| 22435 | The logician's '→' does not mean the English if-then [Quine] |
| Full Idea: The logician drops 'if-then' in favour of '→' without ever entertaining the mistaken idea that they are synonymous. | |
| From: Willard Quine (Mr Strawson on Logical Theory [1953], V) | |
| A reaction: [Quine uses the older horseshoe symbol] The conditional in English is not well understood, whereas the symbol is unambiguous. A warning to myself, since I have a tendency to translate symbols into English all the time. [p.156 'implies' is worse!] |
| 9511 | We write the conditional 'if P (antecedent) then Q (consequent)' as P→Q [Lemmon] |
| 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 ¬ [Lemmon] |
| 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'. |
| 9508 | The sign |- may be read as 'therefore' [Lemmon] |
| 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 [Lemmon] |
| 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) |
| 9510 | That proposition that either P or Q is their 'disjunction', written P∨Q [Lemmon] |
| 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). |
| 9513 | We write 'P if and only if Q' as P↔Q; it is also P iff Q, or (P→Q)∧(Q→P) [Lemmon] |
| 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. |
| 9514 | If A and B are 'interderivable' from one another we may write A -||- B [Lemmon] |
| 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) |
| 12005 | The symbol 'ι' forms definite descriptions; (ιx)F(x) says 'the x which is such that F(x)' [Forbes,G] |
| Full Idea: We use the symbol 'ι' (Greek 'iota') to form definite descriptions, reading (ιx)F(x) as 'the x which is such that F(x)', or simply as 'the F'. | |
| From: Graeme Forbes (The Metaphysics of Modality [1985], 4.1) | |
| A reaction: Compare the lambda operator in modal logic, which picks out predicates from similar formulae. |
| 7799 | Proposition logic has definitions for its three operators: or, and, and identical [Girle] |
| Full Idea: The operators of propositional logic are defined as follows: 'or' (v) is not-A implies B; 'and' (ampersand) is not A-implies-not-B; and 'identity' (three line equals) is A-implies-B and B-implies-A. | |
| From: Rod Girle (Modal Logics and Philosophy [2000], 6.5) |