [See Lemmon 2.5 for details of the proofs]

4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / a. Symbols of PC

13909

Write '(∀x)(...)' to mean 'take any x: then...', and '(∃x)(...)' to mean 'there is an x such that....'

			Full Idea: Just as '(∀x)(...)' is to mean 'take any x: then....', so we write '(∃x)(...)' to mean 'there is an x such that....'
			From: E.J. Lemmon (Beginning Logic [1965], 3.1)
			A reaction: [Actually Lemmon gives the universal quantifier symbol as '(x)', but the inverted A ('∀') seems to have replaced it these days]

13902

'Gm' says m has property G, and 'Pmn' says m has relation P to n

			Full Idea: A predicate letter followed by one name expresses a property ('Gm'), and a predicate-letter followed by two names expresses a relation ('Pmn'). We could write 'Pmno' for a complex relation like betweenness.
			From: E.J. Lemmon (Beginning Logic [1965], 3.1)

13911

The 'symbols' are bracket, connective, term, variable, predicate letter, reverse-E

			Full Idea: I define a 'symbol' (of the predicate calculus) as either a bracket or a logical connective or a term or an individual variable or a predicate-letter or reverse-E (∃).
			From: E.J. Lemmon (Beginning Logic [1965], 4.1)

4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / b. Terminology of PC

13910

Our notation uses 'predicate-letters' (for 'properties'), 'variables', 'proper names', 'connectives' and 'quantifiers'

			Full Idea: Quantifier-notation might be thus: first, render into sentences about 'properties', and use 'predicate-letters' for them; second, introduce 'variables'; third, introduce propositional logic 'connectives' and 'quantifiers'. Plus letters for 'proper names'.
			From: E.J. Lemmon (Beginning Logic [1965], 3.1)

4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / c. Derivations rules of PC

13904

Universal Elimination (UE) lets us infer that an object has F, from all things having F

			Full Idea: Our rule of universal quantifier elimination (UE) lets us infer that any particular object has F from the premiss that all things have F. It is a natural extension of &E (and-elimination), as universal propositions generally affirm a complex conjunction.
			From: E.J. Lemmon (Beginning Logic [1965], 3.2)

13901

Predicate logic uses propositional connectives and variables, plus new introduction and elimination rules

			Full Idea: In predicate calculus we take over the propositional connectives and propositional variables - but we need additional rules for handling quantifiers: four rules, an introduction and elimination rule for the universal and existential quantifiers.
			From: E.J. Lemmon (Beginning Logic [1965])
			A reaction: This is Lemmon's natural deduction approach (invented by Gentzen), which is largely built on introduction and elimination rules.

13903

Universal elimination if you start with the universal, introduction if you want to end with it

			Full Idea: The elimination rule for the universal quantifier concerns the use of a universal proposition as a premiss to establish some conclusion, whilst the introduction rule concerns what is required by way of a premiss for a universal proposition as conclusion.
			From: E.J. Lemmon (Beginning Logic [1965], 3.2)
			A reaction: So if you start with the universal, you need to eliminate it, and if you start without it you need to introduce it.

13906

With finite named objects, we can generalise with &-Intro, but otherwise we need ∀-Intro

			Full Idea: If there are just three objects and each has F, then by an extension of &I we are sure everything has F. This is of no avail, however, if our universe is infinitely large or if not all objects have names. We need a new device, Universal Introduction, UI.
			From: E.J. Lemmon (Beginning Logic [1965], 3.2)

13908

UE all-to-one; UI one-to-all; EI arbitrary-to-one; EE proof-to-one

			Full Idea: Univ Elim UE - if everything is F, then something is F; Univ Intro UI - if an arbitrary thing is F, everything is F; Exist Intro EI - if an arbitrary thing is F, something is F; Exist Elim EE - if a proof needed an object, there is one.
			From: E.J. Lemmon (Beginning Logic [1965], 3.3)
			A reaction: [My summary of Lemmon's four main rules for predicate calculus] This is the natural deduction approach, of trying to present the logic entirely in terms of introduction and elimination rules. See Bostock on that.

4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / d. Universal quantifier ∀

13905

If there is a finite domain and all objects have names, complex conjunctions can replace universal quantifiers

			Full Idea: If all objects in a given universe had names which we knew and there were only finitely many of them, then we could always replace a universal proposition about that universe by a complex conjunction.
			From: E.J. Lemmon (Beginning Logic [1965], 3.2)

4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / e. Existential quantifier ∃

13900

'Some Frenchmen are generous' is rendered by (∃x)(Fx→Gx), and not with the conditional →

			Full Idea: It is a common mistake to render 'some Frenchmen are generous' by (∃x)(Fx→Gx) rather than the correct (∃x)(Fx&Gx). 'All Frenchmen are generous' is properly rendered by a conditional, and true if there are no Frenchmen.
			From: E.J. Lemmon (Beginning Logic [1965], 3.1)
			A reaction: The existential quantifier implies the existence of an x, but the universal quantifier does not.

5. Theory of Logic / B. Logical Consequence / 8. Material Implication

9520	The paradoxes of material implication are P |- Q → P, and ¬P |- P → Q

			Full Idea: The paradoxes of material implication are P |- Q → P, and ¬P |- P → Q. That is, since Napoleon was French, then if the moon is blue then Napoleon was French; and since Napoleon was not Chinese, then if Napoleon was Chinese, the moon is blue.
			From: E.J. Lemmon (Beginning Logic [1965], 2.2)
			A reaction: This is why the symbol → does not really mean the 'if...then' of ordinary English. Russell named it 'material implication' to show that it was a distinctively logical operator.